# Mass, luminosity and magnitude of a galaxy

astroYEEET
Homework Statement:
A galaxy is located at a distance D = 20Mpc from the Sun.
The shape of the galaxy was observed to look like an ellipse with a semi major axis a = 22.6 ”and
a / b = 0.85 ratio
Half of the total light from the galaxy comes from this ellipse and the average brightness of the surface in the ellipse is about 24.7 mag / arcsec2.
A) Calculate the absolute magnitude (mag) that corresponds to the whole galaxy.
B) Determine the total mass of the stars in the galaxy (in solar masses) based on this relation:
mass / brightness = (M / M⊙) / (L / L⊙) = 2
Relevant Equations:
1)S=m+2.5logA where S surface brightness, m apparent magnitude, A area of arcsec^2
2)M-m=5-5logD where M absolute magnitude and D distance
3)(I am not sure if this can be used because it is about two stars not one star(the sun here) and one galaxy)L1/L2=(M1/M2)^3.5
So i first calculated the angular area of the galaxy. a=22.6'' and a/b=0.85 => b = 26.588''( btw I do not know why a/b =0.85 since b is the semi minor axis). Then the area of the elipse is a*b*π =1887.745''. Then using the first equation we get an apparent magnitude of m=16.511.
Using the second equation we get M=-14.994. So if i am correct this is the answer for A and this is how far i have gone.

First, I suspect they mean that b/a = 0.85, and b should be smaller than a. If possible, you should check with the people who gave you the problem and verify if this is an error in the problem. Second, you have forgotten the other half of the light that is outside the ellipse. You should add this in, which will increase the apparent luminosity and decrease the apparent magnitude.

astroYEEET
astroYEEET
First, I suspect they mean that b/a = 0.85, and b should be smaller than a. If possible, you should check with the people who gave you the problem and verify if this is an error in the problem. Second, you have forgotten the other half of the light that is outside the ellipse. You should add this in, which will increase the apparent luminosity and decrease the apparent magnitude.
How do i calculate the luminosity ?

How do i calculate the luminosity ?
Do you know the relation between magnitude and luminosity? So if half the light is included, you need to double the total luminosity to include the other half. What does this do to the magnitude?

astroYEEET
astroYEEET
Do you know the relation between magnitude and luminosity? So if half the light is included, you need to double the total luminosity to include the other half. What does this do to the magnitude?
So luminosity is the energy that is emitted by a star per second.
Then there is absolute and apparent magnitude where apparent magnitude is how bright the star seems to be( which depends and on it's distance) and absolute magnitude is actually the luminosity of a star on a logarithmic scale(It is how bright a star would be if it was from a distance from us of 10 parsecs).
I do not know any formula about the luminosity and the magnitude but i suppose that since luminosity increases the star seems brighter and that corresponds to smaller magnitude.
So how can i find how much does the magnitude decrease from the magnitude i have calculated?

For almost any work in astronomy, you need to understand the quantitative relation between the luminosity and the magnitude. Keep trying to answer my question of what happens to the magnitude when you double the luminosity. You could start here:
https://en.wikipedia.org/wiki/Magnitude_(astronomy)

astroYEEET
astroYEEET
For almost any work in astronomy, you need to understand the quantitative relation between the luminosity and the magnitude. Keep trying to answer my question of what happens to the magnitude when you double the luminosity. You could start here:
https://en.wikipedia.org/wiki/Magnitude_(astronomy)
So as searching i found this formula which relates magnitude and luminosity : m1-m1=2.5log(L2/L1)
and this : Mbol=4.72-2.5log(L/Lsun)

The first one is the one you want. So if you double the luminosity, L2/L1 = 2. Then how does m change?

astroYEEET
astroYEEET
The first one is the one you want. So if you double the luminosity, L2/L1 = 2. Then how does m change?
If the luminosity doubles i suppose the change in magnitude will be log(2)

If the luminosity doubles i suppose the change in magnitude will be log(2)
No. Look at the equation again.

astroYEEET
No. Look at the equation again.
Correct me if i do something wrong
At first Δm=2.5log(1)= 0
But because we have to take into account the other half of the light emitted the luminosity doubles so Δm=2.5log(2)= 0.752

That looks right now. So then what are the apparent and absolute magnitudes?

astroYEEET
astroYEEET
That looks right now. So then what are the apparent and absolute magnitudes?
The apparent magnitude will be 16.511-0.752=15.759
And then the absolute magnitude will be equal to M-m=5-5logD => M=-15.746

The apparent magnitude will be 16.511-0.752=15.759
And then the absolute magnitude will be equal to M-m=5-5logD => M=-15.746
I get a slightly larger number for m. I'm conviced they really mean b/a=0.85. There is no way the semi-minor axis (b) should be larger than the semi-major axis (a). But you'll have to decide whether to take them at their word or correct their mistake. Assuming b>a (wrong!), I think you did this part correctly.

astroYEEET
astroYEEET
I get a slightly larger number for m. I'm conviced they really mean b/a=0.85. There is no way the semi-minor axis (b) should be larger than the semi-major axis (a). But you'll have to decide whether to take them at their word or correct their mistake. Assuming b>a (wrong!), I think you did this part correctly.
Ok so i will correct this. Now how should i solve question B ?

Ok so i will correct this. Now how should i solve question B ?
Once you have the absolute magnitude of the galaxy, you can compare it to the absolute magnitude of Milky Way, which you can look up.

astroYEEET
Once you have the absolute magnitude of the galaxy, you can compare it to the absolute magnitude of Milky Way, which you can look up.
And how do i use the relation that they give me ?

Actually, now that I look at it, you don't need the absolute magnitude of the Miky Way. They have asked you to reference it to the sun. Once you have the absolute magnitude of the galaxy, you can compare it to the absolute magnitude of the sun, which should allow you to calculate Lgalaxy/Lsun. You already told me the relation between magnitude and luminosity in Post #7.

astroYEEET
astroYEEET
Actually, now that I look at it, you don't need the absolute magnitude of the Miky Way. They have asked you to reference it to the sun. Once you have the absolute magnitude of the galaxy, you can compare it to the absolute magnitude of the sun, which should allow you to calculate Lgalaxy/Lsun. You already told me the relation between magnitude and luminosity in Post #7.
Ok so if my calculations are correct the problem is solved. Thanks for your continuous help and i hope one day i will be able to guide others solving their problems. By the way are you a physicist/astrophysicist ?