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Scalar Dark Matter Models with Significant Internal Bremsstrahlung

# Scalar Dark Matter Models with Significant Internal Bremsstrahlung

Federica Giacchino Service de Physique Théorique
Université Libre de Bruxelles
Boulevard du Triomphe, CP225, 1050 Brussels, Belgium
Laura Lopez-Honorez Theoretische Natuurkunde
Vrije Universiteit Brussel and The International Solvay Institutes
Pleinlaan 2, B-1050 Brussels, Belgium
Michel H.G. Tytgat Service de Physique Théorique
Université Libre de Bruxelles
Boulevard du Triomphe, CP225, 1050 Brussels, Belgium
###### Abstract

There has been interest recently on particle physics models that may give rise to sharp gamma ray spectral features from dark matter annihilation. Because dark matter is supposed to be electrically neutral, it is challenging to build weakly interacting massive particle models that may accommodate both a large cross section into gamma rays at, say, the Galactic center, and the right dark matter abundance. In this work, we consider the gamma ray signatures of a class of scalar dark matter models that interact with Standard Model dominantly through heavy vector-like fermions (the vector-like portal). We focus on a real scalar singlet annihilating into lepton-antilepton pairs. Because this two-body final-state annihilation channel is d-wave suppressed in the chiral limit, , we show that virtual internal bremsstrahlung emission of a gamma ray gives a large correction, both today and at the time of freeze-out. For the sake of comparison, we confront this scenario to the familiar case of a Majorana singlet annihilating into light lepton-antilepton pairs, and show that the virtual internal bremsstrahlung signal may be enhanced by a factor of (up to) two orders of magnitude. We discuss the scope and possible generalizations of the model.

###### pacs:
95.35.+d, 12.60.Jv
preprint: ULB-TH/13-09

## I Introduction

The nature of Dark Matter (DM), which is supposed to account for about 80 % of all mass in the universe, is one of the big mysteries of physics. It is also one of the strongest indication for possible physics beyond the Standard Model (SM) of particle physics. Indeed the dominant view is that dark matter is made of new, neutral and stable (or very long-lived) particles. Among the plethora of possible DM candidates, weakly interacting massive particles or WIMPs have many attractive features. First and foremost their abundance may be naturally explained through thermal freeze-out, a mechanism that is very robust, insensitive to unknown, higher scale physics, and points to an almost unique prediction for the annihilation cross section of DM in the early universe, . This feature also paves the way for the strategies for the identification of DM, provided that DM annihilates into –and thus interact with– SM degrees of freedom: at colliders, through direct detection, using low background detectors, or through indirect detection, which is the topic of the present work.

Indirect detection rests on the possibility that DM, which supposedly is accumulated in various parts of the universe, to begin with the central region of our own galaxy, may annihilate into SM particles or messengers, thus contributing to the cosmic flux of particles that reach the Earth or its vicinity. One important issue with this search strategy is that the bulk of the cosmic rays is expected to be of astrophysical origin and thus is somewhat uncertain. This, combined with the fact that the spectral energy distribution of messengers produced from DM annihilation is generically featureless, somewhat limits our ability to non-ambiguously identify DM – of course we may use the data to set exclusion limits and, in practice we do so, since there is no yet any clear signal of DM from the sky.

Possible exceptions to this rule of thumb is offered by so-called smoking guns, that is signatures that in principle have no counterpart of astrophysical origin. An important instance for our purpose is a gamma ray line (i.e. a monochromatic photon) from DM annihilation into Bergstrom and Snellman (1988); Rudaz (1989) or Urban et al. (1992) (see Bringmann and Weniger (2012) for a recent review). Gamma ray lines are actively being searched, most notably by the Fermi satellite  (2013) and the HESS telescope Abramowski et al. (2013), and again the absence of signal has so far only permit to set exclusion limits. Since DM is neutral, its annihilation in should proceed through radiative corrections, and so is a priori suppressed by a factor compared to annihilation into fermion or gauge boson pairs, or , which not only are supposed to determine the relic abundance but also lead to a large ray continuum. However this is not a no-go theorem and following the recent claim of a possible excess of gamma rays around in the Fermi telescope data Bringmann et al. (2012); Weniger (2012), much works have been dedicated to find new ways to circumvent this conclusion Dudas et al. (2012); Cline (2012); Ibarra et al. (2012); Kyae and Park (2013); Buckley and Hooper (2012); Chu et al. (2012); Das et al. (2012); Weiner and Yavin (2012); Tulin et al. (2013); Acharya et al. (2012) (see also Gustafsson et al. (2007); Mambrini (2009); Jackson et al. (2010); Arina et al. (2010); Schmidt-Hoberg et al. (2013); Wang and Han (2013)). Despite the apparent fading of the significance of the signal  (2013), we believe that it remains of interest to look for further alternative scenarios.

In the present work, we specifically focus on virtual internal bremsstrahlung (VIB) as a possible way of producing an enhanced, sharp gamma ray spectral feature from DM annihilation Bergstrom (1989); Flores et al. (1989). VIB is a process by which a gamma ray in the final state is, roughly speaking, emitted by a charged virtual particle 111Distinction from of soft photons through final state radiation (FSR) may be made in a gauge invariant way Bringmann et al. (2008).. While being suppressed by a factor of , it may actually dominate DM annihilation if the two-body process is suppressed. The canonical example is the case of two Majorana particles annihilating into a pair of light fermions  Bergstrom (1989); Flores et al. (1989). Because of Pauli principle, annihilation of the Majorana particles must be either in a s-wave spin 0 state or p-wave spin 1. In the chiral limit , the latter case is the only possibility as the effective coupling between the pair of and the is of the pseudo-vector type (which implies that the final state is ). So annihilation of the Majorana pair is either chirally or p-wave suppressed, or respectively Goldberg (1983). While of little practical importance for freeze-out in the early universe, for which , the suppression is dramatic at the galactic center, where . On the contrary, the emission of a photon in the final state allows for , so 3-body annihilation may proceed in the s-wave channel, which is by far the dominant process at the galactic center.

This beautiful mechanism has been first proposed within the framework of the MSSM Bergstrom (1989), but since then has been considered and studied further in more general terms (see e.g. Baltz and Bergstrom (2003); Bringmann et al. (2008); Ciafaloni et al. (2011); Bell et al. (2011); Barger et al. (2012); Garny et al. (2012)). In particular, a very simple scenario has been proposed in Bergstrom (2012) in an attempt to explain the putative 130 GeV Fermi excess. The dark matter candidate is a right-handed neutrino, and the signal is annihilation into lepton-antilepton pairs together with a gamma. In this work, it has been shown that both the relic abundance and the Fermi measurement could be simultaneously explained, assuming a slight astrophysical boost of the gamma ray signal. In the present work, we consider a scalar dark matter candidate instead, with properties which are otherwise very similar to those of the heavy neutrino of Bergstrom (2012), hence we will adopt this instance as a benchmark to which to compare our model. The basic facts we will use is that the annihilation of a real scalar DM candidate into fermion-antifermions pair is either s-wave but chirally suppressed , like the Majorana case, or d-wave suppressed, . The extra velocity suppression compared to the Majorana case may seem harmless if we consider annihilation in the early universe, but will show otherwise. In particular we will show that VIB annihilation may give to very significant annihilation into gamma rays for candidates that match the measured cosmic abundance. For the sake of the argument we will consider a very simple toy model, and limit ourselves to a leptophilic scalar DM candidate.

We begin with a discussion of the basic features of the model, including its annihilation into lepton-antilepton pairs. Next we give some details of our calculation of its 3-body annihilation in the VIB channel and compare the results with the Majorana case. In particular we show that the VIB signal is strongly enhanced in the scalar case compared to the Majorana case. In the last section we discuss the possible generality of this results, possible drawbacks, and prospects. We finish with some conclusions.

## Ii The model

The model we consider is very simple. It consists of a real scalar particle, , which we take to be leptophilic for the sake of our argument. By this we mean that it has Yukawa couplings only to SM leptons. More specifically, in this section we consider couplings to the right-handed ones (). We will discuss other possibilities in Sec. IV.4. We also introduce heavy vector-like leptons (). At this stage it does not matter whether there is one or many (like one per SM generation) heavy leptons. Their Yukawa interactions are thus of the form

 L⊃ylS¯ΨlR+h.c.. (1)

In this specific instance the are thus singlet and obviously their hypercharges are equal . We want to consider the possibility that the is a dark matter candidate. To insure its stability, we assume that the full Lagrangian is invariant under a discrete symmetry,

 S⟶−S

and

 Ψ⟶−Ψ

while all the SM fields are taken to be even under . We will further assume that the symmetry is unbroken and the lightest odd particle is the scalar, . Hence, in this model, the annihilation of DM into SM fields goes through heavy vector-like fermions exchange. Such an interaction has already been considered in Boehm et al. (2004) when studying scalar dark matter candidates in the MeV range. Following Fileviez Perez and Wise (2013), we call this scenario the vector-like portal. Notice that, unlike in Fileviez Perez and Wise (2013), the scalar field is taken to be a real singlet, like in other simple models of non-fermionic dark matter Silveira and Zee (1985); Veltman and Yndurain (1989); McDonald (1994); Burgess et al. (2001). Being a scalar singlet has also renormalizable coupling to the SM scalar (aka the Higgs) portal,

 L⊃λS2S2|H|2. (2)

We begin by assuming that this coupling is subdominant, but we comment on relaxing this condition in Sec. IV.4.

### ii.1 Two-body annihilation

This being laid down, we now consider the annihilation process

 S(k1)S(k2)→l(p1)¯l(p2). (3)

For reasons that will become clear, we give a rather pedestrian derivation of a the annihilation amplitude.

From Fig. I the and channels amplitudes are given by

 M(t)f¯f=y2l¯u(p1)PL1⧸p1−⧸k1−MΨPRv(p2)

and

 M(u)f¯f=y2l¯u(p1)PL1⧸p1−⧸k2−MΨPRv(p2)

respectively where is the projector on right (left) helicity. The total amplitude reads

 Mf¯f=12y2l¯u(p1)PL[(⧸k1−⧸k2)(D11−D22)+(⧸p2−⧸p1)(D11+D21)]v(p2)

where

 Dij=1(ki−pj)2−M2Ψ.

Here and in the next section we adopt the notations of Ciafaloni et al. (2011). Using the equations of motion for the fermion and the antifermion, this becomes

 Mf¯f=12y2l¯u(p1)[PL(⧸k1−⧸k2)(D11−D21)−mf(D11+D21)]v(p2). (4)

The first term is velocity suppressed, while the second term is proportional to the fermion mass. Neglecting terms with powers of larger than one, using

 D11−D21 = D11D21×2p1⋅(k2−k1) (5)

and

 D11+D21 = D11D21(2M2S−2M2Ψ−2p1⋅p2), (6)

we get

 Mf¯f=12y2l¯u(p1)[PL(⧸k2−⧸k1)2p1⋅(k1−k2)−mf(2M2S−2M2Ψ−2p1⋅p2)]v(p2)D11D21. (7)

The interesting point is that in the chiral limit, , the amplitude squared becomes proportional to , where denotes the relative velocity of the annihilating particles. Assuming that negligible and working in the non-relativistic limit relevant for annihilation of WIMPs, we get at leading order in

 σv(SS→l¯l)=y4l60πv4M2S1(1+r2)4 (8)

for the annihilation of into a light lepton-antilepton pair. Here refers to the ratio of masses .

