Magnitude to Distance Math

We've discussed what magnitude is and that it is related to distance. How exactly do physicists go from magnitude measurements to measurements of distance? Magnitude is a measure of intensity of the object. We see the intensity of an object as a certain number of counts/sec/distance². So lets consider two stars at distances A and B.

m_a= -2.5 log I_a

The intensity, as stated above, is in units of counts/sec/distance².

I_a = N (counts/sec) / 4(pi)(d²)

4*(pi)(d²) = surface area of a sphere. The intensity from an object comes out as a sphere in all directions. Thus, we want to know the intensity at the distance of a certain sphere. Moving on, if we compare two stars at distances a and b,

m_a= -2.5 log I_a

m_b= -2.5 log I_b

We are trying to find a way to determine distance from magnitude. So we want to get I_a and I_b on the same side of one equation. Let's subtract m_a and m_b

m_a-m_b= -2.5 log (I_a) + 2.5 log (I_b)

One of the laws of logs says that: log(a)-log(b)=log(a/b). Thus

m_a-m_b= -2.5 log (I_b/I_a)

We know that Intensity = N (counts/sec) / 4(pi)(d²). The only thing changing between I_a and I_b is the distance measurement. Thus

m_a-m_b= 2.5 log (d_a/d_b)²

There are two special kinds of magnitudes that scientists look at. Apparent magnitude is the magnitude we see here at earth. So the distance you use for apparent magnitude is what we are looking for (the distance to the galaxy). The other type of magnitude is absolute magnitude. The absolute magnitude is the magnitude of a star at a specific distance. We can use any distance we want as long as we're consistent, but astronomers use 10 pc so lets stick with that. Absolute magnitude is usually signified by M, while an apparent magnitude is signified by m. In our data from SDSS we are given m (apparent magnitude). There are a few different ways to determine M. Every measurement of the Hubble constant relies upon a "standard candle" where we know the intrinsic brightness of a galaxy. Astronomers have come up with many different ways to go about that, and it is a very interesting activity. However, for our purposes we don't have time to go into that sort of detail. One way that astronomers do this is what we previously discussed about picking a velocity dispersion. There is a way to go from velocity dispersion to absolute magnitude, however we are going to assume that choosing one velocity dispersion is enough to narrow our data search. We can then pick an absolute magnitude and use it as a constant throughout all of the activity.

So we know m, M, and d_b. Lets do some math

m-M = 2.5 log (d/10)²

m-M = 5 log (d/10)

Using our log law again

m-M = 5 [log(d)-log(10)]

m-M = 5log(d)-5

m-M = -5+5log(d)

We can now solve for d

d=10^(.2(m-M)+5)

This is the equation put into the excel sheet activity. The apparent magnitude comes from our data and the absolute magnitude is programmed into the spreadsheet.

In summary, this mathmatical derivation goes through three main points. First, we are agreeing with scientists that an absolute magnitude is the magnitude we would see if a galaxy were 10 pc away. Second, this activity fundamentally uses the idea that we "see" things on an order of 1/R². Finally, this is a great use of logs, their application, and their laws.