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# An Improved Model for the Prediction of Radiant Heat from Fireballs

By
William E. Martinsen and Jeffrey D. Marx

Presented At 1999 International Conference and Workshop on Modeling Consequences of Accidental Releases of Hazardous Materials San Francisco,California September 28 – October 1, 1999

## Abstract

This paper presents an integrated model for predicting the radiant heat effects of fireballs created by near-instantaneous releases of superheated flammable liquids. Films and videos of fireballs show that a fireball grows quickly in size, rises into the air after some time delay, then dissipates when the available fuel is consumed. Most of the published fireball radiation models ignore this behavior and simply assume the diameter, location, and surface emissive power of the fireball are constant over the full duration of the event. In contrast, the model described in this paper provides a more realistic representation of the true behavior of fireballs by employing equations that account for fireball growth, lift-off, and changing radiative characteristics. Thermodynamic changes that occur during the release of superheated liquids are incorporated into the model, making it suitable for predicting the radiant heat effects of fireballs formed as a result of cold catastrophic failures of pressure vessels, as well as fireballs created by BLEVE incidents. Predictions of the time-varying radiant heat flux incident upon targets located outside the fireball are shown to agree well with results from moderate-scale experiments.

## Text

### Introduction

The sudden release of superheated flammable liquid from a storage tank or process vessel is the beginning of a complex event that often ends in the formation of a short-lived fireball. The event starts with a major failure of the container. Because the pressure in the container is greater than atmospheric pressure, much of the liquid is quickly expelled into the atmosphere. In response to this rapid drop in pressure, a portion of the liquid flashes to vapor nearly instantaneously. This vapor expands rapidly, shattering some of the remaining liquid into small drops, thereby creating a turbulent aerosol cloud consisting of vapor, liquid drops, and air. The aerosol cloud quickly increases in size, entraining more air as it grows.

Ignition of this aerosol cloud results in a fireball that exists until the vapor and liquid fuel within the cloud are consumed. The fireball can emit a large amount of radiant energy during its brief life, and is capable of causing injuries and damage over an area several times greater than the size of the fireball. Therefore, when conducting a hazards or risk analysis of process vessels or storage tanks that contain superheated flammable liquids, it is important to be able to accurately model the radiant heat effects of fireballs. Most fireball radiation models ignore the dynamic nature of fireballs and simply treat them as static events. This simplification often causes such models to overpredict the extent of potentially damaging or injurious radiant heat hazard zones. The fireball model introduced in this paper provides better predictions of hazard zones by treating fireballs as dynamic events.

### The Basics of Fire Radiation

All fires emit thermal radiation. When modeling thermal radiation from fires involving flammable liquids and gases, the flame is typically represented as a simple geometric shape (cone, tilted cylinder, etc.) that emits radiation from its surface. The following equation is then used to calculate the intensity of the thermal radiation at any specific “target” location outside the flame.

The seriousness of injuries and extent of damage that can be caused by thermal radiation from a fire depend on the intensity of the incident radiation, , and the duration of exposure to that level of heat flux. Since fireballs exist for only a few seconds, the duration of exposure is commonly set equal to the duration of the fireball.

Methods for predicting the thermal radiation consequences of fireballs typically include elements or models to define or calculate the following factors.

• Surface emissive power of the fireball
• Geometric view factor
• Atmospheric transmittance
• Fireball duration

Input parameters for these models will either be calculated by submodels or be based on assumptions. For example, the following parameters are required for input to most geometric view factor models.

• Fireball shape (assumed)
• Fireball size (calculated)
• Mass of fuel involved in the fireball (required by fireball size equation)
• Location of the fireball relative to the target (assumed)
• Orientation of the target relative to the fireball (assumed)

### Current Models

A fireball is a dynamic phenomenon. Typically, a fireball begins its life as a small ball of fire located near grade. Over the next few seconds, the fireball grows in size, soon reaching its maximum diameter. The fireball becomes buoyant and lifts off the ground as the heat of the fire vaporizes liquid droplets and increases the bulk temperature of the remaining mixture of vapor, air, and reaction products. As it rises, the limited fuel supply is consumed and the fireball ceases to exist.

In spite of this dynamic behavior, most fireball radiation models currently in use treat the fireball as a static event in which the fireball has a constant size, emits radiant heat at a constant rate, and is located at a fixed position relative to grade. This type of model generally consists of eight primary elements or submodels.

#### Fireball Shape

Nearly all fireball models treat the fireball as a sphere. Videos and other observations of accidental and experimental fireballs indicate that most fireballs can be reasonably described as being spherical over most of their duration.

#### Fireball Diameter

Static models typically assume the fireball reaches its maximum diameter instantaneously and maintains that size for the full duration of the fireball. Nearly all of the static models calculate the fireball diameter by using an equation of the following form, which relates maximum fireball diameter to the mass of fuel involved in the fireball.

#### Mass of Fuel Involved

The mass of fuel involved in the fireball is an integral part of the equations for fireball diameter and duration. This mass can be the same as the mass contained within the vessel prior to its rupture, but it can also be less than that amount—particularly if the amount of superheat is small.

Various authors have proposed different relationships between the mass released and the mass involved in the fireball. Crocker and Napier [1988] proposed the following guidelines.

Mass involved equals mass released if adiabatic flash exceeds 50%. Otherwise, mass involved equals twice the adiabatic flash.

Based on the work of Hasegawa and Sato [1977], Roberts [1981/82] proposed the following rules.

Mass involved equals mass released if adiabatic flash exceeds 35%; and zero if adiabatic flash equals zero. Linear interpolation is suggested between these two limits.

The CCPS book, Guidelines for Evaluating the Characteristics of Vapor Cloud Explosions, Flash Fires, and BLEVEs [CCPS, 1994], suggests the following variation of Roberts’ rules.

Mass involved equals mass released if adiabatic flash exceeds 1/3. Otherwise, mass involved equals three times the adiabatic flash.

This variation provides the most conservative results since the mass of fuel it puts in the fireball always equals or exceeds the amount calculated by either of the other two methods.

#### Surface Emissive Power

Equations for calculating the surface emissive power (SEP) typically relate the SEP to the pressure in the vessel at the time it bursts. Two methods are in common use. Roberts [1981/82] developed the following equation, which predicts the fraction of the total available heat energy that is radiated by the fireball.

At first glance, this equation appears to indicate that the surface emissive power, , is proportional to the mass of fuel involved in the fireball. However, if the fireball duration and diameter are both proportional to the cube root of the mass of fuel involved, the SEP predicted by this method is independent of the mass of fuel. Conversely, if either the diameter or duration is not proportional to , then the SEP prediction will be affected by changes in the mass.

The other common method uses a single equation developed by Moorhouse and Pritchard [1982]. As shown below, it provides a direct relationship between surface emissive power and the pressure at which the vessel burst—for pressures no greater than 2 MPa (~300 psi).

#### Location of Fireball

Most of the static fireball models assume the fireball is tangent to grade, but some use a higher location. For example, the TNO model [TNO, 1983] places the center of the fireball at 1.5 radii above grade.

#### View Factor

The view factor ( in Equation 1) is based on the geometric relationship between the spherical fireball and a flat target located outside the fireball. In some static models, the vector that is normal to the surface of the target points directly at the center of the spherical fireball, as illustrated in Figure 1a. (This orientation results in the maximum possible values for the view factor.) The following equation is used to calculate view factors for this orientation.

In other static models, the target is assumed to have a vertical orientation. Thus, the vector that is normal to the surface of the target will be horizontal. If the fireball is tangent to grade and the target is located at grade, the normal vector will be tangent to the fireball, as illustrated in Figure 1b. The following equation is used to calculate view factors for this orientation.

For otherwise identical situations, this difference in orientation of the target results in a difference in the predicted values of the view factor . This can lead to a considerable difference in predicted values of incident heat flux, , when the target is close to the fireball.

#### Atmospheric Transmittance

Water vapor and carbon dioxide in the atmosphere will absorb some thermal radiation as it passes through the air between the fireball and the target. Several methods have been put forward for calculating this effect. They range in complexity from the use of a constant value (e.g., 0.75) for transmissivity, to equations in which transmissivity is a function of absolute humidity, carbon dioxide concentration in air, and path length (e.g., [Wayne, 1991]).

### An Example of a Static Model

In 1989, The Institution of Chemical Engineers published its Thermal Radiation Monograph [IChemE, 1989], in which it recommended the use of a static model consisting of the following elements.

Predictions of incident radiant heat flux from this type of model will resemble a square wave, as illustrated in Figure 2, which is based on the following assumptions.

• Fuel = butane
• = 2,000 kg
• = 1.51 MPa
• The absolute humidity is zero (i.e., =1.0).
• The horizontal distance between the target and the ruptured vessel is 50 m.
• The target is located at grade and the vector normal to the surface of the target points directly at the center of the fireball.

In contrast, Figure 2 also presents the time-varying incident radiant heat flux data from an actual fireball experiment conducted by British Gas [Johnson, Pritchard, and Wickens, 1990]. Like the scenario modeled above, this experiment released 2,000 kg of butane at a pressure of 1.51 MPa, and the horizontal distance between the target (a heat flux radiometer) and the ruptured vessel was 50 m. The radiometer was located 1.1 m above grade, and the flat sensing element was tilted 30° upward from vertical. Thus, the vector normal to the sensing element was inclined 30° upward from horizontal and intersected the vertical line that passed through the center of the test vessel.

### The Dynamic Model

Dynamic fireball models attempt to model the time-varying behavior of fireballs in order to more accurately predict the thermal radiation consequences of actual fireballs. Like static models, dynamic models are based on certain assumptions (e.g., a sphere is a reasonably accurate representation of a fireball) and several empirical formulas. Unlike static models, a good dynamic model should account for time-varying phenomena that have been observed during planned and unplanned fireball incidents.

The dynamic fireball model developed by the authors is described below.

#### Fireball Shape

The fireball is assumed to be spherical in shape. Although no fireball will be perfectly spherical, the appropriateness of the assumption of a spherical shape is borne out by observations of still photographs and videos of actual fireballs.

#### Mass of Fuel Involved

Equating the mass of fuel involved in the fireball to the mass of fuel released is overly conservative for any fireball created by a release of liquid that is only slightly superheated. Instead, the guidelines proposed by the CCPS [1994] are used.

Mass involved equals mass released if adiabatic flash exceeds 1/3. Otherwise, the mass

involved equals three times the adiabatic flash (calculated at the vessel burst pressure).