The suppression by in the chiral limit is a bit unusual. For the sake of comparison, the corresponding annihilation of a pair of a gauge singlet Majorana into light leptons through a heavy charged scalar (), with coupling

 L⊃glΦ†χlR+h.c. (9)

is given by

 σv(χχ→l¯l)=g4l48πv2M2χ1+r4(1+r2)4 (10)

which shows the usual p-wave suppression and (see also Garny et al. (2012) Eq.A.5). In the following, will always refer to the ratio of the mass of the next-to-lightest particle odd under the (NLZP) symmetry divided by the dark matter mass and it is always larger than one. For identical Yukawa couplings, DM masses and ratios , we have the following ratio of the averaged cross sections into lepton-antilepton pairs

 ⟨σv⟩ll|S⟨σv⟩ll|χ=4⟨v4⟩5⟨v2⟩(1+r4)% \lx@stackrel<∼0.16 (11)

where the bound is obtained assuming and . In other words, for identical DM masses, assuming thermal freeze-out, it takes a larger Yukawa coupling for a scalar than for a Majorana to match the observed relic abundance of DM. This behavior will be studied in more details in Sec. IV.

### ii.2 Chiral suppression from an effective operator perspective

The d-wave, , suppression of the annihilation cross section of real scalars in fermion-antifermion pairs is easy to explain. The annihilation of a pair of scalar in a s-wave corresponds to a CP even state. Thus, the fermion-antifermion final state must be described by a CP even scalar bilinear operator e.g. . However the Yukawa interaction of (1) involves chiral (here right-handed) SM fermions. Hence, the s-wave annihilation is chirally suppressed. In other words, the amplitude of (4) derives from the dimension 5 operator:

 OS=mfS2¯ll. (12)

In principle, we could have annihilation in a p-wave, at least based on the constraint from CP. However there is no CP-odd bilinear operator involving two identical real scalars (i.e. no current) hence the next possibility is a dimension 8 operator of the following form

 OT=∂μS∂νSΘμνR (13)

where is the stress-energy tensor of the Dirac field ,

 ΘμνR=i2¯lR(γμ→∂ν−γν←∂μ)lR

Clearly the (traceless part of) has which implies that the annihilation of a pair of real scalars is d-wave suppressed in the chiral limit.

The tentative conclusion of this section is that 2-body annihilation of a pair of real scalar into SM fermions is suppressed compared to the case of a Majorana particle. We will show that this will have concrete implications.

## Iii Virtual Internal Bremsstrahlung

In this section we turn to radiative processes, and in particular to internal bremsstrahlung. This is of interest for two reasons. First, as it is well-known, the annihilation cross section in the s-wave is no longer suppressed, which implies that the 3-body final state process may be more important than the 2-body process, despite the suppression of the former by a factor . This is typically the case for annihilation in light fermions (e.g. leptons) and when the relative velocity is non-relativistic, like at the galactic center (). Second, the emission from (essentially) the virtual intermediate particle, depending on the ratio of its mass and that of DM, may show a sharp spectral feature, which may even mimic a monochromatic gamma ray line. As mentioned in the introduction, the literature on this topic is vast. In this section we specifically follow the detailed and pedagogical approach of Ciafaloni et al. (2011) to derive the annihilation cross section of the scalar to . We warn the reader that 1/ the result is disappointingly simple – the spectrum of gamma rays and leptons/antileptons is the same as in the case of a Majorana particle– and 2/ this fact was already discussed in the literature Barger et al. (2012) (see also Boehm and Uwer (2006) in the case of light dark matter). Our excuse to re-iterate is that we believe that it is of some interest of being a bit more detailed (and compare to the calculation of the Majorana case of, say, Ciafaloni et al. (2011)) and also that the precise normalization is of prime importance for indirect searches – so that it is perhaps worth providing an independent check.

### iii.1 Derivation of 3-body cross-section

The relevant t-channel diagrams are shown in Fig. II. There are also the three exchange diagrams (u-channel), which add up in the scalar case (while they subtract in the Majorana case). As before, and denote the momenta of the annihilating dark matter particles, and denote the momenta of the outgoing leptons and we write the momentum of the photon,

 S(k1)S(k2)→¯lR(p1)lR(p2)γ(k)

and write

 M⋅ϵ∗=ey2l[M(t)a+M(u)a+M(t)b+M(u)b+M(t)c+M(u)c]

where the superscript refers to the -channel diagram. Having in mind s-wave annihilation, we may set , so that in this limit and after some obvious manipulations, the first two amplitudes, which are associated to inner emission, may be written as

 M(t)a=¯u(p1)PL[(M2F+M2S)⧸ϵ∗−2ϵ∗⋅K⧸K]v(p2)D1D2≡M(u)a (14)

where

 Di=1(pi−K)2−M2Ψ.

Similarly the final state radiation amplitudes may be written as

 M(t)b=¯u(p1)PL[2ϵ∗⋅p1+⧸ϵ∗⧸k]⧸Kv(p2)D2D(p1+k)≡M(u)b (15)

where the final state fermion propagator

 D(p1+k)=1(p1+k)2−m2f≡12p1⋅k

displays the usual infrared divergent behaviour (here we work in the chiral limit ), and

 M(t)c=¯u(p1)PL⧸K[2p2⋅ϵ∗+⧸k⧸ϵ∗]v(p2)D1D(p2+k)≡M(u)c. (16)

Now it is easy to see that the potentially IR divergent pieces cancel from the FSR (Eqs. (15) and (16)) amplitudes. Using and the equation of motion , and , the amplitude reduces to

 Mb = ¯u(p1)PL[2ϵ∗⋅p1⧸k+ϵ∗⧸k⧸p1]v(p2)D2D(p1+k) (17) =

Similarly the amplitude is simply given by

 Mc=¯u(p1)PL⧸ϵ∗v(p2)D1.

Using

 D1+D2=−(2M2Ψ−2M2S+2K⋅(p1+p2))D1D2,

we may combine the six amplitudes to get

 Mtot=¯u(p1)PL[k⋅(p1+p2)⧸ϵ∗−ϵ∗⋅(p1+p2)⧸k]v(p2)D1D2. (18)

The total amplitude is manifestly gauge-invariant ( for as it should be), but the derivation makes clear that both the internal and FSR processes are necessary for this to occur. It is also IR divergence free, as expected on general grounds (see Bringmann et al. (2008)). Indeed, IR divergences in FSR, which are are expected to cancel with similar contributions from the interference of a tree level amplitude and its one-loop radiative corrections (see e.g. Peskin and Schroeder (1995)). In the case at hand, just like in the case of the annihilation of Majorana particles, the tree level amplitude in an s-wave initial state vanishes in the chiral limit.

### iii.2 Spectrum and cross section dependence in r for VIB

The amplitude (18) bears little resemblance to the one of the Majorana case (see e.g. Eq.(4.12) in Ciafaloni et al. (2011)). Yet it gives rise to precisely the same gamma ray spectrum. Defining the 3-body annihilation cross section as

 vdσ2→3 = |M|2128π3dxdy (19)

where refers to the relative velocity of the particles and are the reduced energy parameters and , with the Mandelstam variable corresponding to the center-of-mass energy squared, we obtain the following amplitude squared for the 3-body annihilation with emission of a photon

 |MS|2=32παy4lM2S4(1−y)(2+2x2+2x(y−2)−2y+y2)(1−r2−2x)2(3+r2−2x−2y)2 (20)

with , in agreement with Barger et al. (2012).

As above, it is of interest to compare this expression to the one obtained in the same limit in the Majorana case (see Barger et al. (2012) and also Eq.A.1 in Garny et al. (2012))

 14∑spin|Mχ|2=4παg4lM2χ4(1−y)(2+2x2+2x(y−2)−2y+y2)(1−r2−2x)2(3+r2−2x−2y)2 (21)

where . Clearly the dependence on and are precisely the same. It is interesting to notice that, all other things being kept constant, the cross section is larger by a factor of 8 in the scalar case compared to the Majorana case. A factor of comes clearly from the spin average. The extra factor of is related to the projection of the Majorana pair into a spin zero initial state (i.e. there is a factor of in the amplitude). Let us emphasize that the relevant normalization scalar versus Majorana is not important for our argumentation, what matters is 2-body versus 3-body.

In Fig. III we show the dependence of the photon spectrum

 dNγlldx=MdmσγlldσγlldEγ (22)

in the parameter . The spectra have been obtained integrating the amplitude (20) over . We see that for larger values of the amplitude of the spectra is lower but the spread is larger. The sharp feature in the spectrum also moves to smaller with increasing .

Let us emphasize that is independent of the scalar/Majorana nature of the dark matter candidate and this concordance has already been elucidated in Barger et al. (2012). It is related to the fact that the scalar and Majorana initial states, while both , differ only by their parity, the scalar case being clearly CP even. We have nothing new to add here, but merely repeat their argument, which stems from the fact that the amplitude (18) may be derived from the effective operator

 OS=(∂μ¯lRγνlR+¯lRγν∂μlR)Fμν

while in the Majorana case it is related to

 Oχ=(∂μ¯lRγνlR+¯lRγν∂μlR)~Fμν,

where is the dual of , which only amounts to exchanging the role of the and the of the photon and thus gives rise to the same spectrum.

Although the 3-body cross section differ for Majorana and scalar dark matter by numerical factors, we know that the dependence is universal and, in the appendix, we give a formulation of that emphasize this fact. Notice that a useful approximate expression of has also been derived previously in Bringmann et al. (2008).

### iii.3 Annihilation into γγ versus VIB

For the sake of completeness, and because of their close relation to internal Bremsstrahlung, we discuss in this section the annihilation of DM in gamma ray pairs. As is well-know, the s-wave annihilation of a pair of Majorana particles in monochromatic gamma rays may be derived from a chiral anomaly argument Rudaz (1989); Bergstrom (1989). Concretely, in the chiral limit and for , the amplitude for annihilation may be obtained by simply replacing, in the box diagram corresponding , the scalar propagator by an effective contact interaction between the Majorana particles and the light fermions. Since this effective coupling is of axial-vector type, the resulting triangular diagram, which only involves light fermions, is precisely the one that arises in the derivation of the chiral anomaly. This implies in particular that the amplitude is non-vanishing even for massless fermions. For and in the chiral limit, the annihilation cross section is simply given by Rudaz (1989)

 σv(χχ→γγ)=α264π3g4lM2χ1r4.

The general expression, still in the chiral limit but valid for all (including ) as obtained from the calculation of the box diagrams Bergstrom (1989), is given for reference in the appendix.

Naively we would expect a similar result to hold for the annihilation of scalar particles, with the proviso that the initial state is CP even in this case, while it is CP odd for Majorana particles, thus by replacing the chiral anomaly with the trace anomaly. Concretely, in the limit , we would replace the heavy fermion propagator in the box diagrams by the effective contact interaction of (13) and then use the trace anomaly, to estimate the annihilation into gamma ray lines. The most immediate consequence of this argument is that the annihilation amplitude should be finite in the chiral limit, just like in the Majorana case.222As a way of comparison, notice that this is very different from the contribution of light fermions to the decay of the Higgs in two photons, which vanishes in the chiral limit, see e.g. Djouadi (2008). While this turns out to be the case, the argument seems to be incorrect or, at the very least, it does not give the dominant contribution to the process. Indeed the trace anomaly would lead to a cross section that scales like (the effective operator of (13) is dimension 8), while a calculation of the full box diagram gives a results which actually scales like , again as in the Majorana case. More precisely, following the result quoted in Tulin et al. (2013), which is based on a calculation made in Bertone et al. (2009), we get in the chiral limit and for ,

 σv(SS→γγ)=α236π3y4lM2S1r4,

which, all other things being kept constant, differs from the corresponding expression in the Majorana case simply by a factor of . We are currently re-doing the calculations of the box diagram made in Bertone et al. (2009), for its own sake of and for the possible implications for direct detection (see next section). For the time being, we tentatively conclude that there is a subtle distinction between the Majorana and scalar cases. This is further illustrated by the fact that the exact dependence of on , which is given for reference in the appendix, diverges at in the chiral limit, as illustrated in Fig. IV.

