# Surface brightness [closed]

I have to compute the surface brightness as a function of the radius from the following set of data: {right ascension ($\alpha$), declination ($\delta$), magnitude (m)}. I also know the center of the set $\left(\alpha_c, \delta_c \right)$.

The surface brightness in logarithmic units (mag/arcsec$^2$) can be computed by (1): $$\mu = m + 2.5\log_{10}\Omega$$

and the solid angle can be computed as $d\Omega = \frac{dS}{d^2} = \frac{d^2 \sin\theta d\theta d\phi}{d^2}$, where $d$ is the distance to the stars.

What I did is to compute the integrated magnitude in a certain the region given by $\alpha_2 > \alpha > \alpha_1$ and $\delta_2 > \delta > \delta_1$. In which case the solid angle should be given by: $\Omega = (\alpha_2 - \alpha_1)(\sin\delta_2 - \sin\delta_1)$ because $\delta$ is the complementary angle of $\theta$.

My question is: which radius I should consider if I want to plot $\mu(r)$?. I have consider the angular separation between ($\alpha_c, \delta_c$) and ($\alpha_1, \delta_c$), and also the angular separation between ($\alpha_c, \delta_c$) and ($\alpha_c, \delta_1$) but with neither of this options reproduce previous results.

I also count the stars which angular separation is $r < r_{test}$ (edited: in rings of radius $r_{test}$), compute the integrated magnitude, use $\Omega = 2\pi(1 - \cos(r_{test}))$ and finally compute $\mu(r_{test})$.

Both approaches gives more or less the same results, what am I doing wrong here?

By wrong, I mean that I am getting larger values for the surface brightness that previous published results.

## closed as off-topic by Rob Jeffries, sammy gerbil, Yashas, Kyle Kanos, Jon CusterMar 26 '17 at 16:44

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• It's very difficult to answer this question when you're not explaining what you're looking for, and why what you have is "wrong"... can you elaborate or specify? – DilithiumMatrix May 4 '16 at 23:17
• @DilithiumMatrix. I just want to compute the surface brightness of a dwarf galaxy and then fit it to a King's or Plummer's profile. The results that I obtained from what I described are ~20% bigger surface brightness than previous results when using the same data. I am using some of this data – Jorge May 5 '16 at 1:10

Generally, people calculate the (average) brightness in rings around the center, then plot that brightness as a function of the projected radius (i.e. angle, e.g. in arcseconds). You can check to make sure your solid angle is correct (for example you say $\delta$ is the complementary angle to $\theta$, so perhaps it should be $\cos \delta$ instead of $\sin$?).
A $20%$ difference is pretty small; is it possible that's due to some sort of calibration issue? I.e. background subtraction? Flat fielding? etc etc.
• and if I am not wrong the solid angle should be function of $\sin\delta$ of $\cos\theta$. I was guessing my problem was computing the projected radius, which I assume was just the radius of the rings in arcseconds... unless this is false – Jorge May 5 '16 at 4:08