# Estimating Star Counts from Density Laws

• June_cosmo
In summary, the problem discussed the challenges of determining star density from star counts and the use of density laws to estimate counts. Two density laws were considered, one with a non-zero constant out to a distance of 800 and another with an exponential law and a scale height of 300. The problem also provided information on counting stars of a specific type with known absolute magnitude and a faint magnitude limit. A table was created listing apparent magnitude, distance in pc, volume, density, and total number of stars for both density laws. The log of star counts was also plotted as a function of magnitude for both laws.
June_cosmo

## Homework Statement

(This is a long problem but I think the question is not hard.)

Determining the star density from star counts is tough, but estimating counts from a density law is simpler. In practice, this method of fit-ting observed number counts to an assumed density law is becoming a more common approach given the large datasets available. In this problem we will explore counts along the z-axis. For this problem, consider two density laws (to be clear, number densities being considered in this problem which is the total number of something being counted per unit volume). First, 1(||)=0 out to some distance where 0=0.1 −3 is a non-zero constant for ||≤800 and 0=0 for ||>800 . Second, an exponential law in which (||)=0^(-||/ℎ) the same constant 0 and a scale height ℎ=300 . Assume you have isolated a type of star in your counts that you know has an absolute magnitude =3.5 and that you can count stars to a faint magnitude limit =20.5. Finally, assume that we are conducting the counts in a solid angle of 0.05 steradians centered on a Galactic pole.

With this information you can make a table listing the following information: (i) apparent magnitude from 6.0-20.5 increments of 0.5 mag; (ii) the distance in pc of stars of the type we are counting each magnitude bin; (iii) the volume of space in ^3 contained within each magnitude bin; (iv) the density of stars in ^3 in each magnitude/distance bin according to both density laws and (v) the total number of stars in each magnitude/distance bin according to both density laws. Plot the log of the resulting star counts as a function of magnitude for both density laws.

## Homework Equations

m = M+5(logD-1),where M is the absolute magnitude and m the apparent magnitude, D is the distance between the observer and the star

## The Attempt at a Solution

for (ii), since we don't know the corresponding M,how do we know the distance? Did I miss something...

June_cosmo said:
for (ii), since we don't know the corresponding M,how do we know the distance? Did I miss something...
How do you interpret this statement:
June_cosmo said:
Assume you have isolated a type of star in your counts that you know has an absolute magnitude =3.5 and that you can count stars to a faint magnitude limit =20.5. F
?
Sounds to me that you have selected stars with absolute magnitude 3.5, i.e. those are the only stars to be counted. You are in a position to count all of those with apparent magnitude 20.5 or less.

haruspex said:
How do you interpret this statement:

?
Sounds to me that you have selected stars with absolute magnitude 3.5, i.e. those are the only stars to be counted. You are in a position to count all of those with apparent magnitude 20.5 or less.
Oh I didn't see your reply before. Thanks! You're right. That makes a lot more sense!

## 1. How do density laws help in estimating star counts?

Density laws, also known as the laws of stellar density, provide a mathematical relationship between the number of stars in a given region and their spatial distribution. By using these laws, scientists can estimate the number of stars in a particular area based on the density of stars in nearby regions.

## 2. Can star counts be accurately estimated using density laws?

While density laws provide a helpful framework for estimating star counts, they are not always accurate. This is because the distribution of stars in a region can be affected by various factors such as the presence of gas and dust, as well as the gravitational pull of nearby objects. Therefore, it is important to also consider other observational data when estimating star counts.

## 3. What are some common density laws used in estimating star counts?

Some of the most commonly used density laws include the exponential law, where the number of stars decreases exponentially with distance from the center of a galaxy, and the power law, which states that the number of stars decreases with distance according to a power function. Other laws, such as the King, Plummer, and Sérsic profiles, are also frequently used in estimating star counts.

## 4. How do scientists account for the presence of variable stars in estimating star counts?

Variable stars, which change in brightness over time, can affect star counts as they may not always be visible during observations. To account for this, scientists use statistical methods to estimate the frequency and brightness of variable stars in a given region, and then adjust the star counts accordingly.

## 5. Can star counts be used to determine the age of a galaxy?

Yes, star counts can provide valuable information about the age and evolution of a galaxy. By studying the distribution of stars within a galaxy, scientists can infer its formation and development over time. For example, a galaxy with a high number of young, massive stars is likely to be relatively young, while a galaxy with a larger proportion of older stars may be more mature.

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