Using these results, we show as an illustration the differential photon spectrum associated to the (blue lines), (green lines) and (red lines) channels as well as their sum (grey lines) in Fig. V. The spectra have been normalized by the sum of the annihilation cross-sections into the three final states, i.e.

 dNidx = 1⟨σv⟩γd⟨σv⟩idxfor i=γγ,γZ,γ¯ll (23)

where and . We have chosen GeV and we have considered annihilation into one single lepton specie, with the Yukawa coupling set to one. For these parameters, we have obtained the cross sections and listed in Table 1. We have estimated the cross-sections using

 ⟨σv⟩γZ≃2tan[θW]2(1−(MZ2Mdm)2)3⟨σv⟩γγ, (24)

with is the Weinberg angle, as in the case of Higgs decay into two photons or (see e.g. Djouadi (2008)). From these numbers, we see once more that, for a fixed set of model parameters , and , the radiative processes are always larger for scalar particles, especially for Bremsstrahlung.

The photon raw spectra for the 3-body final state has been obtained in Sec. III.2 while in the case the spectra is just a delta function at , multiplied by two to account for photon multiplicity. For the final state, one expects a broader feature, due to the Z width, around . We follow Bertone et al. (2009); Jackson et al. (2010) that described the resulting photon spectrum with an intrinsic width that depends on the final state massive boson. All raw spectra are then convoluted with a gaussian kernel in order to account for the finite energy resolution of the detector. In Fig. V, we consider a relative energy resolution of .

From the normalized spectra presented in Fig. V the differences between Majorana and scalar dark matter are not obvious. The general form of the spectra is actually very similar for both scalar (continuous curves) and Majorana (dotted curves) particles especially for in which case the Bremsstrahlung drives the main features of the full spectrum. For , the scalar photon line spectrum is still dominated by the VIB characteristics while in the Majorana case a double line structure become more distinguishable (in the Majorana case, we have obtained results which are consistent with those of Garny et al. (2013)). Notice that, for a more complete description of the photon spectrum, one should also include the continuous gamma spectrum from the line as well as from the production of the final state leptons, in particular in the case of tau leptons. This however affects the spectra for small energies only, and this only slightly, see for instance Bringmann et al. (2012).

## Iv Possible phenomenological implications

In this section, we first discuss the implication of d-wave suppression of the annihilation cross section on the dark matter relic abundance. This will lead us to the conclusion that larger Yukawas are needed in the scalar dark matter case than in the Majorana one in order to account for the total amount of dark matter. This in turn will imply that the cross section for radiative 3-body processes can become as important as the 2-body process in the scalar case. Finally in order to illustrate our findings we present a numerical analysis comparing the viable parameter space for the simple scalar and Majorana dark matter models defined in Sec. II.

### iv.1 Relic abundance for d-wave suppressed annihilation

To begin with, we consider the simplest scenario and assume that the relic abundance of pair of scalar in the early universe is determined by the d-wave suppressed 2-body process discussed in Sec. II. The thermally averaged cross section is given by

 ⟨σv⟩=∑fy4l60π⟨v4⟩M2S1(1+r2)4.

Following Srednicki et al. (1988) (or Gondolo and Gelmini (1991)) and using

 ⟨σv⟩=∫d3k1d3k2e−(E1+E2)/Tσv∫d3k1d3k2e−(E1+E2)/T=√x34π∫∞0dve−xv2/4σv

with , we get

 ⟨v2⟩=6xfand⟨v4⟩=60x2f.

Using for the relative temperature at freeze-out one gets and . One has also to take into account the fact that the velocity has a slight numerical impact on the dependence of the relic abundance. If the thermal averaged cross section scale like , with for s-wave, p-wave,… dominated cross section, taking into account that the Boltzmann equation for the relic abundance for takes the form

 dYdmdx=−λx2+nY2dm

where is a constant, and denote the comoving dark matter number density then Kolb and Turner (1990)

 Y∞dm≈(n+1)xn+1fλ.

Hence, to reach the same relic abundance , the averaged annihilation cross section in for instance a pure d-wave channel must be larger by a factor of wrt the s-wave case i.e. . Considering the velocity expansion of the annihilation cross section to the next order compared to e.g. Griest and Seckel (1991); Kolb and Turner (1990), writing , we have

 Ωdmh2≃1.07109xfMpl/GeV√g∗(a+3b/xf+20c/x2f) (25)

where GeV is the Planck mass and is the number of relativistic degrees of freedom at the time of freeze-out.

Lest the reader think that we are splitting hairs, consider again the ratio of (11) but now expressed in terms of the relic abundances,

 Ωdm|SΩdm|χ=∑lg4l(5×3)(1+r4)x2f∑ly4l(4×20)xf=(∑lg4l∑ly4l)316xf(1+r4).

The factor that multiplies the couplings is typically , hence larger Yukawa couplings are required to reach same abundance for the scalar than for the Majorana. It is pretty clear that this implies larger Bremsstrahlung emission in the case of scalar DM.

### iv.2 Enhanced three-body processes for scalar dark matter

In the Majorana case, the 3-body annihilation cross section is always small compared to the 2-body one, which is relevant for the abundance in the early universe. The ratios are shown in the left hand side of Fig. VI for Majorana (blue dashed line) and scalar (black continuous line) dark matter. The limiting values for are

 ⟨σv⟩γll⟨σv⟩ll∣∣∣χ≈0.015and⟨σv⟩γll⟨σv⟩ll∣∣∣S≈0.76 (26)

taking . It is by itself remarkable that, for the scalar candidate, the radiative process may be almost as important as the 2-body one. We will come back to this in the next section. In the meantime we may define the “boost factor” ()

 BF=⟨σv⟩γll⟨σv⟩ll∣∣∣S×⟨σv⟩ll⟨σv⟩γll∣∣∣χ, (27)

that is equal to 50 for and gives the relative enhancement of the Bremsstrahlung signal of the scalar DM candidate compared to the Majorana one. It should be clear from the behaviour of the cross section that is actually a minimum and this is further illustrated in the right hand side of Fig. VI.

The enhancement of the Bremsstrahlung signal from a scalar WIMP is our main result, but as such it is of no immediate use, as other processes may determine the relic abundance. In particular one has to take into account co-annihilation processes, which are important in the case of nearly degenerate particles, Griest and Seckel (1991). Also a singlet scalar candidate may have renormalizable coupling to the SM scalar (SMS). To study these effects, we have implemented the scalar and Majorana models in Micromegas Belanger et al. (2011) with the help of Feynrules Christensen and Duhr (2009). Our analysis is detailed in the next section.

### iv.3 Numerical analysis

We have considered the scalar and Majorana dark matter models which interaction with the SM fermions is dictated by Eqs. (1) and (9), neglecting extra interactions through the SM scalar portal from Eq. (2), i.e. setting . In addition, we have assumed that the dark matter couples to one single lepton specie, the electron. The result of a random scan over the parameter space is shown in Figs VII and VIII, see also the appendix for more details. We also give the 3-body annihilation cross section (Yukawa coupling) versus the mass of the DM in Fig. VII (resp. Fig. VIII) both for the scalar (left) and Majorana (right) candidates. All the points match the observed cosmological relic abundance. Notice that we have taken into account the bremsstrahlung contribution to the effective annihilation cross section relevant for the computation of the relic abundance. In the case of the scalar dark matter such process can modify the dark matter abundance up to a 15% for the largest values of the Yukawa couplings while in the case of the Majorana dark matter it may be safely neglected (it is always %).

The color code corresponds to different values of , which is the relative mass difference between the DM particle and the heavy charged particle. The blueish points (roughly the lower right points in the plots) correspond to dark matter candidates which mass is nearly degenerate, roughly , with the mass of the electrically charged heavy particle ( and ). In the latter case, the annihilation of the charged particles

 ¯ΨΨorΦ†Φ → ¯qq,¯ll,γγ (28)

and the co-annihilation processes

 ¯ΨSorΦ†χ → ¯eγ (29)

are important for the determination of the relic abundance. The greenish points (roughly the top regions) correspond to candidates for which the dark matter annihilation into lepton-antilepton pairs,

 SSorχχ→¯ee (30)

become progressively more important and so does Bremsstrahlung. Increasing further the ratio of masses , the relative weight of the 3-body process diminishes, particularly for the Majorana case, see Sec. III.2. The maximal value of is reached for GeV and TeV given our assumption on the Yukawas , see Fig. VIII. In our numerical analysis as well as writing the co-annihilation annihilation processes as in (29) and (30), we have assumed that the coupling of Eqs. (1) and (9) is made for . For dark matter coupling to three light flavours, the results of Fig. VIII should be rescaled by a factor of . From Fig. VIII, it is clear that significant Bremsstrahlung requires rather large Yukawa couplings , especially for scalar candidates. This simply reflects the fact that the annihilation is d-wave for scalars and p-wave for Majorana, as discussed in Sec. II.

For a fixed value of and , one clearly distinguish two different regimes in the Yukawa- plane. For the lowest values of the Yukawas, the dark matter models accounting for dark matter abundance are clearly independent of or . At some point the Yukawa function of begins to bend and finally reaches a regime with . This behaviour can be understood by comparing the dependence of the processes (28)-(30) in or . For the smallest values of the coupling and , it is the annihilation processes of and through and boson exchanges in (28) that dominate over all other processes and play the major role in fixing the dark matter abundance. The corresponding cross section is , with the weak coupling, and it is independent of (see Appendix). Thus no Yukawa or dependence is to be expected in this regime. For larger values of the Yukawas, the coannihilation cross section of (29) or begin to compete with charged particle annihilation and the abundance begins to depend on the Yukawa coupling. The relative importance of both of those regimes is weighted by Boltzmann factors, and for (29) and (28) respectively Griest and Seckel (1991), so that the dependence in becomes more pronounced for larger values of . For the largest values of the Yukawas, the processes which are or fix the dark matter density. One should be aware that the standard treatment of freeze-out mechanism and coannihilation processes Griest and Seckel (1991), as implemented in numerical code like Micromegas Belanger et al. (2011), rests on the assumption that the dark matter and heavy charged particles are in chemical equilibrium. Here, as in Garny et al. (2013), we have simply checked under which conditions the processes and are in equilibrium at the epoch of thermal freeze-out: this should be the case provided .

Taken the Bremsstrahlung spectral features seriously, one may wonder if these candidates are compatible with constraints from the current gamma ray experiments, in particular Fermi-LAT and HESS. To this end, we also report in Fig VII, with a series of upper bounds on for that were derived in Garny et al. (2013) using Fermi-LAT and HESS data as well as the future constraints from the GAMMA-400 satellite mission and the Cerenkov Telescope Array (CTA). For simplicity, we report the limits in the plane without adding the contribution from , which are anyway negligible for , see Sec. III.3 and also Garny et al. (2013). As in Bringmann et al. (2012), we find that the largest possible value of the 3-body cross section for a Majorana dark matter giving rise to right dark matter abundance is always substantially smaller than current limits (so that an astrophysical boost would be required to match any excess, say the possible feature around 130). In contrast, in the scalar dark matter case, within the assumptions made so far regarding e.g. , one can easily cross those limits, typically giving rise to a more important gamma-ray flux.