These guidelines should provide reasonably conservative results, without being unduly pessimistic.

#### Fireball Duration

TNO proposed the following relationship between fireball duration and the mass of fuel involved in the fireball [TNO, 1983].

The dynamic model uses the following relationship, which is a slight variation of the TNO equation.

Figure 3 compares these two equations and Equations 6a and 6b from Clay, et al. [1988]. Durations predicted by Equation 12 are nearly identical to those of TNO until the mass exceeds 10,000 kg, and are slightly shorter than predicted by TNO when the mass exceeds 10,000 kg. Compared to the duration predictions of Equations 6a and 6b, durations predicted by Equation 12 are slightly longer for masses less than 5,000 kg or more than 300,000 kg.

The surface emissive power section of the static model presented a brief discussion of the relationship between mass of fuel involved in the fireball and the SEP. According to that discussion, if Equations 7 and 8 are used to calculate the SEP, and the fireball duration and diameter are both proportional to the cube root of the mass of fuel involved, the predicted SEP is independent of the mass of fuel. Thus, if Equation 3 is used to calculate the maximum diameter and Equation 6a is used for duration, the SEP is independent of the mass of fuel. However, if Equation 6b is used to calculate the duration of fireballs involving more than 37,000 kg of fuel, the SEP suddenly becomes dependent on the mass of fuel. This change in behavior of the SEP when the mass of fuel reaches 37,000 kg is highly unlikely. Therefore, we have chosen to use Equation 12 for calculating fireball duration. (Additional discussions related to our choice of Equation 12 are presented in the following sections on fireball diameter and surface emissive power.)

#### Fireball Diameter and Location

During the early portion of a fireball’s existence, the dynamic model treats the fireball as a sphere that increases in diameter with time, remaining tangent to grade as it grows. At the end of the growth phase, the fireball has reached its maximum diameter (which is predicted by Equation 3) and begins to rise into the air.

The fireball is assumed to achieve its maximum diameter at the end of the first third of its duration. This is also the time at which lift-off is assumed to occur (i.e., )

These assumptions are based on:

• experimental data from Hasegawa and Sato [1978] and Maillette and Birk [1995], which indicate that peak radiation output occurs at the end of the first third of the fireball’s duration;
• the work of Roberts [1981/82], who noted that peak radiation output occurs when the fireball has grown to its maximum diameter; and
• experimental data from Hasegawa and Sato [1977] and Hardee and Lee [1973], which show the fireball begins to rise into the air once it has reached its maximum diameter.

Hardee and Lee [1973] conducted experiments in which they ignited premixed quantities of gaseous propane and air. According to those experiments, the diameter of the fireball was proportional to the cube root of time, until the diameter reached its maximum. Moorhouse and Pritchard [1982] reviewed other experiments and reached the same conclusion. To check this hypothesis, we conducted a frame-by-frame analysis of three BLEVE fireballs from the NFPA video, “BLEVE Update.” The three accidents involved propane tanks ranging in size from a portable cylinder to a railcar. The results of this analysis are presented in Figure 4. In each of these incidents, the growth rate of the fireball was roughly proportional to the cube root of time. In addition, the time at which the fireball reached its maximum size and the time at which it lifted off were approximately equal.

Hardee, Lee, and Benedick [1978] suggested an equation of the following form for calculating the diameter of a fireball during its growth period.

Shield [1994] observed that the fireball ceased to exist once the center of the fireball rose to a height of three times the maximum radius. He also observed that the velocity of a rising fireball was approximately constant [Shield, 1993]. Thus, the center of the fireball moves upward at a constant rate from its pre-lift-off position (one maximum radius above grade) to three times that elevation in the last two-thirds of the fireball’s existence. With this correlation established, the location of the fireball at any given time can be determined.

#### Surface Emissive Power

The SEP is based on Equation 7 [Roberts, 1981/82], which predicts the fraction of the total available heat

energy that is emitted as radiation.

This equation increases the surface emissive power as the mass of fuel involved in the fireball increases. This trend is supported by experimental data, such as that compiled in Table 6.3 of the CCPS [1994] reference. However, fire research indicates that the SEP will not exceed some upper limit, regardless of the mass or pressure involved. Limits ranging from 300 kW/m2 [Moorhouse and Pritchard, 1982] to 450 kW/m2 [Roberts, 1981/82] have been proposed. Based on the British Gas experiments [Johnson, Pritchard, and Wickens, 1990], we believe 400 kW/m2 is a realistic upper limit for SEP. Hence, the dynamic model uses the SEP calculated by Equation 16, or 400 kW/m2, whichever is smaller.

The resulting SEP is applied to the fireball during its growth period. Experimental data from Hasegawa and Sato [1977] and Maillette and Birk [1995] show that SEP decreases once the fireball has reached its maximum diameter. The dynamic fireball model accounts for this behavior by linearly decreasing the SEP from its maximum value to zero over the last two-thirds of the fireball’s duration.

#### View Factor

Any of the view factor equations for a sphere radiating to a point outside the sphere can be used with this model. The preferable equation would vary with the situation to be modeled and the desires of the modeler. For general purpose use, the dynamic model places the target at grade, with the vector normal to the surface of the target passing through the center of the fireball. Equation 10 is then used to calculate the view factor. As the fireball rises, the target is continually rotated such that the vector normal to the surface of the target continues to pass through the center of the fireball at all times. This is a conservative assumption since it assures that, at any given time, the calculated view factor will be the maximum possible for a point target located a given distance from the fireball.

#### Atmospheric Transmittance

The equations from Wayne [1991] are used to calculate atmospheric transmittance. They account for the absorption of thermal radiation by water vapor and carbon dioxide in the atmosphere, and predict an increase in absorption as absolute humidity and path length increase.

The model sets the path length equal to the distance between the target and the nearest point of the fireball surface. (This is a conservative assumption since thermal radiation from all other points on the surface of the fireball would need to travel a greater path length.) Because the location of the fireball is constantly changing, the path length between the target and the sphere also changes with time.

### Model Validation

Several of the equations used in the dynamic fireball model are empirical relationships based on data from small – to medium – scale experiments, which ensures reasonably good agreement between model predictions and experimental data for variables such as maximum fireball diameter. Comparisons of experimental data and model predictions of incident heat flux as a function of time are more meaningful and of greater interest. However, very few reports on small- or medium-scale fireball experiments contain the level of detail required to make such comparisons, and no such data are available for large-scale (e.g., railcar) experiments.

One of the most complete sources of test data for medium-scale fireball tests is a report by Johnson, Pritchard, and Wickens [1990]. It contains data from a series of five well-instrumented fireball experiments in which superheated liquids were released nearly instantaneously from ruptured pressure vessels. Four of the tests involved commercial grade butane (>90% n-butane), in quantities of 1,000 and 2,000 kg; the other involved 2,000 kg of commercial grade propane. Electric immersion heaters were used to heat the lading in each test, simulating the effects an external fire would have on the temperature and pressure within the vessel. A linearshaped charge was used to initiate catastrophic failure of the vessel.

Each test was filmed with regular- and high-speed film cameras, and videotaped. Several wide-angle radiometers were used to measure incident heat fluxes at distances of 50 to 250 m from the test vessel, and a single narrow-angle radiometer was used to measure the fireball SEP.

The report on these tests [Johnson, Pritchard, and Wickens, 1990] includes a plot of radiant heat flux versus time for each radiometer for each test. Figure 5 presents radiant heat flux data from one of the radiometers used in Test No. 1, along with predictions from the IChemE static model and from our dynamic model. (For the purpose of this comparison, the target in the static model was located at grade, at the same distance as the radiometer, and tilted such that the vector normal to the surface of the target passed through the center of the fireball. The target in the dynamic model was located at the same distance and elevation as the radiometer, and inclined at the same angle. For both models, the equations from Wayne [1991] were used to calculate the atmospheric transmittance.) The flux versus time curve predicted by the dynamic model follows the general shape of the experimental curve, with only a small overprediction of the maximum incident heat flux and a slight offset in time. In contrast, the prediction from the static model bears little relationship to the test data.

The thermal damage or injury potential of a fireball is related to the incident heat flux and the duration of the fireball. One way of expressing the combination of intensity and time is to use dosage (i.e., absorbed energy), which is equivalent to the area under the radiant heat flux versus time curve. The report by Johnson, Pritchard, and Wickens [1990] reports a dosage for each wide-angle radiometer for each test. Figure 6 is a plot of the dosage data from Tests 2-5 versus the dosages predicted by the IChemE static model. (For the purpose of this comparison, the targets were located at grade, at the same distances as the radiometers, and were tilted such that the vector normal to the surface of each target passed through the center of the fireball. Atmospheric transmittance was calculated using the method proposed by Wayne [1991].) As shown in Figure 6, the static model overpredicts dosages, with the amount of overprediction being greatest at high dosages, i.e., at points closest to the fireball. (Although there are thirty-one data points, only twenty-five points are included in Figure 6. Six of the thirty-one predicted dosages exceeded 350 kJ/m2, which put them outside the boundaries of Figure 6.)

Figure 7 is a plot of the same dosage data versus predictions from the dynamic fireball model (with the target being moved and tilted such that it replicated the distances, elevations, and angles of inclination of the radiometers in the tests). A comparison of Figures 6 and 7 illustrates the degree of improvement provided by the dynamic fireball model.

Thus, in comparison to static fireball models, the dynamic model provides more realistic estimates of the extent of potentially damaging or injurious radiant heat hazard zones produced by fireballs, which makes it preferable to static models when conducting a hazards or risk analysis.

