|celestia.Sci Test: Brightness Profile of Elliptical Galaxies
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|Author:||t00fri [ Thu, 28-03-13, 21:23 GMT ]|
|Post subject:||celestia.Sci Test: Brightness Profile of Elliptical Galaxies|
Elliptical galaxies form a large fraction of all galaxies in the Universe!
They are old, have correspondingly B-V color ~ 1, and therefore are colored in orange shades.
A good rendering of ellipticals in celestia.Sci is therefore very important. The code is completely new in comparison with the latest Celestia 1.7.x SVN version.
I would say the rendering is almost photorealistic, at least compared to the resolution of the SDSS survey. Here is a familiar example:
Displayed are the orange E2 galaxy M 60 (NGC 4649) and superimposed the small bluish SBc spiral NGC 4647. The image is from SDSS, and the SDSS color profile is used in the comparisons with celestia.Sci below:
In celestia.Sci this nice galaxy couple looks as follows:
You see that the rendering is pretty close to the SDSS photo (no stars, here)!
There are many examples of whole clusters of elliptical galaxies in excellent agreement with the SDSS survey, but this is for another thread...
The purpose here is to demonstrate that the light distribution of ellipticals as rendered in celestia.Sci matches perfectly the famous de Vaucouleurs 'R^(1/4)' law as it should be.
For the less initiated reader: Here is an example of the measured major axis surface brightness profile, mu(r), of the E1 elliptical galaxy M 105 (Ngc 3379) (solid line) compared to the de Vaucouleurs law (dashed line):
mu(r) = mu_e + 8.32678 [(r/r_e)^(1/4) - 1].
The parameters are: mu_e = 22.24 [mag/arcsecs^2] and r_e = 56.8" (2.7 kpc).
r is the distance from the galaxy center in arcsecs.
You can see that the fit is excellent (as usual)! The example is from deVaucouleurs' great lecture:
http://www.google.com/url?sa=t&rct=j&q= ... 2042,d.Yms
After some quite complex shader-based rendering code, it is most interesting to check the resulting brightness distribution directly in the visualization!
This is what I did to test it: I chose a suitable E0 elliptical galaxy, blew it up, converted it to grayscale as a color average and saved it. Loaded into GIMP, I measured with the color-'pipette' the actual brightness (0..255) as function of the distance (pixels) from the galaxy center. Then I plotted the result in Maple and compared it with the theoretical result, i.e. the deVaucouleur brightness law:
Here is the plot about which I am quite happy:
The blue points correspond to my measurements of the brightness by means of the GIMP 'pipette', while the red curve displays deVaucouleurs' law. Since the display is 8bit, the brightness is clamped at 255 as you can see near the origin.
In conclusion: The rendering of elliptical galaxies in celestia.Sci is not just pretty to look at, but also, the brightness distribution corresponds rather precisely to the successful deVaucouleurs law!
Note that for well-known reasons, we want a logarithmic mapping (i.e. linearly with magnitude) of a galaxy's (surface) brightness to the pixel values of the visualization! Hence I considered above log(I_deVau(r) / I_0). This important fact I forgot to emphasize above...
|Author:||ElChristou [ Thu, 28-03-13, 21:40 GMT ]|
Good job really convincing!
|Author:||t00fri [ Fri, 29-03-13, 19:56 GMT ]|
Let me go on a bit with some explicit calculations within the deVaucouleurs 'R^1/4' framework.
The general deVaucouleurs form for the flux of light from an elliptic galaxy has this exponential form as function of r, the distance from the galaxy center:
Here Ie, r_e are parameters and ν is a constant to be determined as follows:
r_e is defined as the observed galaxy radius which encloses half the galaxy's total light flux. Hence, in order to fix the constant ν, we have to integrate the flux I_deVauc(r) from r=0 to r=r_e and divide this integral by the same integral extending from r=0 to r=infinity. This ratio we then require to be 1/2. This condition corresponds to the definition of r_e, leading to a determination of the constant ν.
Note, the integrands are fluxes of dimension [intensity/area ], hence we need to integrate over the circular area
dA = d(Pi*r^2) = 2*Pi * r dr
The first integral is solved in seconds by Maple (or Mathematica):
The second integral extending to r=infinity (<=> total light flux) is even simpler:
Next we form the ratio of the two, which is a complicated function, ONLY depending on the unknown constant ν. We determine it again in seconds by requiring the ratio to be 1/2 and solving the complicated equation numerically again with Maple:
Indeed we find the unique solution
ν = 3.330712713
Fortunately, this value agrees perfectly with the one in deVaucouleurs' lecture. So we are on the safe side...
Taking the logarithm of base 10, the deVaucouleurs law takes the following form as relevant for visualizing elliptical galaxies in celestia.Sci:
Or expressed equivalently in units [mag/arcsec^2], the surface brightness reads
In celestia.Sci, I preferentially use the so-called isophotal μ(R_25) = 25 radius. Insertion of this definition into the previous equation with r=R_25 gives after solving
It is convenient to use instead the central surface brightness mu_0 instead of mu_e. The final result then reads
With ν = 3.330712713 from above, we get e.g. for M 105 (NGC 3379) with mu_0 ~ 14 [mag/arcsecs^2]:
We see that in case of elliptical galaxies one has to carefully study which kind of radius definition was used in a given scientific catalog...
So this time no nice pictures but some mathematical insight instead
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