For the sake of illustration we consider two benchmark models (see Table 2) that could be relevant for the possible excess of gamma rays around 130 GeV in the Fermi-LAT data Bringmann et al. (2012); Weniger (2012). Both candidates have a relic abundance . The relative contributions of the various processes to their annihilation cross section at freeze-out Griest and Seckel (1991) are given by for dark matter annihilation into the 3-body channels, for annihilation into 2-body and for coannihilation processes, while the annihilation of the heavy charged particles contribute for . The dark matter mass is taken to be 150 GeV, following Bringmann et al. (2012) that pointed out that the best-fit 3-body cross section is cm/s with GeV — in the case of a gamma ray line, the best fit is for about cm/s with a mass of GeV. From Fig. VII, it is clear that can be reached for a scalar dark matter candidate, while an extra, possibly astrophysical boost factor would be required for a Majorana candidate.

In Table 2, the Majorana benchmark candidate is chosen so as to maximize the 3-body annihilation cross section. In order for this candidate to saturate the data, a boost factor of about 10 would be needed. The scalar benchmark has also GeV but this time the Yukawa coupling is chosen so that the candidate could account for a gamma ray at GeV, see the resulting spectra in Fig. IX. Unfortunately no distinction can be made between the shapes of scalar and Majorana benchmark spectra, even for optimistic resolutions such as , we thus only show the scalar dark matter one. The points associated to our benchmark models correspond to the filled black diamonds in Figs. VII and VIII. For reference, we also report the GeV candidate put forward in Bergstrom (2012), which is assumed to couple to the three families of leptons with universal Yukawa coupling for and to a heavy scalar () with mass such that .333Notice that the value of obtained here is two times larger than the number reported in Bergstrom (2012). We expect that this is due to a factor of two rescaling of in Bergstrom (2012) which is relevant when compared to gamma ray flux constraints on .

### iv.4 Prospects and possible extensions

So far we have considered a specific scenario, assuming the coupling of a real scalar DM candidate to a heavy vector-like leptonic particle which we take to be singlet under . This we have done for the sake of simplicity and illustration of the possible enhancement of the VIB in the case of a scalar compared to a Majorana DM candidate. A systematic analysis of the possible extensions of this scenario is beyond the scope of the present work, and will be presented elsewhere. In this section we briefly summarize some possible forthcoming results.

One possible improvement would be to consider the continuous gamma ray emission associated to weak gauge bosons (which we have neglected here) especially generalizing the interaction (1) to the case of Yukawa couplings to doublet vector-like leptons. In the latter case, new contribution to 3-body processes should be taken into account.444 Notice that in our numerical analysis we took into account the annihilation into for the calculation of the relic abundance which typically give rise to Beside the obvious boost of the signal that we may expect in the scalar case compared to the Majorana case (which would couple to new a scalar doublet), the results will however match the existing analysis of Garny et al. (2012), since the scalar and Majorana dark matter spectra are expected to be precisely identical.

Possibly more interesting would be to consider Yukawa couplings to heavy vector-like quarks. In this case, there are potentially two aspects that are worth being studied in more details. To begin with, we have the possibility of the VIB of a gluon. As emphasized in the previous section, in the case of scalar particles annihilation, the VIB of a photon is substantial, possibly similar in magnitude with the 2-body process, at least for annihilation in the early universe. The ratio of the VIB of a gluon to that of a photon being given by

 ⟨σv⟩g¯qq⟨σv⟩γ¯qq=N2c−12NcαsQ2α (31)

which is about for up-like quarks, and for down-like quarks. These features imply that the VBI of a gluon will be potentially more relevant in determining the relic abundance than the corresponding two body process, at least for light quarks. This in turn would imply that the annihilation rate into gamma rays is fixed by (31) a possibility very much in the spirit of the scenario considered in Chu et al. (2012). We will address such processes with gluons and quarks in final states in a work in progress, as well as the constraints from the measurements of the antiproton flux in cosmic rays, see e.g. Ibarra et al. (2012); Chu et al. (2012).

A second interesting aspect in scalar dark matter models coupled to heavy vector-like quarks is direct detection. A priori the analysis should be similar to the Majorana case discussed in Ibarra et al. (2012). In particular, for the scalar effective coupling to quarks, the relevant interaction should be related to the effective operator (12), with the obvious substitution of leptons by quarks. We expect though some subtle differences for the contribution of the so-called twist-2 operator, which is formally related to the operator (13). As is well known, in the Majorana case the proper determination of direct detection collision cross section requires the evaluation of the box diagram in kinematic regimes in which the direct use of the trace anomaly may give wrong results Drees and Nojiri (1993). The equivalent process for scalar DM through a vector-like portal has not yet been studied and will be addressed in a future work.

In our work, we have assumed that the quartic coupling of the real scalar to the Standard Model scalar may be neglected. An obvious generalization of the results presented here would be to relax such an assumption. For the time being, we just emphasize the trivial fact that this coupling may help decreasing the VIB cross section. Indeed, as shown in Fig. VII, there are many WIMP scalar candidates with a too large flux into gamma rays, and this all the way up to GeV. If another annihilation channel is opened, like through the SM scalar, the Yukawa coupling may be smaller, and these candidates may become viable. This is not quite the same for the Majorana candidates, for which the signal is systematically below the current constraints, and thus which would require some astrophysical boost to saturate the observations. As an example, let us take from our scans a scalar candidate with GeV, and which relic abundance but, cm/s is excluded by Fermi-LAT constraint. By allowing for and decreasing to 1.5, we can still account for while getting a cm/s below the limits and near . Notice though that in the latter case, new channels open and give rise to a continuous gamma ray component with a 2-body annihilation cross section that is no more d-wave suppressed. In addition, direct detection searches can also test such a scenario given that the scattering cross-section on a proton amounts to cm and is nearly excluded by the Xenon100 limits Aprile et al. (2012). See also Cline et al. (2013) for recent study of latest and future constraints on singlet scalar dark matter and SMS portal.

In a broader perspective, one may consider a much more extended parity odd sector. For definiteness, consider the so-called Inert Doublet Model (IDM) Deshpande and Ma (1978); Ma (2006); Barbieri et al. (2006). In its simplest incarnation it consists of the addition of a single scalar doublet, , odd under a parity, with no expectation value. A simple and very interesting extension consists in the addition of an odd right-handed neutrino field Ma (2006) with a Majorana mass term, a model in which the mass of the SM neutrinos is generated radiatively at one loop. In the same spirit, we may consider the possibility of introducing heavy vector-like doublets, be them lepton or quark-like, all odds under . Of course this scenario implies the introduction of a large number of new parameters, to begin with the Yukawa couplings, but it also broadens the range of possibilities for the IDM, in particular regarding not only gamma ray features,555Notice that the vanilla IDM already presents VBI gamma ray features Garcia-Cely and Ibarra (2013). but also potentially antimatter in cosmic rays, new direct detection channels, and particle physics signatures with flavour changing processes such as 666Notice that such processes are potentially present in the singlet scalar scenario discussed here but are tuned to be irrelevant by taking the matrix Yukawa couplings, to be diagonal in lepton flavour space. See for instance Cheung and Seto (2004) for an analysis in Majorana dark matter case and Ishiwata and Wise (2013) for a recent study of flavour changing process in the vector-like portal scenario.

## V Conclusions

In this work we have discussed a simple dark matter model that may lead to a significant gamma ray spectral feature. It consists of real scalar DM particle that interacts with SM leptons through heavy vector-like charged fermions (the so-called vector-like portal). The most striking feature of this model is the possibility of an enhanced annihilation of dark matter in a process with virtual internal bremsstrahlung (VIB). This rest on the fact that, 1/ in the chiral limit the annihilation cross section in lepton antilepton pairs is d-wave suppressed, i.e. and, taking all other things constants (dark matter mass, mass of the intermediate particle and the Yukawa couplings) 2/ the bremsstrahlung cross section is relatively larger than in the case of annihilation of Majorana particles. These two features taken together imply that the VIB feature is much more enhanced for real scalar dark matter compared to Majorana candidates.

For the sake of illustration, we have studied in more details a concrete, albeit simplistic case of a real scalar with a Yukawa coupling to the SM right-handed electron (or equivalently for all practical purpose with universal coupling to all lepton). Our main result may be read from Fig. VII that shows the annihilation cross section into 3–body final state , and thus the associated gamma ray flux, is comparatively much stronger for scalar than for Majorana dark matter, provided that these candidates account for all the dark matter cosmological abundance. For completeness, we have also compared the annihilation rates in monochromatic gamma rays of real scalar and Majorana candidates. In the near future, we intend to generalize this framework to the case of heavy vector-like quarks.

## Acknowledgement

We acknowledges stimulating discussions with Thomas Hambye and Céline Boehm. F.G and M.T. work is partially supported by the FNRS-FRS and the IISN. LLH is supported by an “FWO Vlaanderen” post-doctoral fellowship project number 1271513 and recognizes partial from the Strategic Research Program “High Energy Physics” of the Vrije Universiteit Brussel. All the authors acknowledge support from the Belgian Federal Science Policy through the Interuniversity Attraction Pole P7/37.
Note added : while our work was being completed, the article Toma (2013) was published on the ArXiv, finding similar results for real scalar dark matter.

## Appendix

In this appendix, we provide some more expressions of annihilation cross-sections involved in the determination of the relic abundance or the gamma ray signal. Notice though that our numerical results of Sec. IV.3 were obtained using Micromegas code which automatically compute all the two body processes at tree level so that their expressions given below in the non-relativistic limit are only given for the sake of completeness. We implemented the dark matter 3-body annihilation cross-sections into and in Micromegas. Also, notice that annihilation into two gammas were neglected in the computation of the relic abundance.

In Sec. IV, we neglected s- and p-wave contributions to the scalar dark matter annihilation cross-section. Here we give the s- to d-wave contributions to the 2-body annihilation cross section , to leading order in and ,

 σv(SS→¯ll)=y4f4π1M2S(1+r2)2⎡⎣m2fM2S−46m2fv2(1+2r2)M2S(1+r2)2+115v4(1+r2)2⎤⎦. (32)

Concerning the 3-body cross section , we emphasize that the dependence is universal rewriting as

 ⟨σv⟩γll=y4iα32π2KiM2iF(r) (33)

with and for . Equation (33) was obtained integrating the amplitude (20) over the domain Chen and Kamionkowski (1998)

 0≤x=EγMi≤1\rm and(1−x)≤y=EfMi≤1.

The function is identical for the Majorana and scalar cases and it is shown in Fig. X.

It behaves like , as expected, which is also the behaviour of the 2-body cross section in the scalar case. On the contrary, in the Majorana case the 2-body cross section behaves like . Notice that a useful approximate expression of has also been derived previously in Bringmann et al. (2008) in the case of Majorana dark matter. The latter reads

 ⟨σv⟩γll ≈ αKiy4i8π2M2χ((r2+1)(π2/6−log[r2+12r2]2−2Li


text:
# Sending many unique emails at same time

I have an asp.net core app that sends emails with a link. When they click the link they can start taking an online test without logging in. The link contains a number that is validated by the controller who then returns their test.