### References

• CCPS (1994), Guidelines for Evaluating the Characteristics of Vapor Cloud Explosions, Flash Fires, and BLEVEs. Center for Chemical Process Safety of the American Institute of Chemical Engineers, 345 East 47 Street, New York, New York 10017, 1994.
• Clay, G. A., R. D. Fitzpatrick, N. W. Hurst, D. A. Carter, and P. J. Crossthwaite (1988), “Risk Assessment for Installations where Liquefied Petroleum Gas (LPG) is Stored in Bulk Vessels Above Ground.” Journal of Hazardous Materials, Volume 20, 1988: pp. 357-374.
• Crocker, W. P., and D. H. Napier (1988), “Assessment of Mathematical Models for Fire and Explosion Hazards of Liquefied Petroleum Gases.” Journal of Hazardous Materials, Volume 20, 1988: pp. 109-135.
• Hardee, H. C., and D. O. Lee (1973), “Thermal Hazard from Propane Fireballs.” Transportation Planning and Technology, Vol. 2, 1973: pp. 121-128.
• Combustion Science and Technology, Volume 7, 1978: pp. 189-197.
• Hasegawa, K., and K. Sato (1977), “Study on the Fireball Following Steam Explosion of n-Pentane.” Proceedings of the Second International Symposium on Loss Prevention and Safety Promotion in the Process Industries, Heidelberg, Germany, September, 1977: pp. 297-304.
• Hasegawa, K., and K. Sato (1978), “Experimental Investigation of the Unconfined Vapor Cloud Explosions and Hydrocarbons.” Tech. Mem. Fire Res. Inst., 12, Tokyo, 1978.
• IChemE (1989), Thermal Radiation Monograph; Calculation of the Intensity of Thermal Radiation from Large Fires. Institution of Chemical Engineers, 1989.
• Johnson, D. M., M. J. Pritchard, and M. J. Wickens (1990), Large Scale Catastrophic Releases of Flammable Liquids. Commission of the European Communities, Report EV4T.0014.UK(H), 1990.
• Maillette, J., and A. M. Birk (1995), “Effects of Tank Failure Mode and Lading Properties on Propane Fireball Geometry and Fire Hazard.” AIChE International Conference and Workshop on Modeling and Mitigating the Consequences of Hazardous Materials, September, 1995: pp. 721-735.
• Moorhouse, J., and M. J. Pritchard (1982), “Thermal Radiation Hazards from Large Pool Fires and Fireballs—A Literature Review.” Institution of Chemical Engineers Symposium Series, No. 71, 1982: pp. 397-428.
• Roberts, A. F. (1981/82), “Thermal Radiation Hazards from Releases of LPG from Pressurized Storage.” Fire Safety Journal, Volume 4, 1981/82: pp. 197-212.
• Shield, S. R. (1993), “A Model to Predict Radiant Heat and Blast Hazards from LPG BLEVES.” AIChE 29th National Heat Transfer Conference, August, 1993: pp. 139-149.
• Shield, S. R. (1994), Consequence Modeling for LPG Distribution in Hong Kong. Thornton Research Centre, United Kingdom, Research Note TNRN.95.7001, December 13, 1994.
• TNO (1983), LPG Integraal. Report No. 1112, May, 1983.
• Wayne, F. D. (1991), “An Economical Formula for Calculating Atmospheric Infrared Transmissivities.” Journal of Loss Prevention in the Process Industries, Vol. 4, January, 1991: pp. 86-92.


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# glTexEnvi using GL_REPLACE?

This topic is 6166 days old which is more than the 365 day threshold we allow for new replies. Please post a new topic.

## Recommended Posts

From the OpenGL Blue Book, the glTexEnvi() function definition is the following: void glTexEnvi(GLenum target, GLenum pname, GLint param) PARAMETERS . . . param - Specifies a single symbolic constant, one of GL_MODULATE, GL_DECAL, or GL_BLEND. but in the book "OpenGL Game Programming, I see glTexEnvi() used with GL_REPLACE in the "param" parameter lots of times. The first time I noticed it, I thought "that''s probably an error, if I check the source it should be corrected" and so I did... but it was just the same as in the book. And when I tried to compile it, I expected an error but it runs just fine! Can someone explain me why and what does that parameter do? Thanks Raúl

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GL_REPLACE basically tells the renderer to skip the current color and just use the texel colors--if you use GL_MODULATE then if you make a call to glColorx() your textures will be filtered to that color too.

i.e. if you call glColor3f(0.0, 0.0, 1.0) and then draw a texture triangle, the texture will only contain it''s blue elements. The same triangle drawn after you engage GL_REPLACE will appear in it''s "true" colors regardless of the current color.

At least that''s my understanding.

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So for every fragment of a polygon needs a color.
The polygon starts with a colour C.
The texture has a colour T.
And F is what the color ends up as.

GL_REPLACE:
The polygon ends up the colour T.
F = T

GL_MODULATE:
The polygon''s original colour and the texture colour are multiplied together.
F = CT

GL_DECAL:
This is just like GL_REPLACE, unless you specify an alpha for your texture. Then part of the original polygon colour is made to shine through the texture. Kind of like a bad wallpaper job where you can see the paint behind it.

GL_BLEND:
This is a bad wallpaper job all the time.

Hope this explains it. This is kind of a simplified explanation, but check this link for the exact mathematical equations:
http://www.pik-net.pl/marekm/glspec1.1/node88.html#4591

And you may be look in version 1.1 of the bluebook and version 1.2 of the redbook.

Somebody correct me if I''m wrong, but GL_REPLACE may have been added in the 1.2 spec.

SL

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Thanks for your explanations.

Screaming Lunatic, I thought about that (GL_REPLACE being added in the 1.2 spec) but since I don''t have access to the more recent versions of the book...

Well, thanks again, everybody!

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# Solving $x^k+(x+1)^k+(x+2)^k+\cdots+(x+k-1)^k=(x+k)^k$ for $k\in\mathbb N$

Letting $k$ be a natural number, can we solve the following $k$-th degree equation ? $$x^k+(x+1)^k+(x+2)^k+\cdots+(x+k-1)^k=(x+k)^k \tag{\star}.$$

The following two are famous: $$3^2+4^2=5^2, 3^3+4^3+5^3=6^3.$$

I've tried to find the other integers which satisfy $(\star)$, but I can't find any nontrivial solution. Then, I suspect that the following might be proven true.

My conjecture: There is no integer which satisfies $(\star)$ except $(k,x)=(2,-1),(2,3),(3,3)$.

The followings are what I found:

1. In $k=4$ case, there is no integer which satisfies $(\star)$.

Proof: Suppose that there exists an integer $x$ which satisfies $(\star)$ Then, considering in mod $4$, we reach a contradiction in each remainder, so the proof is completed.

2. Supposing that the one's digit of $k$ is $1$, then there is no integer which satisfies $(\star)$.

Proof: In $k=1$ case, it's obvious. Letting $k=10n+1$ ($n$ is a natural number), let's consider in mod $5$. Let $a_l=l^k$ (mod $5$) ($l$ is an integer).

(i) The $n=1,3,5,\cdots$ cases : $$a_{5m}\equiv 0, a_{5m+1}\equiv 1, a_{5m+2}\equiv 3, a_{5m+3}\equiv 2, a_{5m+4}\equiv 4.$$ (ii) The $n=2,4,6,\cdots$ cases : $$a_{5m}\equiv 0, a_{5m+1}\equiv 1, a_{5m+2}\equiv 2, a_{5m+3}\equiv 3, a_{5m+4}\equiv 4.$$ Suppose that there exists an integer $x$ which satisfies $(\star)$. Letting $l=x+k-1$, we get $$(0+1+2+3+4)\times 2n+a_l \equiv a_{l+1}\ \Rightarrow\ a_l\equiv a_{l+1}.$$ in both (i) and (ii). However, this doesn't happen because of the above. Hence, the proof is completed.

3. Suppose that $k+1$ is a prime number. If there exists an integer $x$ which satisfies $(\star)$, then $k=2$.

Proof: Let's consider when $k=p-1$ ($p$ is a prime number which is more than or equal to $5$). We are going to consider in mod $p$. By Fermat's little theorem, note that $$a^{p-1}\equiv 0 (a\equiv0), 1(a\not\equiv 0).$$ Suppose that there exists an integer $x$ which satisfies $(\star)$.

(i) The $x\not\equiv1$ case (the multiples of $p$ exists in the integers from $x$ to $x+p-2$): we get $$0+1\times (p-2)\equiv 1.$$ However, this doesn't happen because $p\ge5$.

(ii) The $x\equiv1$ case (there is no multiples of $p$ in the integers from $x$ to $x+p-2$): we get $$1\times (p-1)\equiv 0.$$ However, this doesn't happen. Hence, the proof is completed.

I crossposted to MO.

https://mathoverflow.net/questions/141969/solving-xkx1kx2k-cdotsxk-1k-xkk-for-k-in-mathbb-n

• You should find an inequality between LHS and RHS. For example: artofproblemsolving.com/Forum/… – Tengu Sep 5 '13 at 16:26
• For cubes, the next best thing is that it can be shown there is an infinite number of solutions to $x^3+(x+1)^3+(x+2)^3+\dots+(x+n)^3 = y^3$ where $y \neq x+n+1$ such as $11^3+12^3+13^3+14^3 = 20^3$, etc. – Tito Piezas III Sep 9 '13 at 4:01
• I just recalled that for squares, there is also an infinite number for $x^2+(x+1)^2+(x+2)^2+\dots+(x+n)^2 = y^2$ where $y \neq x+n+1$ such as the famous cannonball stacking problem $1^2+2^2+3^2+\dots+24^2 = 70^2$ See Kevin Brown's short article. – Tito Piezas III Sep 9 '13 at 5:05
• @TitoPiezasIII: I've known the last relational equation you wrote. Thank you for your information. – mathlove Sep 9 '13 at 15:11
• @harrypham: Thank you for your information. It must be a good hint, but I haven't got any non-trivial result. – mathlove Sep 12 '13 at 15:27

This is more a comment; it shows another approach but does not (yet?) provide a solution

Using the logic of the Bernoulli-polynomials, which allow a polynomial expression for sums of consecutive like powers, we can generate polynomials depending on x and whose order depends on k for which we can then search for an integer root.

First I rewrite the problem as a function, for which a result with a valid x is a root: $$f_k(x) = \sum_{j=0}^{k-1} (x+j)^k - (x+k)^k$$ such that for some k in $f_k(x_k) = 0$ the $x_k$ indicates the root, and for instance
1. $\qquad f_2(3)=0 \qquad \to x_2=3$ is the first solution (for $k=2$) and
2. $\qquad f_3(3)=0 \qquad \to x_3=3$ is the second solution (for $k=3$).

