Differences

This shows you the differences between two versions of the page.

--- wiki:astronomy:observational_astronomy:measurements:radiative_flux [2023/11/11 20:49] – Roy Prouty
+++ wiki:astronomy:observational_astronomy:measurements:radiative_flux [2023/11/14 11:11] (current) – removed Roy Prouty
@@ Line 1: / Line 1: @@
-The radiative energy of a source can be given either in relative terms via some [[~:..:magnitudes|magnitude scale]] or in absolute terms via some measure of [[~:..:radiative_flux|radiative flux]].
-----
-Recall from the discussion on [[~:..:what_the_detector_measures|what detectors measure]] that after calibration of a raw light image to account for the effects of the detector as well as the effects of the atmosphere, the counts that remain associated with each pixel are assumed to be proportional to the amount of radiant energy that fell into the pixel during integration time.
-===== 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.
-$$E = \frac{hc}{\lambda}$$
-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 =====
-The differential amount of energy shown in the numerator of the right hand side tends to vanish as the factors in the denominator tend to zero. The spectral radiance (the left hand side; $I(\lambda, \theta, \phi)$) is the limiting value this ratio approaches. Is can be thought of as a "ray" of light or "pencil" of radiation. The spectral radiance is a convenient theoretical quantity since its dependence on differential solid angle leave it conserved over its optical path (barring any sources or sinks). Therefore, the spectral radiance is the basis of all radiative transfer theory.
-$$I(\lambda, \theta, \phi) = \frac{d^6 E}{d\lambda\cdot dt\cdot d^2A \cos{\theta}\cdot d^2\Omega(\theta,\phi)}$$
-The spectral radiance has units $[I(\lambda, \theta, \phi)]=J\cdot (m\cdot t\cdot m^2\cdot sr^2)^{-1} = W\cdot (m\cdot m^2\cdot sr^2)^{-1}$\\ \\
-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 $[F(\lambda)]=J\cdot (m\cdot t\cdot m^2)^{-1}$. \\ \\
-{{tag>[not-done]}}