I log into my app and create a project. Here I enter a list of emails in a textbox separated by commas. In the backend I put this textbox string into a list like so:

string emailList;
List<string> list = emailList.Split(',').ToList();
//removes whitespace in list:
for (int i = 0; i < list.Count(); i++)
{
list[i] = string.Join("", list[i].Split(default(string[]),
StringSplitOptions.RemoveEmptyEntries));
}

The only way i can see on how to send many unique emails would be to loop over this list and then use the normal SendEmail method in mimekit:

string message = "Press this link to take the test";
int number = 0;

foreach(var email in emailList)
{
//gets a unique number stored in a table for each person:
number = GetNumber(email);
message = $"http://www.mywebsite.com/Test/{number}"; await _emailSender.SendEmailAsync(email, "Subject", message); }  Now the emailSender uses the normal mimekit method to send one email for each in the list. This takes a long time for a big list. Are there any other ways to do it, or can I write this more effectively? What could be the bottlenecks of my solution? Any feedback would be great, from my bad async code to security . I suspect that I use too much async when I should return only a task, but i'm not sure how to do it. I also wonder how I can even test this on a larger scale as I don't have a big emaillist to use. Here is the mimekit code to send the email: public async Task SendEmailAsync(string email, string subject, string message) { var melding = new MimeMessage(); melding.From.Add(new MailboxAddress("Frank", _options.Option1)); melding.To.Add(new MailboxAddress(email)); melding.Subject = subject; melding.Body = new TextPart("plain") { Text = message }; Try { await SendMessage(melding); } catch(Exception ex) { //log error throw new ApplicationException($"something went wrong:'{ex.Message}'.");
}
}
{
using (var client = new SmtpClient())
{
await client.ConnectAsync("smtp.WebSiteLive.net", 587, SecureSocketOptions.None);
client.AuthenticationMechanisms.Remove("XOAUTH2"); // Must be removed for Gmail SMTP
await client.AuthenticateAsync(_options.Option1, _options.Option2);
await client.SendAsync(message);
await client.DisconnectAsync(true);
}
}

any help much apprechiated!!

• It depends on the volume, but once you need to start sending out significant amounts of emails you'll need to look into using an outside service that specifically deals with such issues and offers an API (like for instance sendgrid.com ). See also stackoverflow.com/questions/4369178/… Apr 12 '18 at 9:25
• you are probably right. I just don't want to pay for it, haha. Apr 12 '18 at 11:42
• Sign up for SendGrid via Azure and you get 25000 free e-mails per month. I'm not aware of any minimum spend on Azure to qualify. Apr 12 '18 at 11:46

Populate mails in batches and process batch by batch instead of sending mail one by one, this provides better scalability and also have more control on the usage of number of threads {cancellation[if required]}.

   For example:
Number of emails to send: 103
BatchSize :5 (Make it configurable)
We have in total : 20 batches of size 5 and 1 batch of size 3

Steps are as follows

    a. Process first batch.
b. Wait till batch completes execution
c. Take next batch for process
d. Repeat the process till all the batches.

I have two approaches to achieve this, based on the requirement you can use any of the approach

1. Use async/await and Task => Create separate task for sending each mail in each batch, which uses multiple threads.

Process Batch 1 => 5 users, Create 5 tasks and execute => Wait for all tasks to complete Process Batch 2 => 5 users, Create 5 tasks and execute => Wait for all tasks to complete ...

2. Use only async/await => which will use almost same thread for processing.

Consider the Email Class as below, and _batchSize

    public class EmailClass
{
public string Email { get; set; }
public string Message { get; set; }
public string Subject { get; set; }
}
private int _batchSize = 5;
List<EmailClass> _emailCollection = new List<EmailClass>();

When the user clicks on Send button

   private async void button_Click(object sender, RoutedEventArgs e)
{
for (int ii = 0; ii < 22; ii++)
{
{
Email = "Email_" + ii + "@gmail.com",
Subject = "Subject_" + ii
});
}

await SendBulkMail();
}

Divide the whole bunch of mails, into chunks of _batchSize, process each chunk one by one.

    private async Task SendBulkMail()
{
int remainingBatches = _emailCollection.Count % _batchSize;
int numberOfBatches = (_emailCollection.Count - remainingBatches) / _batchSize;
Debug.WriteLine(string.Format("TotalCount:{0} NumberOfBatches:{1}", _emailCollection.Count, numberOfBatches));
for (int ii = 0; ii < numberOfBatches; ii++)
{
Debug.WriteLine(DateTime.Now.ToString() + " CurrentChunk : " + ii);
await SendBatchMail(0, ii, _batchSize);
}

if (remainingBatches != 0)
{
await SendBatchMail(_emailCollection.Count - remainingBatches, 0, _emailCollection.Count);
}
}

Solution 1: Create Task for each mail while processing each block

    private async Task SendBatchMail(int initalIndex, int batchIndex, int size)
{
for (int jj = initalIndex; jj < size; jj++)
{
var index = _batchSize * batchIndex + jj;
SendEmailAsync(_emailCollection[index].Email, _emailCollection[index].Subject, _emailCollection[index].Message);
}));
}
{
});
}

public void SendEmailAsync(string email, string subject, string message)
{
try
{
SendMessage(email + subject + message);
}
catch (Exception ex)
{
}
}

private void SendMessage(string message)
{
int sleepTime = new Random().Next(5) * 1000;
}

Solution 1 output : Separate threads[may or may not] is used for sending each mail in every batch

    TotalCount:22 NumberOfBatches:4
4/12/2018 11:07:02 PM CurrentChunk : 0
4/12/2018 11:07:02 PM 13 Sending mail : Email_0@gmail.com
4/12/2018 11:07:02 PM 15 Sending mail : Email_1@gmail.com
4/12/2018 11:07:02 PM 15 Message sending :  => Email_1@gmail.comSubject_1http://google.com
4/12/2018 11:07:02 PM 11 Sending mail : Email_2@gmail.com
4/12/2018 11:07:02 PM 11 Message sending :  => Email_2@gmail.comSubject_2http://google.com
4/12/2018 11:07:02 PM 13 Message sending :  => Email_0@gmail.comSubject_0http://google.com
4/12/2018 11:07:02 PM 14 Sending mail : Email_3@gmail.com
4/12/2018 11:07:02 PM 14 Message sending :  => Email_3@gmail.comSubject_3http://google.com
4/12/2018 11:07:02 PM 14 Message sent :  => Email_3@gmail.comSubject_3http://google.com
4/12/2018 11:07:02 PM 14 Sending mail : Email_4@gmail.com
4/12/2018 11:07:02 PM 14 Message sending :  => Email_4@gmail.comSubject_4http://google.com
4/12/2018 11:07:02 PM 14 Message sent :  => Email_4@gmail.comSubject_4http://google.com
4/12/2018 11:07:05 PM 15 Message sent :  => Email_1@gmail.comSubject_1http://google.com
4/12/2018 11:07:05 PM 11 Message sent :  => Email_2@gmail.comSubject_2http://google.com
4/12/2018 11:07:05 PM 13 Message sent :  => Email_0@gmail.comSubject_0http://google.com
4/12/2018 11:07:05 PM CurrentChunk : 1
4/12/2018 11:07:05 PM 16 Sending mail : Email_5@gmail.com
4/12/2018 11:07:05 PM 14 Sending mail : Email_7@gmail.com
4/12/2018 11:07:05 PM 13 Sending mail : Email_6@gmail.com
4/12/2018 11:07:06 PM 15 Sending mail : Email_8@gmail.com
4/12/2018 11:07:06 PM 15 Message sending :  => Email_8@gmail.comSubject_8http://google.com
4/12/2018 11:07:06 PM 11 Sending mail : Email_9@gmail.com
4/12/2018 11:07:06 PM 11 Message sending :  => Email_9@gmail.comSubject_9http://google.com
4/12/2018 11:07:06 PM 14 Message sending :  => Email_7@gmail.comSubject_7http://google.com
4/12/2018 11:07:06 PM 13 Message sending :  => Email_6@gmail.comSubject_6http://google.com
4/12/2018 11:07:06 PM 15 Message sent :  => Email_8@gmail.comSubject_8http://google.com
4/12/2018 11:07:06 PM 16 Message sending :  => Email_5@gmail.comSubject_5http://google.com
4/12/2018 11:07:06 PM 16 Message sent :  => Email_5@gmail.comSubject_5http://google.com
4/12/2018 11:07:10 PM 11 Message sent :  => Email_9@gmail.comSubject_9http://google.com
4/12/2018 11:07:10 PM 14 Message sent :  => Email_7@gmail.comSubject_7http://google.com
4/12/2018 11:07:10 PM 13 Message sent :  => Email_6@gmail.comSubject_6http://google.com

Solution 2 : Use only async/await pattern

    private async Task SendBatchMail(int initalIndex, int batchIndex, int size)
{
for (int jj = initalIndex; jj < size; jj++)
{
var index = _batchSize * batchIndex + jj;
}
{
});
}

public async Task SendEmailAsync(string email, string subject, string message)
{
try
{
await SendMessage(email + subject + message);
}
catch (Exception ex)
{
}
}

{
{
int sleepTime = new Random().Next(5) * 1000;
});
}

Solution 2 Output (Almost one thread is used for processing of all batches)

    4/12/2018 11:01:05 PM CurrentChunk : 0
4/12/2018 11:01:05 PM 10 Sending mail : Email_0@gmail.com
4/12/2018 11:01:05 PM 10 Message sending :  => Email_0@gmail.comSubject_0http://google.com
4/12/2018 11:01:05 PM 10 Sending mail : Email_1@gmail.com
4/12/2018 11:01:05 PM 10 Message sending :  => Email_1@gmail.comSubject_1http://google.com
4/12/2018 11:01:05 PM 10 Sending mail : Email_2@gmail.com
4/12/2018 11:01:05 PM 10 Message sending :  => Email_2@gmail.comSubject_2http://google.com
4/12/2018 11:01:05 PM 10 Sending mail : Email_3@gmail.com
4/12/2018 11:01:05 PM 10 Message sending :  => Email_3@gmail.comSubject_3http://google.com
4/12/2018 11:01:05 PM 10 Sending mail : Email_4@gmail.com
4/12/2018 11:01:05 PM 10 Message sending :  => Email_4@gmail.comSubject_4http://google.com
4/12/2018 11:01:05 PM 10 Message sent :  => Email_2@gmail.comSubject_2http://google.com
4/12/2018 11:01:05 PM 10 Message sent :  => Email_3@gmail.comSubject_3http://google.com
4/12/2018 11:01:05 PM 10 Message sent :  => Email_4@gmail.comSubject_4http://google.com
4/12/2018 11:01:08 PM 10 Message sent :  => Email_0@gmail.comSubject_0http://google.com
4/12/2018 11:01:08 PM 10 Message sent :  => Email_1@gmail.comSubject_1http://google.com
4/12/2018 11:01:08 PM CurrentChunk : 1
4/12/2018 11:01:08 PM 10 Sending mail : Email_5@gmail.com
4/12/2018 11:01:08 PM 10 Message sending :  => Email_5@gmail.comSubject_5http://google.com
4/12/2018 11:01:08 PM 10 Sending mail : Email_6@gmail.com
4/12/2018 11:01:08 PM 10 Message sending :  => Email_6@gmail.comSubject_6http://google.com
4/12/2018 11:01:08 PM 10 Sending mail : Email_7@gmail.com
4/12/2018 11:01:08 PM 10 Message sending :  => Email_7@gmail.comSubject_7http://google.com
4/12/2018 11:01:08 PM 10 Sending mail : Email_8@gmail.com
4/12/2018 11:01:08 PM 10 Message sending :  => Email_8@gmail.comSubject_8http://google.com
4/12/2018 11:01:08 PM 10 Sending mail : Email_9@gmail.com
4/12/2018 11:01:08 PM 10 Message sending :  => Email_9@gmail.comSubject_9http://google.com
4/12/2018 11:01:09 PM 10 Message sent :  => Email_9@gmail.comSubject_9http://google.com
4/12/2018 11:01:10 PM 10 Message sent :  => Email_7@gmail.comSubject_7http://google.com
4/12/2018 11:01:10 PM 10 Message sent :  => Email_8@gmail.comSubject_8http://google.com
4/12/2018 11:01:11 PM 10 Message sent :  => Email_5@gmail.comSubject_5http://google.com
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• Really enjoyed reading your solutions. Thanks alot for your feedback! I will start to rewrite my solution on monday Apr 13 '18 at 8:37
   using (var client = new SmtpClient())
{
await client.ConnectAsync("smtp.WebSiteLive.net", 587, SecureSocketOptions.None);
client.AuthenticationMechanisms.Remove("XOAUTH2"); // Must be removed for Gmail SMTP
await client.AuthenticateAsync(_options.Option1, _options.Option2);
...
await client.DisconnectAsync(true);
}

Look at how much work you're doing per message. If you reuse the client, you save setup and teardown time on every message after the first one.