Fiddling a bit with the adaption of the Bernoulli-polynomials we get the family of polynomials $$\small \begin{array} {r|l} k & f_k(x) \\ \hline \\ 2 & x^2-2 x-3 \\ 3 & 2 x^3-12 x-18 \\ 4 & 3 x^4+8 x^3-12 x^2-112 x-158 \\ 5 & 4 x^5+25 x^4+50 x^3-250 x^2-1355 x-1825 \\ 6 & 5 x^6+54 x^5+285 x^4+180 x^3-4755 x^2-20106 x-26141 \\ 7 & 6 x^7+98 x^6+882 x^5+3430 x^4-4410 *x^3-96726 x^2-353346 x-446782 \\ 8 & 7 x^8+160 x^7+2128 x^6+15232 x^5+40600 x^4-210560 x^3-2165072 x^2-7174784 x-8869820 \\ ... & ... \end{array}$$

The roots for that polynomials are now $$\small \begin{array} {r|lll} k & real & roots & complex & roots \\ \hline 2 & -1.00000 & 3.00000 & . & . \\ 3 & . & 3.00000 & -1.50000+0.866025 i & ... \\ 4 & -1.96857 & 3.32947 & -2.01378+1.99502 i & ... \\ 5 & . & 3.73445 & -2.46439-0.816002 i & ... \\ 6 & -2.95510 & 4.16379 & -2.96548-1.73889 i & ... \\ 7 & . & 4.60171 & -3.46914-2.80013 i & ... \\ ... & ... \end{array}$$

so we see (in teh second column), that after the two relevant real roots $x_2=3$ and $x_3=3$ all other roots $x_k$ in the second column are non-integer, and thus have a small list of where indeed there is and where is not a solution...

The final answer can now only come from the knowledge of the general form of the polynomials and whether then polynomials of that form can have positive integer roots.
There are some criteria which sometimes allow to determine before actually computed, whether there do exist integer/rational roots for certain polynomials (for instance Eisenstein-criterion), unfortunately I'm not firm with that.

[update] I don't really know the significance of this but after the comment of @minar I evaluated the real roots up to index 128.
If we define the minimum of the $\Gamma(x)$ near $x=1.461$ as $\mu \approx 0.8856...$ then the scaling of the positive roots seem to converge to the integer index plus some fixed fractional $$\small {\rho_k \cdot 2 / \mu =... 10.3922 , 11.3881 , 12.3865 , 13.3858 ,...\\ ...,73.3707 , 74.3705 , 75.3702 , 76.3700 ,...\\ ...,127.3577, 128.3574, 129.3572, 130.3570, ...\\}$$

[update2] Hmm, after I've tested some odd k it seems that the following is true:

Let $k$ be an odd number (prime or composite) and $w$ its squarefree primefactor-kernel then the sum $$f_k(x) \equiv \sum_{j=1}^{k-1} (x+j) \pmod k$$ is zero only for such $x$ which are multiples of $w$: $x=t \cdot w \qquad t \in (1,2,3...)$ .

If this is true, then the above estimate of the real root of the sum-quation might come into the play...

• In fact, $f_4$ to $f_{400}$ are even irreducible. – minar Sep 14 '13 at 11:29


text:
# Combustion Formula

Human beings have always been fascinated by using the fire. But it is during the last decades we have actually started to understand the science behind it. This is due to the reason for this breakthrough is the use of modern scientific tools. Such tools are developed and utilized in the scientific work at our division. Combustion is a chemical process involving hydrocarbon and oxygen. In the process of the combustion, fuel is oxidized and large amounts of chemically bound energy are released. In this article, we will learn the concept of combustion and combustion formula with examples. Let us begin learning!

Source: en.wikipedia.org

## Combustion Formula

### Concept of Combustion

Combustion is a chemical process or a reaction between Fuel i.e. Hydrocarbons and Oxygen. When fuel and oxygen react it releases the energy in form of heat and light.  Heat and light energy then result in a flame. The combustion process is used in car motors and rocket engines and in many other types of machinery. Mainly combustions are of 5 different types.

### Examples of Combustion

• Burning of Wood and Coal at our home
• Burning of Petrol and Diesel in our vehicles
• Combustion of CNG, PNG or LPG to cook for on your stovetop
• For the production of energy in thermal power plants
• Fireworks

Thus, combustion encompasses a great variety of phenomena with wide application in the industry as well as in science. It also happens in our home also. And its applications are based on the knowledge of physics, chemistry, and mechanics. Their interrelationship becomes particularly evident in treating flame propagation.

This energy heats the products. Combustion of a fuel such as methane with air leads to a flame temperature of around 1900 ºC. In the combustion of methane with air. Its main products are carbon dioxide and water. But, the formation of these products is very complex. This is because of hundreds of different species like H, O, OH, CH, CH3, etc. are intermediates in the combustion process.

Besides carbon dioxide and water, a large number of pollutants are also produced, for instance, nitrogen oxides, carbon monoxide, and soot particles. The efficiency and cleanliness of the combustion process depending on several parameters. Some parameters are oxygen supply, temperature history, and mixing properties.

Therefore we may conclude that it is a rapid chemical reaction of a substance with oxygen and with the production of heat and light. Also, it is an exothermic reaction accompanied by the development of heat due to which temperature rises considerably.

### Combustion Formula

The heat produced during combustion = Heat of formation of the products – Heat of the formation of reactants.

The complete Combustion process is oxidation of hydrocarbon producing only $$CO_2 and H_2O$$. For example clean combustion of the burning of the candle wax.

Incomplete Combustion is hydrocarbon oxidation that produces carbon monoxide i.e. CO in addition to $$CO_2$$. For example burning coal, where a lot of soot and carbon monoxide is released.

## Solved Examples

Q.1 Write down the Combustion formula for Methane.

Solution: $$CH_4 + 2 O_2 → CO_2 + 2 H_2O$$ + Energy

Q.2: Write down the combustion formula for Ethane:

Solution: $$2 C_2H_6 + 7 O_2 → 4 CO_2 + 6 H_2O + Energy$$

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Typo Error>
Speed of Light, C = 299,792,458 m/s in vacuum
So U s/b C = 3 x 10^8 m/s
Not that C = 3 x 108 m/s
to imply C = 324 m/s
A bullet is faster than 324m/s

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I have realy intrested to to this topic

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umer

m=f/a correct this

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B. Akshaya

M=f/g

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Interesting studies

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Yashdeep tiwari

It is already correct f= ma by second newton formula…


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• ### Multi-band high resolution spectroscopy rules out the hot Jupiter BD+20 1790b - First data from the GIARPS Commissioning(1805.01281)

May 3, 2018 astro-ph.SR, astro-ph.EP
Context. Stellar activity is currently challenging the detection of young planets via the radial velocity (RV) technique. Aims. We attempt to definitively discriminate the nature of the RV variations for the young active K5 star BD+20 1790, for which visible (VIS) RV measurements show divergent results on the existence of a substellar companion. Methods. We compare VIS data with high precision RVs in the near infrared (NIR) range by using the GIANO - B and IGRINS spectrographs. In addition, we present for the first time simultaneous VIS-NIR observations obtained with GIARPS (GIANO - B and HARPS - N) at Telescopio Nazionale Galileo (TNG). Orbital RVs are achromatic, so the RV amplitude does not change at different wavelengths, while stellar activity induces wavelength-dependent RV variations, which are significantly reduced in the NIR range with respect to the VIS. Results. The NIR radial velocity measurements from GIANO - B and IGRINS show an average amplitude of about one quarter with respect to previously published VIS data, as expected when the RV jitter is due to stellar activity. Coeval multi-band photometry surprisingly shows larger amplitudes in the NIR range, explainable with a mixture of cool and hot spots in the same active region. Conclusions. In this work, the claimed massive planet around BD+20 1790 is ruled out by our data. We exploited the crucial role of multi- wavelength spectroscopy when observing young active stars: thanks to facilities like GIARPS that provide simultaneous observations, this method can reach its maximum potential.
• We highlight the power of the Gaia DR2 in studying many fine structures of the Hertzsprung-Russell diagram (HRD). Gaia allows us to present many different HRDs, depending in particular on stellar population selections. We do not aim here for completeness in terms of types of stars or stellar evolutionary aspects. Instead, we have chosen several illustrative examples. We describe some of the selections that can be made in Gaia DR2 to highlight the main structures of the Gaia HRDs. We select both field and cluster (open and globular) stars, compare the observations with previous classifications and with stellar evolutionary tracks, and we present variations of the Gaia HRD with age, metallicity, and kinematics. Late stages of stellar evolution such as hot subdwarfs, post-AGB stars, planetary nebulae, and white dwarfs are also analysed, as well as low-mass brown dwarf objects. The Gaia HRDs are unprecedented in both precision and coverage of the various Milky Way stellar populations and stellar evolutionary phases. Many fine structures of the HRDs are presented. The clear split of the white dwarf sequence into hydrogen and helium white dwarfs is presented for the first time in an HRD. The relation between kinematics and the HRD is nicely illustrated. Two different populations in a classical kinematic selection of the halo are unambiguously identified in the HRD. Membership and mean parameters for a selected list of open clusters are provided. They allow drawing very detailed cluster sequences, highlighting fine structures, and providing extremely precise empirical isochrones that will lead to more insight in stellar physics. Gaia DR2 demonstrates the potential of combining precise astrometry and photometry for large samples for studies in stellar evolution and stellar population and opens an entire new area for HRD-based studies.
• ### On the chemical abundances of Miras in clusters: V1 in the metal-rich globular NGC 5927(1802.07314)

Feb. 20, 2018 astro-ph.SR
We present the first spectroscopic abundance determination of iron, alpha-elements (Si, Ca and Ti) and sodium for the Mira variable V1 in the metal-rich globular cluster NGC 5927. We use high-resolution (R~ 28,000), high signal-to-noise ratio (~200) spectra collected with WINERED, a near-infrared (NIR) spectrograph covering simultaneously the wavelength range 0.91--1.35 micron. The effective temperature and the surface gravity at the pulsation phase of the spectroscopic observation were estimated using both optical (V) and NIR time-series photometric data. We found that the Mira is metal-rich ([Fe/H]=-0.55 \pm 0.15) and moderately alpha-enhanced ([alpha/Fe]=0.15 \pm 0.01, sigma=0.2). These values agree quite well with the mean cluster abundances based on high-resolution optical spectra of several cluster red giants available in the literature ([Fe/H]=-0.47 \pm 0.06, [alpha/Fe]=+0.24 \pm 0.05). We also found a Na abundance of +0.35 \pm 0.20 that is higher than the mean cluster abundance based on optical spectra (+0.18 \pm 0.13). However, the lack of similar spectra for cluster red giants and that of corrections for departures from local-thermodynamical equilibrium prevents us from establishing whether the difference is intrinsic or connected with multiple populations. These findings indicate a strong similarity between optical and NIR metallicity scales in spite of the difference in the experimental equipment, data analysis and in the adopted spectroscopic diagnostics.
• ### GIARPS@TNG GIANO-B & HARPS-N together for a wider wavelength range spectroscopy(1707.04553)

