The plate scale, $S = \delta s / \delta \theta$, is the displacement, $\delta s$ at the telescope focal plane for a given difference in incoming light angle, $\delta \theta$. For a given telescope focal length:
$$\sin (\delta \theta) = \frac{\delta s}{f}$$
Using the small-angle approximation:
$$\sin(\delta \theta) \approx \delta \theta$$
so
$$\frac{\delta s}{f} \approx \delta \theta$$
$$\frac{\delta s}{\delta \theta} \approx \frac{1}{f}$$
$S \approx \frac{1}{f}$ with S radians / length
or, converting radians to arcseconds
$S \approx 206265/f$ arcsec / length
The pixel scale, $P$, for a given telescope focal length, $f$, and sensor pixel length or width, $w$, is the angular distance on the sky between sensor pixels.
$P = Sw \approx \frac{w}{f}$ in radians / pixel assuming w and f are the same units
converting to arcsec:
$P = Sw \approx 206265 \times \frac{w}{f}$ in arcsec / pixel assuming w and f are the same units
and adding factors to make w in μm and f in mm:
$P = Sw \approx 206.265 \times \frac{w}{f}$ in arcsec / pixel assuming w in μm and f in mm
The AT65EDQ has a nominal 420 mm focal length, while the ASI533MC Pro has 3.76 μm square pixels.
According to PixInsight plate solve:
All From1
Photon flux can be calculated from Jy by:
1 Jy = $1.51\times 10^7$ photons s$^{-1}$m$^{-2}$ $\left(\frac{\Delta\lambda}{\lambda}\right)^{-1}$
Info About Each Photometric Band2:
| Band | Central Wavelength (nm) | $\frac{\Delta \lambda}{\lambda}$ | Flux at m=0 (Jy) |
|---|---|---|---|
| U | 360 | 0.15 | 1810 |
| B | 440 | 0.22 | 4260 |
| V | 550 | 0.16 | 3640 |
| R$_c$ | 640 | 0.23 | 3080 |
| I$_c$ | 790 | 0.19 | 2550 |
The PDF for the observed ADC camera value is:
$$\mathrm{ADC} = A \times \big[\mathrm{Poisson}(N_\mathrm{signal}+B_\mathrm{thermal})+\mathrm{Gaussian}(0,r)\big]$$
where ADC is the ADC readout value, $A$ is the ADC electron to count amplification factor, $N_\mathrm{signal}$ is the true number of signal electrons, $B_\mathrm{thermal}$ is the true thermal background in electrons, and $r$ is the readout noise in electrons.
To estimate $A$, $B_\mathrm{thermal}$, and $r$, take many exposures of various lengths with $N_\mathrm{signal}=0$, i.e. with the shutter closed. By investigating the variance of the ADC value v. the ADC value, one may estimate $A$ and $r$. For a given mean ADC value, determined by exposure length and temperature, the variance in the ADC output is related to $A$ and $r$, by
$$\sigma^2_\mathrm{ADC} = A \times ADC + (Ar)^2$$
Thus one may perform a linear fit to estimate the slope, $A$, and the intercept $(Ar)^2$, which can be used to estimate $r$.
Once $A$ and $r$ are known, $B_\mathrm{thermal}$ v. exposure length or temperature may be investigated directly, since $$B_\mathrm{thermal} = \mathrm{mean}(ADC)/A$$
Finally, when looking at a signal source with the shutter open, $N_\mathrm{signal}$ may be estimated as
$$N_\mathrm{signal} = ADC/A - B_\mathrm{thermal}$$
with uncertainty:
$$\sigma_{N_\mathrm{signal}} = \sqrt{ADC + r^2}$$
A nice page on DSLRs here: http://theory.uchicago.edu/~ejm/pix/20d/tests/noise/index.html
For long enough time scales that multiple exposures can be stacked, the uncertainty in each pixel should be estimated from the standard deviation of the exposures $/ \sqrt{n-1} $. For shorter time scales where a measurement is required per exposure, uncertainty should be calculated from $\sigma_{N_\mathrm{signal}}$ as shown in the previous section, for each light frame. Master dark, flat, and bias frame uncertainty, estimated from the standard deviation of the exposures $/ \sqrt{n-1} $, should then be propagated through the calibration procedure.
Fast Lomb-Scargle Algorithm: http://adsabs.harvard.edu/abs/2010ApJS..191..247T
Date-compensated Discrete Fourier Transform: http://adsabs.harvard.edu/abs/1981AJ.....86..619F
Weighted Wavelet Z-transform: http://adsabs.harvard.edu/abs/1996AJ....112.1709F
Fast Lomb-Scargle Algorithm:
@ARTICLE{2010ApJS..191..247T,
author = { {Townsend}, R.~H.~D.},
title = "{Fast Calculation of the Lomb-Scargle Periodogram Using Graphics Processing Units}",
journal = {\apjs},
archivePrefix = "arXiv",
eprint = {1007.1658},
primaryClass = "astro-ph.SR",
keywords = {methods: data analysis, methods: numerical, techniques: photometric, stars: oscillations},
year = 2010,
month = dec,
volume = 191,
pages = {247-253},
doi = {10.1088/0067-0049/191/2/247},
adsurl = {http://adsabs.harvard.edu/abs/2010ApJS..191..247T},
adsnote = {Provided by the SAO/NASA Astrophysics Data System}
}
Date-compensated Discrete Fourier Transform:
@ARTICLE{1981AJ.....86..619F,
author = { {Ferraz-Mello}, S.},
title = "{Estimation of Periods from Unequally Spaced Observations}",
journal = {\aj},
year = 1981,
month = apr,
volume = 86,
pages = {619},
doi = {10.1086/112924},
adsurl = {http://adsabs.harvard.edu/abs/1981AJ.....86..619F},
adsnote = {Provided by the SAO/NASA Astrophysics Data System}
}
Weighted Wavelet Z-transform:
@ARTICLE{1996AJ....112.1709F,
author = { {Foster}, G.},
title = "{Wavelets for period analysis of unevenly sampled time series}",
journal = {\aj},
keywords = {STARS: OSCILLATIONS, METHODS: NUMERICAL},
year = 1996,
month = oct,
volume = 112,
pages = {1709},
doi = {10.1086/118137},
adsurl = {http://adsabs.harvard.edu/abs/1996AJ....112.1709F},
adsnote = {Provided by the SAO/NASA Astrophysics Data System}
}