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): \begin{equation} \mu = m + 2.5\log_{10}\Omega \end{equation}

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 Custer Mar 26 '17 at 16:44

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  • $\begingroup$ 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? $\endgroup$ – DilithiumMatrix May 4 '16 at 23:17
  • $\begingroup$ @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 $\endgroup$ – Jorge May 5 '16 at 1:10

Hopefully an observer can chime in with better advice... but:

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.

It's unclear what you mean by "counting stars". Individual stars shouldn't be resolvable in your image, right?

  • $\begingroup$ Thanks for your comments @DilithiumMatrix. I am not dealing with images and I have no idea about backgrounds subtraction or calibrations. I just take other people's result and work with it, for instance the position (right ascension, declination) and the magnitud of the stars that compose the galaxy, therefore I should be getting the same as previous results. What you mention is exactly what I am trying to do, I am taking rings (or squares) around a center, compute the integrated magnitud of all the stars ("counting the stars") inside the region and finally computing the surface brightness. $\endgroup$ – Jorge May 5 '16 at 3:59
  • $\begingroup$ 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 $\endgroup$ – Jorge May 5 '16 at 4:08

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