July 18, 2017 astro-ph.IM
Since 2012, thanks to the installation of the high resolution echelle spectrograph in the optical range HARPS-N, the Italian telescope TNG (La Palma) became one of the key facilities for the study of the extrasolar planets. In 2014 TNG also offered GIANO to the scientific community, providing a near-infrared (NIR) cross-dispersed echelle spectroscopy covering 0.97 - 2.45 micron at a resolution of 50,000. GIANO, although designed for direct light-feed from the telescope at the Nasmyth-B focus, was provisionally mounted on the rotating building and connected via fibers to only available interface at the Nasmyth-A focal plane. The synergy between these two instruments is particularly appealing for a wide range of science cases, especially for the search for exoplanets around young and active stars and the characterisation of their atmosphere. Through the funding scheme "WOW" (a Way to Others Worlds), the Italian National Institute for Astrophysics (INAF) proposed to position GIANO at the focal station for which it was originally designed and the simultaneous use of these spectrographs with the aim to achieve high-resolution spectroscopy in a wide wavelength range (0.383-2.45 micron) obtained in a single exposure, giving rise to the project called GIARPS (GIANO-B & HARPS-N). Because of its characteristics GIARPS can be considered the first and unique worldwide instrument providing not only high resolution in a large wavelength band, but also a high precision radial velocity measurement both in the visible and in the NIR arm, since in the next future GIANO-B will be equipped with gas absorption cells.
• ### The GAPS Programme with HARPS-N at TNG XV. A substellar companion around a K giant star identified with quasi-simultaneous HARPS-N and GIANO measurements(1706.06955)

June 21, 2017 astro-ph.SR
Context. Identification of planetary companions of giant stars is made difficult because of the astrophysical noise, that may produce radial velocity (RV) variations similar to those induced by a companion. On the other hand any stellar signal is wavelength dependent, while signals due to a companion are achromatic. Aims. Our goal is to determine the origin of the Doppler periodic variations observed in the thick disk K giant star TYC 4282-605-1 by HARPS-N at the Telescopio Nazionale Galileo (TNG) and verify if they can be due to the presence of a substellar companion. Methods. Several methods have been used to exclude the stellar origin of the observed signal including detailed analysis of activity indicators and bisector and the analysis of the photometric light curve. Finally we have conducted an observational campaign to monitor the near infrared (NIR) RV with GIANO at the TNG in order to verify whether the NIR amplitude variations are comparable with those observed in the visible. Results. Both optical and NIR RVs show consistent variations with a period at 101 days and similar amplitude, pointing to the presence of a companion orbiting the target. The main orbital properties obtained for our giant star with a derived mass of M=0.97+-0.03M_sun are M_Psini=10.78+-0.12MJ;P=101.54+-0.05days;e=0.28+-0.01 and a=0.422+-0.009AU. The chemical analysis shows a significant enrichment in the abundance of Nai, Mgi, Ali and S i while the rest of analyzed elements are consistent with the solar value demonstrating that the chemical composition corresponds with an old K giant (age = 10.1 Gyr) belonging to local thick disk. Conclusions. We conclude that the substellar companion hypothesis for this K giant is the best explanation for the observed periodic RV variation. This study also shows the high potential of multi-wavelength RV observations for the validation of planet candidates.
• ### High-resolution TNG spectra of T Tauri stars: Near-IR GIANO observations of the young variables XZ Tau and DR Tau(1706.05447)

June 16, 2017 astro-ph.SR
We used the TNG/GIANO instrument to obtain near-IR high-resolution spectra (R~50000) of XZ Tau and DR Tau, two actively accreting T Tauri stars classified as EXors. The analysis of the observed features provides information on the properties of the inner disk, the accretion columns, and the winds. Both sources display composite HI line profiles, with contributions from both accreting gas and high-velocity winds. These lines become more symmetric and narrower with increasing upper energy, which may be interpreted in terms of two components with different decrements or imputed to self-absorption effects. XZ Tau shows a relatively high state of activity with respect to literature observations. The variation of the HeI 1.08um line blue-shifted absorption suggests that the inner wind has undergone a dramatic change in its velocity structure, connected with a recent accretion event. DR Tau has a more stable wind as its HeI 1.08um absorption is stable in time in spite of strong variability of the emission component. The IR veiling can be interpreted as due to blackbody emission at temperatures of 1600K for XZ Tau and 2300K DR Tau, with emitting areas ~30 times larger than the star. For XZ Tau these conditions are consistent with emission from the inner rim of the dusty disk, while for DR Tau the fairly high temperature may suggest an origin from a thick gaseous disk within the dust sublimation radius. Strong and broad metallic lines, mainly from CI and FeI, are detected in XZ Tau, similar to those observed in other EXor sources during burst phases. DR Tau shows weaker and narrower metallic lines, despite its larger accretion luminosity. This suggests that accretion is not the only driver of metallic line excitation. Our observations show the potential of wide-band, high-resolution near-IR spectroscopy to simultaneously probe the phenomena that occur in the star-disk interaction region.
• ### High precision radial velocities with GIANO spectra(1604.01408)

April 5, 2016 astro-ph.SR, astro-ph.IM
Radial velocities (RV) measured from near-infrared (NIR) spectra are a potentially excellent tool to search for extrasolar planets around cool or active stars. High resolution infrared (IR) spectrographs now available are reaching the high precision of visible instruments, with a constant improvement over time. GIANO is an infrared echelle spectrograph at the Telescopio Nazionale Galileo (TNG) and it is a powerful tool to provide high resolution spectra for accurate RV measurements of exoplanets and for chemical and dynamical studies of stellar or extragalactic objects. No other high spectral resolution IR instrument has GIANO's capability to cover the entire NIR wavelength range (0.95-2.45 micron) in a single exposure. In this paper we describe the ensemble of procedures that we have developed to measure high precision RVs on GIANO spectra acquired during the Science Verification (SV) run, using the telluric lines as wavelength reference. We used the Cross Correlation Function (CCF) method to determine the velocity for both the star and the telluric lines. For this purpose, we constructed two suitable digital masks that include about 2000 stellar lines, and a similar number of telluric lines. The method is applied to various targets with different spectral type, from K2V to M8 stars. We reached different precisions mainly depending on the H -magnitudes: for H ~ 5 we obtain an rms scatter of ~ 10 m s-1, while for H ~ 9 the standard deviation increases to ~ 50 - 80 m s-1. The corresponding theoretical error expectations are ~4 m s-1 and 30 m s-1, respectively. Finally we provide the RVs measured with our procedure for the targets observed during GIANO Science Verification.
• ### GIANO-TNG spectroscopy of red supergiants in the young star cluster RSGC3(1510.06870)

Oct. 23, 2015 astro-ph.SR
The Scutum complex in the inner disk of the Galaxy has a number of young star clusters dominated by red supergiants that are heavily obscured by dust extinction and observable only at infrared wavelengths. These clusters are important tracers of the recent star formation and chemical enrichment history in the inner Galaxy. During the technical commissioning and as a first science verification of the GIANO spectrograph at the Telescopio Nazionale Galileo, we secured high-resolution (R=50,000) near-infrared spectra of five red supergiants in the young Scutum cluster RSGC3. Taking advantage of the full YJHK spectral coverage of GIANO in a single exposure, we were able to measure several tens of atomic and molecular lines that were suitable for determining chemical abundances. By means of spectral synthesis and line equivalent width measurements, we obtained abundances of Fe and iron-peak elements such as Ni, Cr, and Cu, alpha (O, Mg, Si, Ca, Ti), other light elements (C, N, F, Na, Al, and Sc), and some s-process elements (Y, Sr). We found average half-solar iron abundances and solar-scaled [X/Fe] abundance patterns for most of the elements, consistent with a thin-disk chemistry. We found depletion of [C/Fe] and enhancement of [N/Fe], consistent with standard CN burning, and low 12C/13C abundance ratios (between 9 and 11), which require extra-mixing processes in the stellar interiors during the post-main sequence evolution. We also found local standard of rest V(LSR)=106 km/s and heliocentric V(HEL)=90 km/s radial velocities with a dispersion of 2.3 km/s. The inferred radial velocities, abundances, and abundance patterns of RSGC3 are very similar to those previously measured in the other two young clusters of the Scutum complex, RSGC1 and RSGC2, suggesting a common kinematics and chemistry within the Scutum complex.
• ### GIANO Y-band spectroscopy of dwarf stars: Phosphorus, Sulphur, and Strontium abundances(1510.06396)

Oct. 21, 2015 astro-ph.SR
In recent years a number of poorly studied chemical elements, such as phosphorus, sulphur, and strontium, have received special attention as important tracers of the Galactic chemical evolution. By exploiting the capabilities of the infrared echelle spectrograph GIANO mounted at the Telescopio Nazionale Galileo, we acquired high resolution spectra of four Galactic dwarf stars spanning the metallicity range between about one-third and twice the solar value. We performed a detailed feasibility study about the effectiveness of the P, S, and Sr line diagnostics in the Y band between 1.03 and 1.10 microm. Accurate chemical abundances have been derived using one-dimensional model atmospheres computed in local thermodynamic equilibrium (LTE). We computed the line formation assuming LTE for P, while we performed non-LTE analysis to derive S and Sr abundances. We were able to derive phosphorus abundance for three stars and an upper limit for one star, while we obtained the abundance of sulphur and strontium for all of the stars. We find [P/Fe] and [S/Fe] abundance ratios consistent with solar-scaled or slightly depleted values, while the [Sr/Fe] abundance ratios are more scattered (by +/-0.2 dex) around the solar-scaled value. This is fully consistent with previous studies using both optical and infrared spectroscopy. We verified that high-resolution, Y-band spectroscopy as provided by GIANO is a powerful tool to study the chemical evolution of P, S, and Sr in dwarf stars.
• ### Trade-off study for high resolution spectroscopy in the near infrared with ELT telescopes: seeing-limited vs. diffraction limited instruments(1407.3047)