If you reuse the client then you'll need to handle errors and be prepared to close it, open a new one, and continue where you left off; so there is a speed-complexity trade-off.

• good point! I could maybe loop the emailmessages inside the client call and use the sendmessage there for each one. Apr 12 '18 at 11:41
string emailList;
List<string> list = emailList.Split(',').ToList();
//removes whitespace in list:
for (int i = 0; i < list.Count(); i++)
{
list[i] = string.Join("", list[i].Split(default(string[]),
StringSplitOptions.RemoveEmptyEntries));
}

This is a really creative way of removing whitespace. But one could just use Trim:

emailList.Split(',').Select(x => x.Trim()).ToList();

string emailList;
List<string> list = emailList.Split(',').ToList();

This naming convention however isn't so inventive anymore. You name the emailString as emailList but use list instead of emails for a list of emails. Exactly the other way round.

• Thanks for feedback! I knew about Trim, just couldn't find a way to use it. I feel dumb not using a more simple solution. You are right about the naming, I will implement that too! Apr 12 '18 at 12:52

format: plain


text:
Cell gradient operator (cell centers to faces)

This property constructs the 2nd order numerical gradient operator that maps from cell centers to faces. The operator is a sparse matrix $$\mathbf{G_c}$$ that can be applied as a matrix-vector product to a discrete scalar quantity $$\boldsymbol{\phi}$$ that lives at the cell centers; i.e.:

grad_phi = Gc @ phi

By default, the operator assumes zero-Neumann boundary conditions on the scalar quantity. Before calling cell_gradient however, the user can set a mix of zero Dirichlet and zero Neumann boundary conditions using set_cell_gradient_BC. When cell_gradient is called, the boundary conditions are enforced for the gradient operator. Once constructed, the operator is stored as a property of the mesh. See notes.

This operator is defined mostly as a helper property, and is not necessarily recommended to use when solving PDE’s.

Returns
(n_faces, n_cells) scipy.sparse.csr_matrix

The numerical gradient operator from cell centers to faces

Notes

In continuous space, the gradient operator is defined as:

$\vec{u} = \nabla \phi = \frac{\partial \phi}{\partial x}\hat{x} + \frac{\partial \phi}{\partial y}\hat{y} + \frac{\partial \phi}{\partial z}\hat{z}$

Where $$\boldsymbol{\phi}$$ is the discrete representation of the continuous variable $$\phi$$ at cell centers and $$\mathbf{u}$$ is the discrete representation of $$\vec{u}$$ on the faces, cell_gradient constructs a discrete linear operator $$\mathbf{G_c}$$ such that:

$\mathbf{u} = \mathbf{G_c} \, \boldsymbol{\phi}$

Second order ghost points are used to enforce boundary conditions and map appropriately to boundary faces. Along each axes direction, we are effectively computing the derivative by taking the difference between the values at adjacent cell centers and dividing by their distance.

Examples

Below, we demonstrate how to set boundary conditions for the cell gradient operator, construct the cell gradient operator and apply it to a discrete scalar quantity. The mapping of the operator and its sparsity is also illustrated. Our example is carried out on a 2D mesh but it can be done equivalently for a 3D mesh.

We start by importing the necessary packages and modules.

>>> from discretize import TensorMesh
>>> import numpy as np
>>> import matplotlib.pyplot as plt
>>> import matplotlib as mpl

We then construct a mesh and define a scalar function at cell centers which is zero on the boundaries (zero Dirichlet).

Create a uniform grid

>>> h = np.ones(20)
>>> mesh = TensorMesh([h, h], "CC")

Create a discrete scalar on nodes

>>> centers = mesh.cell_centers
>>> phi = np.exp(-(centers[:, 0] ** 2 + centers[:, 1] ** 2) / 4 ** 2)

Once the operator is created, the gradient is performed as a matrix-vector product.

Construct the gradient operator and apply to vector

>>> Gc = mesh.cell_gradient
>>> grad_phi = Gc @ phi

Plot the results

Expand to show scripting for plot
>>> fig = plt.figure(figsize=(13, 6))
>>> mesh.plot_image(phi, ax=ax1)
>>> ax1.set_title("Scalar at cell centers", fontsize=14)
>>> mesh.plot_image(
...     stream_opts={"color": "w", "density": 1.0}
... )
>>> ax2.set_yticks([])
>>> ax2.set_ylabel("")
>>> plt.show()

The cell gradient operator is a sparse matrix that maps from cell centers to faces. To demonstrate this, we construct a small 2D mesh. We then show the ordering of the elements in the original discrete quantity $$\boldsymbol{\phi}$$ and its discrete gradient as well as a spy plot.

>>> mesh = TensorMesh([[(1, 3)], [(1, 6)]])

Expand to show scripting for plot
>>> fig = plt.figure(figsize=(12, 10))
>>> mesh.plot_grid(ax=ax1)
>>> ax1.plot(mesh.cell_centers[:, 0], mesh.cell_centers[:, 1], "ro", markersize=8)
>>> for ii, loc in zip(range(mesh.nC), mesh.cell_centers):
...     ax1.text(loc[0] + 0.05, loc[1] + 0.02, "{0:d}".format(ii), color="r")

>>> ax1.plot(mesh.faces_x[:, 0], mesh.faces_x[:, 1], "g^", markersize=8)
>>> for ii, loc in zip(range(mesh.nFx), mesh.faces_x):
...     ax1.text(loc[0] + 0.05, loc[1] + 0.02, "{0:d}".format(ii), color="g")
>>> ax1.plot(mesh.faces_y[:, 0], mesh.faces_y[:, 1], "g>", markersize=8)
>>> for ii, loc in zip(range(mesh.nFy), mesh.faces_y):
...     ax1.text(loc[0] + 0.05, loc[1] + 0.02, "{0:d}".format((ii + mesh.nFx)), color="g")

>>> ax1.set_xticks([])
>>> ax1.set_yticks([])
>>> ax1.spines['bottom'].set_color('white')
>>> ax1.spines['top'].set_color('white')
>>> ax1.spines['left'].set_color('white')
>>> ax1.spines['right'].set_color('white')

>>> ax1.legend(
...     ['Mesh', '$\mathbf{\phi}$ (centers)', '$\mathbf{u}$ (faces)'],
...     loc='upper right', fontsize=14
... )


text:
# Tag Info

17

Generally speaking, mathematical physics questions are on-topic and more than welcome. Note that advanced questions naturally tend to have fewer views, upvotes, and potential answerers. But don't get discouraged by that. You'll just have to be patient. If Phys.SE currently don't have the expertise to answer them, we might get it in the future! Also it is a ...

14

While I do not speak on behalf of the other close voters, the facts are clear: If the velocity is constant, the acceleration is zero. If a quantity is constant, it does not depend on time (in this context). Zero is clearly a constant, and a constant can be zero. Therefore, constant acceleration comprises two cases: Uniformly-changing velocity, which means ...

12

With the remodeling of the review queues, users now have the option to explicitly mark their edits to closed questions as significant enough to warrant reopen reviews. So, as long as the editors refrain from marking their cosmetic edits as significant, there is no longer any reason to refrain from such edits.

11

In case you are talking about this question, it has been reopened by regular reopen review around the same time you posted this question. This site does not work in real-time and after I closed your question I had no further interaction with it, i.e. I neither actively refused to reopen it nor did I reopen it. Note that the first version I unilaterally ...

11

I for one think the question is unclear or too broad. “Inhabitable” is not really defined, and actually hard to define. One can imagine the surface of the Earth is cold but that doesn’t make it uninhabitable: some animals may thrive. The physics side is also vague. You do not specify anything about the black hole or its orbital parameter, so there is ...

10

Let's kick things off with the most important one - the homework close reason, which currently reads Homework-like questions and check-my-work questions are considered off-topic here, particularly when asking about specific computations instead of underlying physics concepts. Homework questions can be on-topic when they are useful to a broader audience. If ...

9

The SE will block your account from posting new questions if your existing questions are judged to be low quality, but the exact criteria they use have never been disclosed - presumably to stop people from gaming the system. I have read your existing questions, and you are obviously enthusiastic about physics. Can I suggest you ask in the Physics Stack ...

9

"Closed seconds after I wrote them" is a mischaracterization - your question was closed 6 minutes after it was posted. (While this might seem like a petty complaint, I want to make it clear I read the question before closing it.) You have presented no citations or support for your claims that "dual numbers represent Newtonian spacetime" ...

7

Yes, under certain technical circumstances it can happen that an answer is posted up to 4 hours after closure. See this meta.SE answer for more information.

6

Rapidly closing a valid and interesting question only because the questioner uses non-standard terminology seems to me to be very hostile. One of the closure reasons is "needs more detail and clarity". If the OP is using terminology in such a way that makes the question unclear, then that's a valid closure reason. Furthermore, I don't see any ...

6

Welcome to Physics! In its current form (v4) it’s unclear to me what your post is asking about. One symptom: it does not appear to contain a question that ends in a question mark. Your self-answer suggests that you have solved the problem you were having by locating a factor of two. That’s typical of check-my-work questions, which our community has decided ...

6

The possibility that someone else might ask a question with the word "tachyon" in it that's on topic doesn't seem germane to the issue of whether your question is, on its own merits, on topic. (That's before considering that some open questions probably should have been closed too but slipped through for some reason or another.) The comment by ...

5

Your question is asking about a name, and the names of ideas in physics are often historical accidents that have little significance. So in general it's a bit pointless asking why a particular name was chosen. In this case the name principle of least action is actually appropriate - well almost. It's because for any trajectory we can calculate the action and ...

5

I get the impression you are confusing the mathematical-physics tag on Physics SE with the one on Mathematics SE. They are not the same. Physics SE : Mathematical physics is the application of mathematics to problems in physics and mathematical methods suitable for such applications, e.g., partial differential equations (PDEs), functional analysis, ...

5

Yes, closing does seem to discourage a lot of askers, especially those who don't understand how Stack Exchange sites work, but it's supposed to encourage them to improve the question, when possible. We close questions that are off-topic, for obvious reasons. We close duplicates because they've already been answered. The point of early closure is not to ...

5

Your meta question is actually not a question. It is a comment that you cannot understand why a main site question of yours was closed. As a result: I dunno how your non-question in meta can be answered, Even if we understand your meta post as a kind of question, it is not possible to answer it in any way except by formulating a speculative opinion on the ...

