Energy, Counts, Flux

– under construction –

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 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 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.

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 electrons at a rate assumed to be linear with respect to the integration-time.

$$E_{\gamma}~d\lambda = \frac{hc}{\lambda}d\lambda$$

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 pixel received 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.

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

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 ($d\lambda$).

Radiant energy ($E_\gamma$) is energy delivered by electromagnetic radiation. For our purposes, it has units of Joules ($[E_\gamma] = J$)¹⁾. While in this documentation, we will find it to be transformed to mechanical or other forms of energy, $E_\gamma$ obeys the normal conservation laws of energy, given an appropriately defined system. For convenience, we will drop the $\gamma$ and refer to the radiant energy as $E$.

Energy is a convenient quantity to start this discussion with because we can lean on a quantum mechanical description of radiant energy to help us understand the effect radiant energy has on a detector.

$$E(\lambda) = n\frac{hc}{\lambda} = \int_{\lambda}^{\lambda+d\lambda}E(\lambda)d\lambda$$

Once this radiant energy is incident on a detector pixel-well, it liberates photoelectrons that are stored in small storage circuits assigned to each pixel-well ²⁾. Also in each pixel-well are electrons resulting from thermal agitation during integration time in addition to a 'jolt' of electrons that were liberated once the pixel-well was energized by the readout circuitry.

At readout time, these electrons are converted to digital units via the analog-to-digital converter (ADC). The resulting digital units are called many things … analog-digital-units (ADU), counts, LUM, and probably others! These counts are what appear in any pixel value readout in SharpCap or similar software.

Remember the extra electrons mentioned above? Their anomalous contribution to the counts value reported to the imaging software can be accounted for. Even after these two sources of extra electrons (now converted to counts) have been removed, there are still non-uniformities introduced by the detector. These three effects are “calibrated” away by calibration frames discussed on the Data Reduction to Telescope page. After these calibrations have been applied, we arrive at a number of counts that should be proportional to the radiant energy incident on the telescope primary optical device.

Now, we are left with counts that should be

While $E$ is a physical quantity of interest in many astronomical endeavors, it is not directly measurable. More directly measurable is the radiant energy that falls into a detector pixel that is sensitive to it per unit time. The detector pixel has an area with unit normal $\theta$ measured from the direction of any incoming energy. The detector pixel is also made of some material that is sensitive to only a specific range of frequencies($\nu$) or wavelengths($\lambda$). For more on how this material works, see the page: Ideas in Semiconductor Physics. So the more directly measurable quantity is the radiative, spectral flux.

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

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

In the cases where a target is spatially resolved on our detector and therefore angularly resolved on the sky, we can go deeper to define the radiative, spectral radiance³⁾. $$I(\lambda, \theta, \phi) = \frac{d^6 E}{d\lambda\cdot dt\cdot d^2A \cos{\theta}\cdot d^2\Omega(\theta,\phi)}$$

The spectral radiance has units $[I(\lambda, \theta, \phi)]=J\cdot (m\cdot t\cdot m^2\cdot sr^2)^{-1}$