It is impossible to measure all of the radiant energy that an astronomical source radiates. This should be obvious, since we generally only see one size of an astronomical source at a time! Energy ($E$) is measured in units $[E]=Joules,J$¹⁾.

The total amount of radiant energy an astronomical source radiates away per unit time is called the luminosity. Luminosity ($L$) is measured in units of power, $[L]=Watts,W$. This radiant power is conserved per unit time. However, again, it is impossible to measure the luminosity of an astronomical source. We only receive a small fraction of this radiant power. The fraction depends on the area of the detector and the distance between the detector and the astronomical source.

Generally, this detected quantity is called the radiant flux. Radiant flux ($F$) is measured in units of $[F]=W(m^2)^{-1}$. The fraction of the luminosity collected by every square-meter of a detector some distance $r$ away from the center of the source is just the luminosity divided by the surface of a sphere of radius $r$. As such, the radiative flux is not conserved it obeys an inverse-square law.

$$F=\frac{L}{4\pi r^2}$$

The radiative flux defined thus is the total flux emitted over all wavelengths. This is referred to as the bolometric flux, $F_{bol}$. It is therefore prudent to define the spectral radiative flux ($F_\lambda$) which carries units of $[F_\lambda]=W(m~m^2)^{-1}$.

$$F_{bol} = \int_0^\infty F_\lambda d\lambda = \int_0^\infty F_\nu d\nu$$

While we focus our discussion on wavelengths, you can transform scales to frequency by differentiating the above integrals. This leaves you with the relation: $F_\lambda d\lambda = F_\nu d\nu$. Together with $c=\lambda\nu$, you can derive algebraic relationships between $F_\nu$ and $F_\lambda$. Be careful with units

Further, conversion of $F_\lambda$ or $F_\nu$ to the photon flux, $N_\lambda$ or $N_\nu$, can be achieved using the relation $E = h\nu = \frac{hc}{\lambda}$, leading to

$N_\nu = \frac{F}{E} = \frac{F}{h\nu}$, in units of $[N_\nu] = photons (s~ m^{-2} Hz)^{-1}$, and

$N_\lambda = \frac{F}{E} = F\frac{\lambda}{hc}$, in units of $[N_\lambda] = photons (s~ m^{-2} m)^{-1}$.

If we restrict ourselves further to only collecting flux from a small piece of an astronomical source that we can spatially resolve ²⁾ on our detector, we can define the spectral radiance. The spectral radiance ($I_\lambda$) has units of $[I_\lambda]=W(s~m~m^2~sr)^{-1}$. The spectral radiance is another convenient quantity, since it is also conserved along its optical path.

$$I_\lambda(\theta, \phi) = \frac{d^6 E}{d\lambda\cdot dt\cdot d^2A \cos{\theta}\cdot d^2\Omega(\theta,\phi)}$$

With this definition, 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 approaches³⁾. 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.

To understand how the solid angle gives the spectral radiance this conserved quality, consider the above schematic along with the definition of spectral radiance and the added spherical coordinate know-how gleaned from this page describing spherical coordinates.

Radiant Energy is conserved. $$E_1 = E_2$$ Spectral Radiance emitted from source $2$ into a solid angle, $d^2\Omega_2$, is received by a unit area, $d^2A_1$ and vice-versa. $$I_1 ~dt ~d\lambda ~d^2A_1~\hat{n}_1\cdot d^2\Omega_2 = I_2 ~dt ~d\lambda ~d^2A_2~\hat{n}_2\cdot d^2\Omega_1$$ Expand-out from the definition of a solid angle from spherical coordinates. $$I_1 ~dt~ d\lambda ~d^2A_1~\hat{n}_1\cdot \frac{d^2A_2~\hat{n}_2}{r^2} = I_2 ~dt ~d\lambda~ d^2A_2~\hat{n}_2\cdot \frac{d^2A_1~\hat{n}_1}{r^2}$$ Some algebra leaves us with the desired result: spectral radiance is conserved (vanilla radiance, too!) $$I_1 ~dt ~d\lambda ~ d^2A_1~\hat{n}_1\cdot d^2A_2~\hat{n}_2 \frac{1}{r^2} = I_2 ~dt ~d\lambda ~d^2A_1~\hat{n}_1\cdot d^2A_2~\hat{n}_2 \frac{1}{r^2}$$

Said slightly differently, the spectral radiance is the amount of energy received by an area ($d^2A$) with some orientation relative to incidence ($\cos{\theta}$) and some a sensitivity to photons over a specific range of wavelengths ($d\lambda$) from a specific angular area of the sky ($d^2\Omega(\theta,\phi)$) during some time ($dt$).

For angularly unresolved sources, we are forced to fall back on the radiant spectral flux, introduced above. that the energy we received in each pixel is the integral of all the spectral radiance that emerged from the source parallel to our line of sight. Since we only can hope to get components of radiance from one hemisphere of the radiance field generated by the source, we can integrate over half of the sphere ($\theta\in[0,\frac{\pi}{2}], \phi\in[0,2\pi]$).

$$\int_{2\pi}I(\lambda, \theta, \phi)\cos{\theta}\cdot d^2\Omega(\theta,\phi) = \int_{2\pi}\frac{d^6 E}{d\lambda\cdot dt\cdot d^2A \cos{\theta}\cdot d^2\Omega(\theta,\phi)}\cos{\theta}\cdot d^2\Omega(\theta,\phi)$$

Note that this is the first angular moment of the spectral radiance field. And it is defined to be the spectral flux.

$$F(\lambda) = \frac{d^4 E}{d\lambda\cdot dt\cdot d^2A_{\perp}}$$

The radiative, spectral flux has units $[F(\lambda)]=J\cdot (m\cdot t\cdot m^2)^{-1}=W\cdot (m\cdot m^2)^{-1}$.

Each pixel effectively integrates this radiant spectral flux 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 proportional to a voltage they generate as counts.

We can discuss this radiant energy in terms of photon flux.

Recall from the discussion on 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.

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 ($[E]=~ J$)⁴⁾, 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.

BLAH THIS PAGE IS NOT DONE. I need to explain how to photometrically calibrate still with this background out of the way … maybe this is another page.