4

I can actually see several reasons why this question doesn't fit, including the engineering reason that it is closed for. I'll go through my opinion on them. For off topic as engineering, I don't think that blurb you quoted gives the whole picture on why something could be seen as an "engineering" as opposed to physics question. The problem is ...

4

Briefly speaking, asking about faster-than-light tachyonic particles is non-mainstream/primarily opinion-based and usually$^1$ off-topic (unless it's about experimental searches), while asking about tachyonic fields can be on-topic. $^1$ No rule without exceptions, especially if the question is well-researched/well-documented, or illustrates in a ...

3

I think that if the question is about mathematics per se, it is out of scope, even if the mathematics is commonly used in physics. It would be in scope if the question was about how to apply or interpret the mathematics in relation to physics. For example, if someone were to ask how to solve a quadratic equation, that would stand on its own mathematically ...

3

Personally, I do not like tachyonic physics as a subject, but I'm afraid I have to disagree with Brick's and Qmechanic's point of view about the non-mainstreamness of the subject and this specific question for the following reasons: the subject "tachyons" is still nowadays present in serious papers published in reputable journals. A fast search on ...

3

I fully agree that this question should not have been closed and should be reopened. A teacher has asked an ambiguous and opinion-based multiple-choice question to a student followed by a wrong or poorly worded reasoning. The understandably confused student then asks on Physics SE what is right. From a physics point-of-view the answer to this is very clear. ...

2

As written, the question is not an ideal fit for the site. The complete answer to the question in the title of the OP is simply “Yes” but the OP is suggests that “ Speculations on solution would really be appreciated.” It is clear from the answers that these answers are primarily opinion-based or at least depend on the interpretation of the statement by the ...

2

Sounds to me like SE doesn't think this is a problem. I suggest you take a look at their response to this issue. Vote your conscience and leave a comment there.

2

I think you're somehow thinking closing questions means they are "undesirable". This is not true of this one: it just needs to be clarified. I did not participate in the VTC but reading it I have to admit that I would have voted to close this question because it lacks clarity. The onus is then on the OP to edit the question and clarify why a ...

2

This might be too late (I didn't notice this Meta post until after several half-lives of viewing activity had already elapsed), but I'll clarify the intent of my original comment. Here's a copy of the comment: Tachyons are not consistent with mainstream physics, so the only way to answer this "what if" question is to use a non-mainstream theory. ...

1

Perhaps it should be noted that the decision of this question is intimately linked with the question: What should Physics do with String theory heritage? It is an open secret among theoretical physicists that much of the String theory community has migrated during the past fifteen years to other questions than quantum gravity (the reasons are the ...

Only top voted, non community-wiki answers of a minimum length are eligible


text:
Chapter 3. The First Law of Thermodynamics

# 3.5 Heat Capacities of an Ideal Gas

### Learning Objectives

By the end of this section, you will be able to:

• Define heat capacity of an ideal gas for a specific process
• Calculate the specific heat of an ideal gas for either an isobaric or isochoric process
• Explain the difference between the heat capacities of an ideal gas and a real gas
• Estimate the change in specific heat of a gas over temperature ranges

We learned about specific heat and molar heat capacity in Temperature and Heat; however, we have not considered a process in which heat is added. We do that in this section. First, we examine a process where the system has a constant volume, then contrast it with a system at constant pressure and show how their specific heats are related.

Let’s start with looking at Figure 3.12, which shows two vessels A and B, each containing 1 mol of the same type of ideal gas at a temperature T and a volume V. The only difference between the two vessels is that the piston at the top of A is fixed, whereas the one at the top of B is free to move against a constant external pressure p. We now consider what happens when the temperature of the gas in each vessel is slowly increased to $T+dT$ with the addition of heat.

Since the piston of vessel A is fixed, the volume of the enclosed gas does not change. Consequently, the gas does no work, and we have from the first law

$d{E}_{\text{int}}=dQ-dW=dQ.$

We represent the fact that the heat is exchanged at constant volume by writing

$dQ={C}_{V}dT,$

where ${C}_{V}$ is the molar heat capacity at constant volume of the gas. In addition, since $d{E}_{\text{int}}=dQ$ for this particular process,

$d{E}_{\text{int}}={C}_{V}dT.$

We obtained this equation assuming the volume of the gas was fixed. However, internal energy is a state function that depends on only the temperature of an ideal gas. Therefore, $d{E}_{\text{int}}={C}_{V}dT$ gives the change in internal energy of an ideal gas for any process involving a temperature change dT.

When the gas in vessel B is heated, it expands against the movable piston and does work $dW=pdV.$ In this case, the heat is added at constant pressure, and we write

$dQ={C}_{p}dT,$

where ${C}_{p}$ is the molar heat capacity at constant pressure of the gas. Furthermore, since the ideal gas expands against a constant pressure,

$d\left(pV\right)=d\left(RT\right)$

becomes

$pdV=RdT.$

Finally, inserting the expressions for dQ and pdV into the first law, we obtain

$d{E}_{\text{int}}=dQ-pdV=\left({C}_{p}-R\right)dT.$

We have found $d{E}_{\text{int}}$ for both an isochoric and an isobaric process. Because the internal energy of an ideal gas depends only on the temperature, $d{E}_{\text{int}}$ must be the same for both processes. Thus,

${C}_{V}dT=\left({C}_{p}-R\right)dT,$

and

${C}_{p}={C}_{V}+R.$

The derivation of Equation 3.10 was based only on the ideal gas law. Consequently, this relationship is approximately valid for all dilute gases, whether monatomic like He, diatomic like ${\text{O}}_{2},$ or polyatomic like ${\text{CO}}_{2}{\phantom{\rule{0.2em}{0ex}}\text{or NH}}_{3}\text{.}$

In the preceding chapter, we found the molar heat capacity of an ideal gas under constant volume to be

${C}_{V}=\frac{d}{2}R,$

where d is the number of degrees of freedom of a molecule in the system. Table 3.3 shows the molar heat capacities of some dilute ideal gases at room temperature. The heat capacities of real gases are somewhat higher than those predicted by the expressions of ${C}_{V}$ and ${C}_{p}$ given in Equation 3.10. This indicates that vibrational motion in polyatomic molecules is significant, even at room temperature. Nevertheless, the difference in the molar heat capacities, ${C}_{p}-{C}_{V},$ is very close to R, even for the polyatomic gases.

Table 3.3
Molar Heat Capacities of Dilute Ideal Gases at Room Temperature
Type of Molecule Gas ${C}_{p}$
(J/mol K)
${C}_{V}$
(J/mol K)
${C}_{p}-{C}_{V}$
(J/mol K)
Monatomic Ideal $\frac{5}{2}R=20.79$ $\frac{3}{2}R=12.47$ $R=8.31$
Diatomic Ideal $\frac{7}{2}R=29.10$ $\frac{5}{2}R=20.79$ $R=8.31$
Polyatomic Ideal $4R=33.26$ $3R=24.94$ $R=8.31$

### Summary

• For an ideal gas, the molar capacity at constant pressure ${C}_{p}$ is given by ${C}_{p}={C}_{V}+R=dR\text{/}2+R$, where d is the number of degrees of freedom of each molecule/entity in the system.
• A real gas has a specific heat close to but a little bit higher than that of the corresponding ideal gas with ${C}_{p}\simeq {C}_{V}+R.$

### Conceptual Questions

How can an object transfer heat if the object does not possess a discrete quantity of heat?

Most materials expand when heated. One notable exception is water between $0\phantom{\rule{0.2em}{0ex}}\text{°}\text{C}$ and $4\phantom{\rule{0.2em}{0ex}}\text{°}\text{C,}$ which actually decreases in volume with the increase in temperature. Which is greater for water in this temperature region, ${C}_{p}$ or ${C}_{V}$ ?

Show Solution

Typically ${C}_{p}$ is greater than ${C}_{V}$ because when expansion occurs under constant pressure, it does work on the surroundings. Therefore, heat can go into internal energy and work. Under constant volume, all heat goes into internal energy. In this example, water contracts upon heating, so if we add heat at constant pressure, work is done on the water by surroundings and therefore, ${C}_{p}$ is less than ${C}_{V}$.

Why are there two specific heats for gases ${C}_{p}$ and ${C}_{V}$, yet only one given for solid?

### Problems

The temperature of an ideal monatomic gas rises by 8.0 K. What is the change in the internal energy of 1 mol of the gas at constant volume?

Show Solution

100 J

For a temperature increase of $10\phantom{\rule{0.2em}{0ex}}\text{°}\text{C}$ at constant volume, what is the heat absorbed by (a) 3.0 mol of a dilute monatomic gas; (b) 0.50 mol of a dilute diatomic gas; and (c) 15 mol of a dilute polyatomic gas?

If the gases of the preceding problem are initially at 300 K, what are their internal energies after they absorb the heat?

Show Solution

a. 370 J; b. 100 J; c. 500 J

Consider 0.40 mol of dilute carbon dioxide at a pressure of 0.50 atm and a volume of 50 L. What is the internal energy of the gas?

When 400 J of heat are slowly added to 10 mol of an ideal monatomic gas, its temperature rises by $10\phantom{\rule{0.2em}{0ex}}\text{°}\text{C}$. What is the work done on the gas?

Show Solution

850 J

One mole of a dilute diatomic gas occupying a volume of 10.00 L expands against a constant pressure of 2.000 atm when it is slowly heated. If the temperature of the gas rises by 10.00 K and 400.0 J of heat are added in the process, what is its final volume?

### Glossary

molar heat capacity at constant pressure
quantifies the ratio of the amount of heat added removed to the temperature while measuring at constant pressure
molar heat capacity at constant volume
quantifies the ratio of the amount of heat added removed to the temperature while measuring at constant volume


text:
An elliptic equation with no monotonicity condition on the nonlinearity
ESAIM: Control, Optimisation and Calculus of Variations, Volume 12 (2006) no. 4, pp. 786-794.

An elliptic PDE is studied which is a perturbation of an autonomous equation. The existence of a nontrivial solution is proven via variational methods. The domain of the equation is unbounded, which imposes a lack of compactness on the variational problem. In addition, a popular monotonicity condition on the nonlinearity is not assumed. In an earlier paper with this assumption, a solution was obtained using a simple application of topological (Brouwer) degree. Here, a more subtle degree theory argument must be used.