July 11, 2014 astro-ph.IM
HIRES, a high resolution spectrometer, is one of the first five instruments foreseen in the ESO roadmap for the E-ELT. This spectrograph should ideally provide full spectral coverage from the UV limit to 2.5 microns, with a resolving power from R$\sim$10,000 to R$\sim$100,000. At visual/blue wavelengths, where the adaptive optics (AO) cannot provide an efficient light-concentration, HIRES will necessarily be a bulky, seeing-limited instrument. The fundamental question, which we address in this paper, is whether the same approach should be adopted in the near-infrared range, or HIRES should only be equipped with compact infrared module(s) with a much smaller aperture, taking advantage of an AO-correction. The main drawbacks of a seeing-limited instrument at all wavelengths are: \textit{i)} Lower sensitivities at wavelengths dominated by thermal background (red part of the K-band). \textit{ii)} Much higher volumes and costs for the IR spectrograph module(s). The main drawbacks of using smaller, AO-fed IR module(s) are: \textit{i)} Performances rapidly degrading towards shorter wavelengths (especially J e Y bands). \textit{ii)} Different spatial sampling of extended objects (the optical module see a much larger area on the sky). In this paper we perform a trade-off analysis and quantify the various effects that contribute to improve or deteriorate the signal to noise ratio. In particular, we evaluate the position of the cross-over wavelength at which AO-fed instruments starts to outperform seeing-limited instruments. This parameter is of paramount importance for the design of the part of HIRES covering the K-band.
• ### Preliminary results on the characterization and performances of ZBLAN fiber for infrared spectrographs(1407.3052)

July 11, 2014 astro-ph.IM
Present telescopes and future extremely large telescopes make use of fiber-fed spectrographs to observe at optical and infrared wavelengths. The use of fibers largely simplifies the interfacing of the spectrograph to the telescope. At a high spectral resolution (R>50,000) the fibers can be used to achieve very high spectral accuracy. GIANO is an infrared (0.95-2.5\mu m) high resolution (R=50,000) spectrometer[1] [2] [3] that was recently commissioned at the TNG telescope (La Palma). This instrument was designed and built for direct feeding from the telescope [4]. However, due to constraints imposed on the telescope interfacing during the pre-commissioning phase, it had to be positioned on the rotating building, far from the telescope focus. Therefore, a new interface to the telescope, based on IR-transmitting ZBLAN fibers with 85 \mu m core, was developed. In this article we report the first, preliminary results of the effects of these fibers on the quality of the recorded spectra with GIANO and with a similar spectrograph that we set-up in the laboratory. The effects can be primarily associated to modal-noise (MN) that, in GIANO, is much more evident than in optical spectrometers, because of the much longer wavelengths.
• ### The fiber-fed preslit of GIANO at T.N.G(1407.3126)

July 11, 2014 astro-ph.IM
Giano is a Cryogenic Spectrograph located in T.N.G. (Spain) and commisioned in 2013. It works in the range 950-2500 nm with a resolving power of 50000. This instrument was designed and built for direct feeding from the telescope [2]. However, due to constraints imposed on the telescope interfacing during the pre-commissioning phase, it had to be positioned on the rotating building, far from the telescope focus. Therefore, a new interface to the telescope, based on IR-transmitting ZBLAN fibers with 85\mu m core, was developed.Originally designed to work directly at the $f/11$ nasmyth focus of the telescope, in 2011 it has decided to use a fiber to feed it. The beam from the telescope is focused on a double fiber boundle by a Preslit Optical Bench attached to the Nasmith A interface of the telescope. This Optical Bench contains the fiber feeding system and other important features as a guiding system, a fiber viewer, a fiber feed calibration lamp and a nodding facility between the two fibers. The use of two fibers allow us to have in the echellogram two spectrograms side by side in the same acquisition: one of the star and the other of the sky or simultaneously to have the star and a calibration lamp. Before entering the cryostat the light from the fiber is collectd by a second Preslit Optical Bench attached directly to the Giano cryostat: on this bench the correct f-number to illuminate the cold stop is generated and on the same bench is placed an image slicer to increase the efficiency of the system.
• ### The WFPC2 UV Survey: the BSS population in NGC 5824(1311.1332)

Nov. 6, 2013 astro-ph.GA
We have used a combination of high-resolution Hubble Space Telescope WFPC2 and wide-field ground-based observations, in ultraviolet and optical bands, to study the blue straggler star population of the massive outer-halo globular cluster NGC 5824, over its entire radial extent. We have computed the center of gravity of the cluster and constructed the radial density profile, from detailed star counts. The profile is well reproduced by a Wilson model with a small core (r_c \simeq 4.4 arcsec) and a concentration parameter c \simeq 2.74. We also present the first age determination for this cluster. From the comparison with isochrones, we have found t=13\pm0.5 Gyr. We discuss this result in the context of the observed age-metallicity relation of Galactic globular clusters. A total of 60 bright blue stragglers has been identified. Their radial distribution is found to be bimodal, with a central peak, a well defined minimum at r \sim 20 arcsec, and an upturn at large radii. In the framework of the dynamical clock defined by Ferraro et al. (2012), this feature suggests that NGC 5824 is a cluster of intermediate dynamical age.
• ### BSS populations from the WFPC2 UV Survey(1301.2916)

Jan. 14, 2013 astro-ph.GA, astro-ph.SR
We have used a combination of high-resolution Hubble Space Telescope WFPC2 and wide-field ground-based observations in ultraviolet and optical bands to study the BSS population of the globular clusters NGC 6229 and M75. The combination of different filters allows us to optimally select specific stellar populations. In particular, the UV bands are ideal to study the hot objects such as HB and BSS stars. In both these clusters, BSS are more segregated than the normal stars. In NGC 6229 the BSS projected radial distribution is found to be bimodal, with a high central peak, a well defined minimum at intermediate radii, and an upturn in the outskirts, while no significant upturn in the BSS frequency has been observed in the outskirts of M75, suggesting that these clusters are in different dynamical evolutionary phases.
• ### Dynamical age differences among coeval star clusters as revealed by blue stragglers(1212.5071)

Dec. 20, 2012 astro-ph.SR
Globular star clusters that formed at the same cosmic time may have evolved rather differently from a dynamical point of view (because that evolution depends on the internal environment) through a variety of processes that tend progressively to segregate stars more massive than the average towards the cluster centre. Therefore clusters with the same chronological age may have reached quite different stages of their dynamical history (that is, they may have different dynamical ages). Blue straggler stars have masses greater than those at the turn-off point on the main sequence and therefore must be the result of either a collision or a mass-transfer event. Because they are among the most massive and luminous objects in old clusters, they can be used as test particles with which to probe dynamical evolution. Here we report that globular clusters can be grouped into a few distinct families on the basis of the radial distribution of blue stragglers. This grouping corresponds well to an effective ranking of the dynamical stage reached by stellar systems, thereby permitting a direct measure of the cluster dynamical age purely from observed properties.
• ### The BSS population in NGC 6229(1202.1094)

Feb. 6, 2012 astro-ph.GA
We have used a combination of high-resolution Hubble Space Telescope WFPC2 and wide-field ground-based observations in ultraviolet and optical bands to study the blue straggler star (BSS) population of the outer-halo globular cluster NGC 6229, over its entire radial extent. A total of 64 bright BSS (with m255 6 21.30, corresponding to m555 6 20.75) has been identified. The BSS projected radial distribution is found to be bimodal, with a high central peak, a well defined minimum at intermediate radii (r \sim 40"), and an upturn in the outskirts. From detailed star counts even in the very inner region, we compute the centre of gravity of the cluster and the most accurate and extended radial density profile ever published for this system. The profile is reasonably well reproduced by a standard King model with an extended core (rc \simeq 9.5") and a modest value of the concentration parameter (c \simeq 1.49). However, a deviation from the model is noted in the most external region of the cluster (at r > 250" from the centre). This feature needs to be further investigated in order to assess the possible presence of a tidal tail in this cluster.
• ### On the radial extent of the dwarf irregular galaxy IC10(1009.3917)

Sept. 20, 2010 astro-ph.SR
We present new deep and accurate space (Advanced Camera for Surveys -- Wide Field Planetary Camera 2 at the Hubble Space Telescope) and ground-based (Suprime-Cam at Subaru Telescope, Mega-Cam at Canada-France-Hawaii Telescope) photometric and astrometric data for the Local Group dwarf irregular IC10. We confirm the significant decrease of the young stellar population when moving from the center toward the outermost regions. We find that the tidal radius of IC10 is significantly larger than previous estimates of $r_t \lesssim$ 10\min. By using the $I$,\vmi\ Color Magnitude Diagram based on the Suprime-Cam data we detect sizable samples of red giant (RG) stars up to radial distances of 18-23$'$ from the galactic center. The ratio between observed star counts (Mega-Cam data) across the tip of the RG branch and star counts predicted by Galactic models indicate a star count excess at least at a 3$\sigma$ level up to 34-42\min\ from the center. This finding supports the hypothesis that the huge H{\size{I}} cloud covering more than one degree across the galaxy is associated with IC10 \citep{huchtmeier79,cohen79}. We also provide new estimates of the total luminosity ($L_V\sim9\times$$10^7 L_\odot, M_V$$\sim$-15.1 mag) that agrees with similar estimates available in the literature. If we restrict to the regions where rotational velocity measurements are available (r$\approx13'$), we find a mass-to-light ratio ($\sim$10 $M_\odot$ $L_\odot$) that is at least one order of magnitude larger than previous estimates. The new estimate should be cautiously treated, since it is based on a minimal fraction of the body of the galaxy.
• ### On the stellar content of the starburst galaxy IC10(0905.2580)

May 15, 2009 astro-ph.SR
We investigate the stellar content of the starburst dwarf galaxy IC10 using accurate and deep optical data collected with the Advanced Camera for Surveys and with the Wide Field Planetary Camera 2 on board the Hubble Space Telescope. The comparison between theory and observations indicates a clear change in age distribution when moving from the center toward the external regions. Moreover, empirical calibrators and evolutionary predictions suggest the presence of a spread in heavy element abundance of the order of one-half dex. The comparison between old and intermediate-age core He-burning models with a well defined overdensity in the color-magnitude diagram indicates the presence of both intermediate-age, red clump stars and of old, red horizontal branch stars.
• ### On the radial distribution of horizontal branch stars in ngc2808(0903.4531)

