This work is based on the following project: https://github.com/maksymbrl/solar-limb/blob/master/report/AstroLab_Report.pdf. The main idea was to write a self-consisten GUI application which will find the effective temperature of the Sun through analyzing images taken in the different wave-bands.
Here, I briefly state main concepts and equations used. For more complete description please refer to the https://github.com/maksymbrl/solar-limb/blob/master/report/AstroLab_Report.pdf
In general, to solve radiative trasfer equation
is tedious task. Luckily, some simplifications can be made in study of stellar atmospheres:
- Stellar atmosphere can usually be well approximated as a plane-parallel system -> Plane-parallel approximation;
- We assume the emission being thermal;
- We assume that the atmosphere is homogeneous and isotropic;
Thus, we have:
and the source function is the product of power series:
Assuming Eddington approximation with the local thermodynamic equilibrium, we get
Since photosphere is defined as the layer on which effective temperature is equivalent to real temperature T, we get
Starting from the Planck's law expressed in terms of wavelength
we get
- We divide the Sun for 9 concentric rings with the center in the solar center and number each region with symbol k (each region has
points).
- The k-th region extends from the minimum distance (from the center)
to the maximum
.
- The mean distance
from the center and its associated error
are defined as:
- We firstly find an intensity of each pixel
in the k-th region and then get an average intensity
in this ring. As error we take the standard deviation over the
of the k-th region
Note that is not a measure error, but depends on the way in which we define the regions on the diameter.
- Referring to the central intensity
and restricting ourselves to a second order polynomial fit, we can write
- In order to avoid non-physical behaviors, we limit ourselves with the second order expansion. We are interested in relative intensities
and the least square polynomial fit allows us to find the fit coefficients (
).
In the whole discussion we refer to relative intensities. In particular all the intensity values we treat are expressed in digital units and we usually normalize all the values relatively to the center of the disk. But, to obtain the source function we need physical central intensity expressed in
units.
| Band | Wavelength (nm) | |
|---|---|---|
| B | 420 | |
| V | 547 | |
| R | 648 | |
| I | 871 |
- Then we extrapolated the temperatures for
for each filter and found a values for the effective temperatures.
- We applied the uncertainty propagation theory for calculation of the associated errors on the wavelength errors
(the FWHM of the filter transparency profile), the central intensities
and the fit coefficients
.
- In the end, we used a weighted mean for various band filters as
together with associated uncertainties , supposing them being small, i.e., neglecting the higher orders of smallness
to obtain the final effective temperature
- Averaged data for R filter
- Snapshot of the Sun for R filter
- Resulting table