DOI: 10.1051/cocv:2006022
Classification: 35J20,  35J60
Keywords: mountain-pass theorem, variational methods, Nehari manifold, Brouwer degree, concentration-compactness
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title = {An elliptic equation with no monotonicity condition on the nonlinearity},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
pages = {786--794},
publisher = {EDP-Sciences},
volume = {12},
number = {4},
year = {2006},
doi = {10.1051/cocv:2006022},
zbl = {1123.35021},
mrnumber = {2266818},
language = {en},
url = {http://archive.numdam.org/articles/10.1051/cocv:2006022/}
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Spradlin, Gregory S. An elliptic equation with no monotonicity condition on the nonlinearity. ESAIM: Control, Optimisation and Calculus of Variations, Volume 12 (2006) no. 4, pp. 786-794. doi : 10.1051/cocv:2006022. http://archive.numdam.org/articles/10.1051/cocv:2006022/

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[3] P. Caldiroli, A New Proof of the Existence of Homoclinic Orbits for a Class of Autonomous Second Order Hamiltonian Systems in ${ℝ}^{N}$. Math. Nachr. 187 (1997) 19-27. | Zbl

[4] P. Caldiroli and P. Montecchiari, Homoclinic orbits for second order Hamiltonian systems with potential changing sign. Comm. Appl. Nonlinear Anal. 1 (1994) 97-129. | Zbl

[5] V. Coti Zelati, P. Montecchiari and M. Nolasco, Multibump solutions for a class of second order, almost periodic Hamiltonian systems. Nonlinear Ord. Differ. Equ. Appl. 4 (1997) 77-99. | Zbl

[6] V. Coti Zelati and P. Rabinowitz, Homoclinic Orbits for Second Order Hamiltonian Systems Possessing Superquadratic Potentials. J. Amer. Math. Soc. 4 (1991) 693-627. | Zbl

[7] K. Deimling, Nonlinear Functional Analysis. Springer-Verlag, New York (1985). | MR | Zbl

[8] M. Estaban and P.-L. Lions, Existence and non existence results for semilinear elliptic problems in unbounded domains. Proc. Roy. Soc. Edinburgh 93 (1982) 1-14. | Zbl

[9] B. Franchi, E. Lanconelli and J. Serrin, Existence and Uniqueness of Nonnegative Solutions of Quasilinear Equations in ${𝐑}^{N}$. Adv. Math. 118 (1996) 177-243. | Zbl

[10] L. Jeanjean and K. Tanaka, A Note on a Mountain Pass Characterization of Least Energy Solutions. Adv. Nonlinear Stud. 3 (2003) 445-455. | Zbl

[11] L. Jeanjean and K. Tanaka, A remark on least energy solutions in ${ℝ}^{N}$. Proc. Amer. Math. Soc. 131 (2003) 2399-2408. | Zbl

[12] P.L. Lions, The concentration-compactness principle in the calculus of variations. The locally compact case. Ann. Instit. Henri Poincaré 1 (1984) 102-145 and 223-283. | Numdam | Zbl

[13] J. Mawhin and M. Willem, Critical Point Theory and Hamiltonian Systems. Springer-Verlag, New York (1989). | MR | Zbl

[14] P. Rabinowitz, Homoclinic Orbits for a class of Hamiltonian Systems. Proc. Roy. Soc. Edinburgh Sect. A 114 (1990) 33-38. | Zbl

[15] P. Rabinowitz, Minimax Methods in Critical Point Theory with Applications to Differential Equations, C.B.M.S. Regional Conf. Series in Math., No. 65, Amer. Math. Soc., Providence (1986). | MR | Zbl

[16] P. Rabinowitz, Théorie du degrée topologique et applications à des problèmes aux limites nonlineaires, University of Paris 6 Lecture notes, with notes by H. Berestycki (1975).

[17] G. Spradlin, Existence of Solutions to a Hamiltonian System without Convexity Condition on the Nonlinearity. Electronic J. Differ. Equ. 2004 (2004) 1-13. | Zbl

[18] G. Spradlin, A Perturbation of a Periodic Hamiltonian System. Nonlinear Anal. Theory Methods Appl. 38 (1999) 1003-1022. | Zbl

[19] G. Spradlin, Interacting Near-Solutions of a Hamiltonian System. Calc. Var. PDE 22 (2005) 447-464. | Zbl

[20] E. Serra, M. Tarallo and S. Terracini, On the existence of homoclinic solutions to almost periodic second order systems. Ann. Instit. Henri Poincaré 13 (1996) 783-812. | Numdam | Zbl

[21] G. Whyburn, Topological Analysis. Princeton University Press (1964). | MR | Zbl

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# Three processes arrive at time zero with CPU bursts of 16, 20, and 10 milliseconds. If the scheduler has prior knowledge about the length of the CPU bursts, the minimum achievable average waiting for these three processes in a non-preemptive scheduler (rounded to the nearest integer) is ______ milliseconds.

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## Answer (Detailed Solution Below) 12

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## Detailed Solution

Key Points

The scheduler has prior knowledge about the length of the CPU bursts and it is non-preemptive.

Therefore shortest job first algorithms while giving minimum achievable average waiting

Gantt chart:

 P3 P1 P2

0            10           26            46

Process table:

 Process Id Arrival Time (AT in ms) Burst Time (BT in ms) Completion Time (CT) Turn Around Time (TAT) Waiting Time (WT) P1 0 16 26 26 10 P2 0 20 46 46 26 P3 0 10 10 10 0

Average waiting time = $$\frac{0~+10+26}{3} = 12\; ms$$

Average waiting time = 12 ms

Important Points:

TAT = CT – AT

WT = TAT – BT


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# High performance VLSI implementation for H.264 Inter/Intra prediction

Alle, Mythri and Biswas, J and Nandy, SK (2007) High performance VLSI implementation for H.264 Inter/Intra prediction. In: International Conference on Consumer Electronics:ICCE 2007, Digest of Technical Papers, 10-14 January 2007, Las Vegas, NV, pp. 1-2.

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## Abstract

We provide a hardware realization of motion compensation and reconstruction (MCR) module for H.264 baseline profile. We synthesize the MCR module using UMC library in $0.13\hspace{2mm}\mu$ CMOS technology. Our implementation occupies an area of 94756 gates and operates at a frequency of 250 MHz.

Item Type: Conference Paper Copyright 2007 IEEE. Personal use of this material is permitted. However, permission to reprint/republish this material for advertising or promotional purposes or for creating new collective works for resale or redistribution to servers or lists, or to reuse any copyrighted component of this work in other works must be obtained from the IEEE. CMOS integrated circuits;image reconstruction;motion;compensation;video coding;VLSI Division of Information Sciences > Supercomputer Education & Research Centre 14 Aug 2007 19 Sep 2010 04:38 http://eprints.iisc.ernet.in/id/eprint/11677

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# How do rollercoaster designers calculate the ideal curve?

I've been looking into rollercoaster track designs and wondered how they mathematically calculate the ideal curves to fit some specific layout without creating excessive Gs or other rider discomfort or for maximising speeds.

One thing I do know is they don't use perfect arcs, due to G forces especially on the loop-the-loop which is why they take on the tear-drop shape (do normal curves also there for not take perfect curves for the same reason?).

My current guess is they must be using some bezier type spline stuff, but there are many ways to design a bezier curve between two points. So what are they actually using to find optimal curves for rider safety / comfort and speed?

Are there special types of curves they use or do they shape the curves manually in 3D software then run simulations to see if its rideable?

• How were the first ones done before 3D? Jun 5, 2021 at 5:23
• There is a lot of ways to design a curve between two points. But you can eliminate the choices which violate the constraints like max g, max curvature, max speed, min speed, self intersecting etc. The remaining choices will be a much smaller set.
– AJN
Jun 5, 2021 at 7:18

Although, I've never had the opportunity to work in the design of a rollercoaster, I suspect the idea is that there is a general idea of the rollercoaster, and then every section is checked that it does not exceed the maximum acceleration on the participants.

The main equation that you need to consider is the equation for the centrifugal/centripetal acceleration at a specific point on the roller coaster.

$$a_c = \frac{v^2}{R}$$

where:

• $$a_c$$ is the acceleration of the body (this is usually set to a g value as a maximum e.g. 3g)
• $$v$$ is the velocity at point on the rollercoaster
• $$R$$ is the Radius at point on the rollercoaster

So at point P, in the following image:

• either for a given curve, fit the largest possible circle which is tangent to the trajectory of the rollercoaster, and then estimate the maximum velocity, using $$v_{crit}= v_{max} = \sqrt{aR}$$
$$R_{crit}= R_{min} = \frac{v^2}{a}$$


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# GATE Civil Engineering 2020 SET-2

 Question 1
The ordinary differential equation $\frac{d^2u}{dx^2}-2x^2u+\sin x=0$ is
 A linear and homogeneous B linear and nonhomogeneous C nonlinear and homogeneous D nonlinear and nonhomogeneous
Engineering Mathematics   Ordinary Differential Equation
Question 1 Explanation:
Its solution is of the type u=f(x), i.e., dependent variable is u.
Hence, given equation is Linear and Non-Homogeneous.
 Question 2
The value of $\lim_{x \to \infty } \frac{\sqrt{9x^2+2020}}{x+7}$ is
 A $\frac{7}{9}$ B 1 C 3 D Indeterminable
Engineering Mathematics   Calculus
Question 2 Explanation:
$\lim_{x \to \infty }\frac{3x \sqrt{1+\frac{2020}{x^2}}}{x\left ( 1+\frac{7}{x} \right )}=3$
 Question 3
The integral
$\int_{0}^{1}(5x^3+4x^2+3x+2)dx$

is estimated numerically using three alternative methods namely the rectangular, trapezoidal and Simpson's rules with a common step size. In this context, which one of the following statement is TRUE?
 A Simpsons rule as well as rectangular rule of estimation will give NON-zero error. B Simpson's rule, rectangular rule as well as trapezoidal rule of estimation will give NON-zero error. C Only the rectangular rule of estimation will given zero error. D Only Simpson's rule of estimation will give zero error.
Engineering Mathematics   Numerical Methods
Question 3 Explanation:
Because integral is a polynomial of 3rd degree so Simpson's rule will give error free answer.
 Question 4
The following partial differential equation is defined for $u:u(x,y)$

$\frac{\partial u}{\partial y}=\frac{\partial^2 u}{\partial x^2}; \; \; y\geq 0;\;x_1\leq x\leq x_2$

The set of auxiliary conditions necessary to solve the equation uniquely, is
 A three initial conditions B three boundary conditions C two initial conditions and one boundary condition D one initial condition and two boundary conditions
Engineering Mathematics   Ordinary Differential Equation
Question 4 Explanation:
Given: DE is $\frac{\partial u}{\partial y}=\frac{\partial^2 u}{\partial x^2}; \; y\geq 0;\; x_1 \leq x\leq x_2$
$\because$ y is given as $\geq 0$ so we take it as time. Hence, above equation is nothing but one-D heat equation which requires one initial condition and two boundary condition.
 Question 5
The ratio of the plastic moment capacity of a beam section to its yield moment capacity is termed as
 A aspect ratio B load factor C shape factor D slenderness ratio
Design of Steel Structures   Plastic Analysis
Question 5 Explanation:
Ratio of $\frac{M_p}{M_y}=$ Shape factor
 Question 6
The state of stress represented by Mohr's circle shown in the figure is
 A uniaxial tension B biaxial tension of equal magnitude C hydrostatic stress D pure shear
Solid Mechanics   Properties of Metals, Stress and Strain
Question 6 Explanation:
In pure shear condition, Mohr's circle has its center at origin.
 Question 7
A weightless cantilever beam of span L is loaded as shown in the figure. For the entire span of the beam, the material properties are identical and the cross-section is rectangular with constant width.

From the flexure-critical perspective, the most economical longitudinal profile of the beam to carry the given loads amongst the options given below, is
 A A B B C C D D
Solid Mechanics   Shear Force and Bending Moment
Question 7 Explanation:

$(-PL) + (PL) + (-M_A) = 0$
$M_A = 0$

For most economical,
Maximum cross-section is given where maximum bending moment occurs.

 Question 8
As per IS 456:2000, the pH value of water for concrete mix shall NOT be less than
 A 4.5 B 5 C 5.5 D 6
RCC Structures   Working Stress and Limit State Method
Question 8 Explanation:
1. Minimum pH value of water for concrete = 6.0
As per IS code provision no. 5.4.2, the pH value of water shall not less than 6.0.
 Question 9
The traffic starts discharging from an approach at an intersection with the signal turning green. The constant headway considered from the fourth or fifth headway position is referred to as