March 26, 2009 astro-ph.GA
We present accurate new ultraviolet and optical BVI photometry for the Galactic globular cluster ngc2808, based on both ground-based and archival HST imagery. From this we have selected a sample of ~2,000 HB stars; given the extensive wavelength range considered and the combination of both high-angular-resolution and wide-field photometric coverage, our sample should be minimally biased. We divide the HB stars into three radial bins and find that the relative fractions of cool, hot and extreme HB stars do not change radically when moving from the center to the outskirts of the cluster: the difference is typically smaller than ~2sigma. These results argue against the presence of strong radial differentiation among any stellar subpopulations having distinctly different helium abundances. The ratio between HB and RG stars brighter than the ZAHB steadly increases when moving from the innermost to the outermost cluster regions. The difference is larger than ~4sigma and indicates a deficiency of bright RGs in the outskirts of the cluster.
• ### On the distance and reddening of the starburst galaxy IC10(0810.1210)

Oct. 7, 2008 astro-ph
We present deep and accurate optical photometry of the Local Group starburst galaxy IC10. The photometry is based on two sets of images collected with the Advanced Camera for Surveys and with the Wide Field Planetary Camera 2 on board the Hubble Space Telescope. We provide new estimates of the Red Giant Branch tip (TRGB) magnitude, m_{F814W}^{TRGB}=21.90+-0.03, and of the reddening, E(B-V)=0.78+-0.06, using field stars in the Small Magellanic Cloud (SMC) as a reference. Adopting the SMC and two globulars, Omega Centauri and 47 Tucanae, as references we estimate the distance modulus to IC10: independent calibrations give weighted average distances of mu=24.51+-0.08 (TRGB) and mu=24.56+-0.08 (RR Lyrae). We also provide a new theoretical calibration for the TRGB luminosity, and using these predictions we find a very similar distance to IC10 (mu~24.60+-0.15). These results suggest that IC10 is a likely member of the M31 subgroup.
• ### Relative distances of Omega Centauri and 47 Tucanae(0809.0863)

Sept. 4, 2008 astro-ph
We present precise optical and near-infrared ground-based photometry of two Globular Clusters (GCs): Omega Cen and 47 Tuc. These photometric catalogs are unbiased in the Red Giant Branch (RGB) region close to the tip. We provide new estimates of the RGB tip (TRGB) magnitudes--m_I(TRGB)=9.84+/-0.05, Omega Cen; m_I(TRGB)=9.46+/-0.06, 47 Tuc--and use these to determine the relative distances of the two GCs. We find that distance ratios based on different calibrations of the TRGB, the RR Lyrae stars and kinematic distances agree with each other within one sigma. Absolute TRGB and RR Lyrae distance moduli agree within 0.10--0.15 mag, while absolute kinematic distance moduli are 0.2--0.3 mag smaller. Absolute distances to 47 Tuc based on the Zero-Age-Horizontal-Branch and on the white dwarf fitting agree within 0.1 mag, but they are 0.1--0.3 mag smaller than TRGB and RR Lyrae distances.


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## University Calculus: Early Transcendentals (3rd Edition)

Published by Pearson

# Chapter 7 - Practice Exercises - Page 420: 7

#### Answer

$$\int \frac{\ln(x-5)}{x-5}dx=\frac{\Big(\ln(x-5)\Big)^2}{2}+C$$

#### Work Step by Step

$$A=\int \frac{\ln(x-5)}{x-5}dx$$ Set $u=\ln(x-5)$, we have $$du=\frac{(x-5)'}{x-5}dx=\frac{1}{x-5}dx$$ Therefore, $$A=\int udu=\frac{u^2}{2}+C$$ $$A=\frac{\Big(\ln(x-5)\Big)^2}{2}+C$$

After you claim an answer you’ll have 24 hours to send in a draft. An editor will review the submission and either publish your submission or provide feedback.


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Basic Rhythms

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Explore a number pattern which has the same symmetries in different bases.

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Can you correctly order the steps in the proof of the formula for the sum of a geometric series?

An Alphanumeric

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Freddie Manners, of Packwood Haugh School in Shropshire solved an alphanumeric without using the extra information supplied and this article explains his reasoning.

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By proving these particular identities, prove the existence of general cases.

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A composite number is one that is neither prime nor 1. Show that 10201 is composite in any base.

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Picture Story

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Can you see how this picture illustrates the formula for the sum of the first six cube numbers?

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Make and prove a conjecture about the value of the product of the Fibonacci numbers $F_{n+1}F_{n-1}$.

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With n people anywhere in a field each shoots a water pistol at the nearest person. In general who gets wet? What difference does it make if n is odd or even?

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Take any pair of two digit numbers x=ab and y=cd where, without loss of generality, ab > cd . Form two 4 digit numbers r=abcd and s=cdab and calculate: {r^2 - s^2} /{x^2 - y^2}.

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We continue the discussion given in Euclid's Algorithm I, and here we shall discover when an equation of the form ax+by=c has no solutions, and when it has infinitely many solutions.

Impossible Sandwiches

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In this 7-sandwich: 7 1 3 1 6 4 3 5 7 2 4 6 2 5 there are 7 numbers between the 7s, 6 between the 6s etc. The article shows which values of n can make n-sandwiches and which cannot.

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Can you discover whether this is a fair game?

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Tom writes about expressing numbers as the sums of three squares.

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Some diagrammatic 'proofs' of algebraic identities and inequalities.

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This follows up the 'magic Squares for Special Occasions' article which tells you you to create a 4by4 magicsquare with a special date on the top line using no negative numbers and no repeats.

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Imagine two identical cylindrical pipes meeting at right angles and think about the shape of the space which belongs to both pipes. Early Chinese mathematicians call this shape the mouhefanggai.

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Kyle and his teacher disagree about his test score - who is right?

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Can you convince me of each of the following: If a square number is multiplied by a square number the product is ALWAYS a square number...

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Take any rectangle ABCD such that AB > BC. The point P is on AB and Q is on CD. Show that there is exactly one position of P and Q such that APCQ is a rhombus.

Modulus Arithmetic and a Solution to Dirisibly Yours

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Peter Zimmerman from Mill Hill County High School in Barnet, London gives a neat proof that: 5^(2n+1) + 11^(2n+1) + 17^(2n+1) is divisible by 33 for every non negative integer n.

Triangular Intersection

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What is the largest number of intersection points that a triangle and a quadrilateral can have?

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Suppose A always beats B and B always beats C, then would you expect A to beat C? Not always! What seems obvious is not always true. Results always need to be proved in mathematics.

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A point moves around inside a rectangle. What are the least and the greatest values of the sum of the squares of the distances from the vertices?

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Find all positive integers a and b for which the two equations: x^2-ax+b = 0 and x^2-bx+a = 0 both have positive integer solutions.

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Prove that you cannot form a Magic W with a total of 12 or less or with a with a total of 18 or more.

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The country Sixtania prints postage stamps with only three values 6 lucres, 10 lucres and 15 lucres (where the currency is in lucres).Which values cannot be made up with combinations of these postage. . . .

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Problem solving is at the heart of the NRICH site. All the problems give learners opportunities to learn, develop or use mathematical concepts and skills. Read here for more information.

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Pythagorean Triples I

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The first of two articles on Pythagorean Triples which asks how many right angled triangles can you find with the lengths of each side exactly a whole number measurement. Try it!

Sums of Squares and Sums of Cubes

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Proof of Pick's Theorem

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Follow the hints and prove Pick's Theorem.

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Professor Korner has generously supported school mathematics for more than 30 years and has been a good friend to NRICH since it started.

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Some puzzles requiring no knowledge of knot theory, just a careful inspection of the patterns. A glimpse of the classification of knots and a little about prime knots, crossing numbers and. . . .

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Picturing Pythagorean Triples

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This article discusses how every Pythagorean triple (a, b, c) can be illustrated by a square and an L shape within another square. You are invited to find some triples for yourself.

Pareq Exists

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Matter of Scale

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A Long Time at the Till

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Notty Logic

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Have a go at being mathematically negative, by negating these statements.

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# K-homology classes of Dirac operators on Hermitian manifolds

Given a compact Hermitian manifold $M$, we have three canonical pseudo-differential operators on the sections of complexified de Rham complex, namely

1) (d + d$^*,\Omega^{*})$

2) ($\partial$ + $\partial^*,\Omega^{(*,0)})$

3) $(\overline{\partial} + \overline{\partial}^*,\Omega^{(0,*)})$

Question 1: What is the relation between the K-homology classes (see example 3.4) of these three operators? Can we at least say that all three give non-equivalent classes? In general, given two differential operators which are "representatives" of the same K-homology class, will tensoring them each by a line bundle always land in the same class? I.e. does tensoring by a line bundle descent to an operation on K-homology?

Assume in addition $M$ is spin, which according to Atiyah (see this question) means that $\Omega^{(0,n)}$ admits a line bundle. Then we also have an associated Dirac operator, which is the tensor product of the third operator with the square root endowed with a connection.

Question 2: N-lab says that the class of the Dirac is equal to the class of the operator in (1) above.

Question 3: If so, is this operation a permutation?

I am not entirely familiar with $KK$-theory, so please correct me if there are mistakes. I think ultimately you are trying to show the topological $K$-theory class you get from taking the horizontal derivative, vertical derivative, or the total deriative is the same for a double-complex. This can be done by standard comparsion theorems for de Rham cohomology. The usual way to do it is via diagram chasing, or if you strongly prefer using spectral sequences.
Here is a "direct" way to see it analytically. They are all self-adjoint elliptic operators, and their index should all be zero, essentially because $$Ind(P)=\dim (\ker P)-\dim(\ker P^{*})$$ There are some analytic subtlies to sort out ($C^{\infty}_{c}(X)$ is dense in the Sobolev space $H^{s}(X)$, which is an extension of the map on $C^{\infty}(X)$, etc). But the result should be the same. My guess is that you meant something more refined like the signature operator or the operator associated with a spin bundle.


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# Contents

## Idea

H-spaces are simply types equipped with the structure of a magma (from classical Algebra). They are useful classically in constructing fibrations.

## Definition

A H-Space consists of

• A type $A$,
• A basepoint $e:A$
• A binary operation $\mu : A \to A \to A$
• for every $a:A$, equalities $\mu(e,a)=a$ and $\mu(a,e)=a$

## Properties

Let $A$ be a connected H-space. Then for every $a:A$, the maps $\mu(a,-),\mu(-,a):A \to A$ are equivalences.


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Browse Questions

# If the sum of the slopes of the lines given by $x^2-2cxy-7y^2=0$ is four times their product c has the value.

$\begin{array}{1 1}(A)\;-2\\(B)\;-1 \\(C)\;2\\(D)\;1 \end{array}$

Can you answer this question?

