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--- wiki:observational_astronomy:energy:energy_flux_counts [2023/11/10 12:19] – Roy Prouty
+++ wiki:observational_astronomy:energy:energy_flux_counts [2023/11/10 12:28] (current) – removed Roy Prouty
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-====== What are we measuring? ======
-Q: What do the values our imaging software give us mean?\\
-A: Well, what do our pixels report to us? Answer: Counts. These are digital units derived from the on-chip analog-to-digital conversion process that is thought to be (to a first-order approximation) linear with respect to the voltage generated by liberated electrons in the pixels of the detector.\\ \\
-So we've passed the bill from the counts to the voltage and the associated liberated electrons. Grant me that the piling up of electrons over the integration-time generates a voltage. After that, what liberates the electrons? Answer: photons, thermal agitations, and other electromagnetic forces. For a more delicate discussion of this, refer to the [[[[wiki:detectors:semiconductor_physics|Ideas in Semiconductor Physics]] page. \\ \\
-The thermal agitations and other electromagnetic forces arise from the detector itself. These contributions to the liberated electrons and therefore counts must be calibrated away (see [[wiki:observational_astronomy:data_reduction_telescope|Data Reduction I: Remove Detector Effects]]). After that, the counts speak to only the liberated electrons due to interactions with photons incident on the detector.\\ \\
-=== Radiative Energy ===
-The energy associated with the liberation of these electrons is the radiative energy we seek to measure. This energy interacts with the semiconductor substrate and liberates photoelectrons at a rate assumed to be linear with respect to the integration-time.
-$$E_{\gamma}~d\lambda = \frac{hc}{\lambda}d\lambda$$
-This energy carries the normal units Joules ($[E]=~ J$)((Also of relevance is the $erg$.  $1 ~erg \equiv 100~ nJ$)), but be careful! These counts are proportional to the received radiant energy (in the form of photons for this discussion) incident on a //single pixel//. This single pixel receives photons from a specific angular area of the sky, the pixel has an area over which it received photons, an orientation of that area, a time during which it was 'allowed' to receive photons, and a sensitivity to photons over a specific range of wavelengths.\\ \\
-So these counts are proportional not to the total energy of a target observed by the telescope, but to the differential amount of that energy called (in the broadest sense) the spectral radiance.
-=== Spectral Radiance ===
-$$I(\lambda, \theta, \phi) = \frac{d^6 E}{d\lambda\cdot dt\cdot d^2A \cos{\theta}\cdot d^2\Omega(\theta,\phi)}$$
-The spectral radiance has units $[I(\lambda, \theta, \phi)]=J\cdot (m\cdot t\cdot m^2\cdot sr^2)^{-1}$\\ \\
-Said slightly differently, the spectral radiance is the amount of energy received from a specific angular area of the sky ($d^2\Omega(\theta,\phi)$), over an area ($d^2A$) with some orientation ($\cos{\theta}$), during time during which it was 'allowed' to receive energy ($dt$), and a sensitivity to photons over a specific range of wavelengths ($d\lambda$).\\ \\
-Each pixel effectively integrates this spectral radiance with respect to wavelength, time, perpendicular area, and solid angle. The resulting measure of radiant energy is what is responsible for liberating photoelectrons in the CMOS semiconductor substrate which are eventually readout as counts.\\ \\
-Since each of the pixels are assumed to be sensitive to photons of the same range of wavelengths, were exposed to photons for the same amount of time, have the same orientation and area, and look at congruent solid angles of the sky, we can appropriately sum-up the bits of radiant energy to think about the total radiant energy that fell on a pixel during our integration time.\\
-=== Spectral Flux ===
-Since we do know the range of wavelengths, the integration time, and pixel size, we can define the spectral flux:
-$$F(\lambda) = \frac{d^4 E}{d\lambda\cdot dt\cdot d^2A_{\perp}}$$
-The radiative, spectral flux has units $[F(\lambda)]=J\cdot (m\cdot t\cdot m^2)^{-1}$. \\ \\
-==== Magnitudes ====
-The spectral flux is convenient since the magnitude system has a long history in publication and serves as a good benchmark for any star-observing optical system.
-=== Magnitude Definition ===