Let the lines be $y=m_1X$
$y=m_2x$ then
$m_1+m_2=-\large\frac{-2c}{7}$
$m_1m_2=-\large\frac{1}{2}$
Given $m_1+m_2=4m_1m_2$
=> $\large\frac{2c}{7}=\frac{-4}{7}$
$2c=\large\frac{-28}{7}$
$2c=-4$
$c=-2$
$c=2$
Hence C is the correct answer.
answered Apr 8, 2014 by


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# Tag Info

36

An animation is worth a million words:

26

It does, but the effects are negligible in the regions we think about. If you think about a volume of air as a box of atoms bouncing around, you can apply an oscillating pressure gradient across that box and show that it behaves close enough to an ideal wave propagation medium that you can get away with using such an ideal model. The variations you are ...

18

In this case it's probably best to be pragmatic. A pulse can be described as a superposition of sine waves that extend infinitely into space and time. But it's just that: a mathematical description that is useful for your purposes. There is not necessarily a physical meaning connected to it. Nevertheless, in quantum mechanics the wave-description of ...

16

They do, its just usually negligible in practice. There's also scattering because the particles are not all the same (H2O, N2, O2, etc.)--but that, too, is usually negligible. Its mainly because there are so many particles in a single wave. Consider that the wave must be extremely short before it becomes noticeable (megahertz).

11

This will be a purely mathematical treatment. It needs to be combined with some practical playing around to really "get" it. Traveling wave Let's start with the description of a harmonic traveling wave in one-dimension. Here "harmonic" just means the mathematical form of the wave is sinusiodal in both time and space. For concreteness we'll using talk ...

6

Dispersion in waves arises from both material property variation with frequency and from the geometry of the fields in question. That wave dispersion will arise from material property variation is obvious. But wave geometry and boundary conditions also matter. Simple example: a conductive waveguide with rectangular cross-section with sidelengths $a$ and $b$...

6

I just wanted to add to a previous (very accurate) answer: you can think of it as an Fourier expansion of the actual (physical) wave profile. It is not a real life process, it is a mathematical approximation. The wave pulse can be thought of as a superposition of plane waves, which happens to interfere destructively in entire space, except for the localized ...

6

My high school physics teacher was saying that “this is because of interference of sound waves. During the day, there are a lot of sounds and they cancel each other due to interference. But, during the night, there are few sounds and they can reach to our ears without canceling each other”. You need a better high school physics teacher. Temperatures tend ...

6

Any physical phenomenon is potentially capable to cause some change to any other phenomenon, more or less directly. If it was not the case, the physical world could be divided into completely independent realms; there would not be the one single world we call Nature. Practically though, many if not most of the actually existing interactions between systems ...

5

I can answer half your question in that a sound can change the path of light. A change in the density of the air produces a change in the refractive index of the air and so a Schlieren photograph can make this visible. Here is a YouTube video to show a sound wave produced by clapping.

5

how is a standing wave related to the atomic orbit (It is my understanding that the atomic orbit is both a mathematical function that describes the probability of an electron being at a certain place, but it is also the image of this function in terms of real space, i.e. the actual 3 dimensional volume around the nucleus that a particular electron calls "...

4

Imagine something oscillating in space and time, for example a plane wave propagating across the axis $x$. This propagation is expressed via the so-called phase $$\phi(x,t)=\omega \cdot t - k\cdot x = \dfrac{2\pi}{T}\cdot t -\dfrac{2\pi}{\lambda}\cdot x \tag{01}$$ and the magnitude of the plane wave as $$E(x,t)=A\cos\phi(x,t) \tag{02}$$ As the ...

4

This is a lot more subtle problem than is indicated in any of the comments. The problem is not just the issue of how the sum of non-causal signals can approximate a causal one, but how is it possible that while all real-life signals must start and stop at some time they must also be band-limited beyond some frequency, but as we know these two are ...

4

The battery on your smart phone only provides approximately 3 W of power to the phone's transmitter. This is enough power to reach a nearby cell tower, but totally insufficient to send a signal for several hundred miles. Due to this, the nearest cell tower is used to relay your signal to another cell tower that is located close to the party that you are ...

4

Do radio waves from the Sun reach Earth? Of course they do. It's just another form of electromagnetic radiation. If so, do they penetrate the atmosphere or are they reflected, absorbed, or scattered? That depends on frequency (or wavelength). The atmosphere reflects, absorbs, or scatters most incoming electromagnetic radiation. There's a window in ...

3

The photon is an elementary particle in the standard model of particle physics. It does not have a wavelength. It is characterized in the table as a point particle with mass zero and spin one. Its energy is given by E=h*nu, where nu is the frequency of the classical electromagnetic wave which can be built up by photons of the same energy. This is where ...

3

The maximum and minimum are "local" values. As you move closer to A (at 0.2 m you are MUCH closer to A than to B) the amplitude of A is much larger - so although there may be destructive interference between A and B at that point, this is by no means perfect interference, and the resulting amplitude is still quite large (lot of A minus a little of B). ...

3

In this type of problem one has to take great care in defining intensities. In this case there are 4 different intensities: 1. $I_{ss}$, the intensity received by the static observer as perceived by himself, 2. $I_{ms}$, the intensity received by the moving observer as perceived by a static observer. 3 $I_sm$, the intensity received by the static observer as ...

3

I would tend to agree that background noise is a factor, but rather than reducing, adding to the sound you are trying to make sense of. So part of that may be how your brain is able to filter the information from the background noise. But at night the temperature is lower and according to this tutorial on sound propagation (which does cite reliable ...

2

To add to the existing answer, I think there is a nomenclature issue. When you say "bass" people understand "low frequencies" but what you probably mean is "beat". Rapid changes in amplitude, like a beat, carry a lot of high frequencies. You do hear mostly the beat from other peoples' headphones, ans it's annoying. You can think about the extreme case: the ...

2

Yes, sound waves in a gas, liquid or solid can affect the light passing through it, as the motion of the atoms due to sound waves changes the atomic spacing, and this changes the index of refraction slightly. So the light would be diffracted and some amount of the light would experience a frequency shift up and a frequency shift down by the sound wave ...

2

I'd like to add to ACuriousMind's Answer. His/Her answer emphasizes that Maxwell's equations and other equations that give rise to the Huygens spherical wave kernel in their Green's functions are inherently acausal and we must force causality by hand through the appropriate boundary conditions. In antenna problems, we simply discard the advanced wave part ...

2

Basically it is whatever you need to multiply a distance by to find a phase difference (in radians). For a traveling wave, the wave number is the amount of phase difference per unit length. For a physical sine wave, it is the ratio between the maximal slope of the wave surface and the amplitude. In other words, it measures how dramatic the local ...

2

The Poynting vector is just an intensity vector, it is just $\vec S = \vec E \times \vec B$, one might need some factors of $c$ in there, I have been in $c = 1$ land too long …. Due to the time dependence, the amplitudes of the electric fields are time dependent. If one would call that flickering, it would be the frequency of the light. I don't think that ...

2

You have the correct conclusion, and I think you have the correct analysis, but I don't fully understand your presentation. Think of it this way: If the displacement occurs only between "neutral" and "forward", then the average density (air molecules per volume, or spring coils per length) over the entire system (air chamber or spring) must increase. But ...

2

In general, a mixed wave of the form $$f(x,t)=\cos(x-t)+r\cos(x+t)$$ will not have nodes, at least in the sense of points $x_0$ for which $f(x_0,t)\equiv 0$ for all times $t$. In general, there's relatively little to say beyond what the picture will convey: As you turn $r$ up from 0 to 1, you first start inducing modulations into the amplitude of the ...

2

He is referring to the second paragraph of 31-3: However, we shall discuss the formula we have obtained, in various possible circumstances. First of all, for most ordinary gases (for instance, for air, most colorless gases, hydrogen, helium, and so on) the natural frequencies of the electron oscillators correspond to ultraviolet light. These ...

2

My theory is that since water, and indeed all liquids, are incompressible, they both form at the same time. As the pebble hits the surface of the water and pushes it down, the surrounding liquid is pushed up (relative to the original surface level of the liquid), as you stated. Any displacement of a liquid must be accounted for.

2

The D'Alembert solution has a simple interpretation. Using your notation, it reads $$\phi(x, t) = \frac{g(x-t) + g(x+t)}{2} + \frac12 \int_{x-t}^{x+t} h(x') dx'.$$ where $g(x)$ is the initial position, $h(x)$ is the initial velocity, and $v = 1$. The intuition is as follows: the first term above solves the wave equation with initial position $g(x)$ but zero ...

2

A standing wave is basically two opposing waves of equal amplitude, as shown in the diagram below (where n is a positive integer): You can see this more clearly if you look at the top line where n=3, and follow it as it goes down, up, down. That's wave one. Then, if you look at the bottom line in the same case as it goes up, down, up, that is the opposing ...

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


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This week’s homework provides you with an introduction to modeling occupancy data collected for a single-species across multiple seasons. You’ll work with data collected on northern spotted owls (Strix occidentalis caurina) and reported on by MacKenzie et al. (2003).

### Background

You will conduct modeling that will show you some of the important ways that you can examine changes in occupancy and assess equilibrium with this model type. The data set consists of detection/non-detection data for northern spotted owls on multiple sites in multiple seasons (first year = 1997). We will model rates of (1) occupancy, (2) detection, (3) colonization, and (4) extinction. Here, we’ll use the parameters $$\psi$$ for occupancy rate, $$p$$ for detection rate, $$\gamma$$ for colonization rate (going from un-occupied to occupied from 1 year to the next), and $$\epsilon$$ for the local extinction rate (from 1 year to the next). Given this, $$1-\epsilon$$ is the rate at which occupied sites remain occupied, and $$1-\gamma$$ is the rate at which un-occupied sites remain un-occupied. Given the background you have in robust design, you should be able to see that the probability that a site is occupied at time $$t+1$$ may or may not depend on its occupancy status in time $$t$$, i.e., it can be either 1st-order Markovian $$(\gamma \ne 1-\epsilon)$$ or random $$(\gamma = 1-\epsilon)$$.

Set up a new analysis in Program MARK using “Robust Design Occupancy” as the data type. After you do this, a new window will open that allows you to choose which parameterization to use for the modeling. We’ll start with a parameterization that works with initial occupancy ($$\psi_{1997}$$), $$\gamma_t$$, $$\epsilon_t$$, and $$p_{session, t}$$.