# The Math

### Planet-Star contrast for a Lambertian sphere

Contrast between and planet and its star (i.e. the ratio of planet flux to star flux, $F_p/F_s$) in reflected light depends on numerous factors including the size of the planet, planet-star separation, characteristics of the atmosphere, presence/absence of clouds, system geometry (phase). However as a first pass we can approximate the contrast by treating the planet as a Lambertian sphere, where apparent brightness is constant and incident starlight is reflected isotropically. Then, contrast as a function of phase ($\alpha$) is: $$C(\alpha) = \frac{2}{3} A_g(\lambda) \left(\frac{R_p}{r}\right)^2 \left[\frac{\sin\alpha + (\pi - \alpha)\cos\alpha}{\pi} \right]$$ where
$C(\alpha)$ is planet-star contrast
$A_g(\lambda)$ is geometric albedo
$R_p$ is planet radius
$r$ is planet-star separation in the orbit plane ("true" separation)
And phase as a function of orbital elements is given by:
$$\alpha = \cos^{-1} \left(\sin(i) \;\times\; \sin(\theta + \omega_p)\right)$$ where
$\omega_p$ is argument of periastron
$i$ is inclination, with i=90 being edge on and i = 0,180 being face on
$\theta$ is the true anomaly with $$\theta = 2 \tan^{-1} \left(\sqrt{\frac{1+e}{1-e}} \tan(E/2) \right)$$ where
$e$ is the eccentricity
$E$ is the eccentricity anomaly
with $$M = E - e \sin E$$ $$M = 2\pi \frac{\Delta t}{P}$$ where
$M$ is the mean anomaly
$\Delta t$ is the time since periastron passage
$P$ is the orbital period
(see Cahoy et al. 2010 Eqn 1, Sobolev 1975)

This plot shows 100s of the nearest ($<$70 pc) known RV-detected planets in the Exoplanet Archive (as of Aug 2023), plotted as a function of separation, contrast, and phase for GMagAO-X on the GMT. For planets without inclinitation values in the Archive, we used inclination = $60^{o}$, the average inclination for a uniform half-sphere. If radius was not available in the Exoplanet Archive, we used a Mass/Radius relation; if mass was not available we used Msini. Separation is in units of $\lambda$/D, the fundamental length scale for direct imaging (1 $\lambda$/D $\approx$ FWHM of PSF core). The separation, phase, and contrast shown are the "typical" (contrast-weighted average) values (How that is computed)

This plot is interactive. Hovering your mouse over each point will give the planet information. The sliders show the effect of changing the geometric albedo $A_g$, the wavelength $\lambda$, and the primary mirror size of the telescope. The default values are for observations of $A_g$ = 0.3 Lambertian spheres in $i^{\prime}$ ($\lambda$ = 800nm) with the GMT (diameter = 25.4m). The grey dashed line marks $\lambda$/D = 2, a typical inner working angle for direct imaging; planets leftward of this line will not be detectable.

Bokeh Plot

### S/N Ratios for Atmospheric Speckle Limited Observations

#### Get photons per sec from a model spectrum

PICASO and other models return the flux from the surface of the object, the planet in this case. So we need to scale the flux to that arriving at the observer. $$F = I \Omega$$ where $F$ = flux at Earth, $I$ = model intensity, and $$\Omega = \frac{R_p^2}{D^2}$$ where $R_p$ = planet radius and $D$ = distance

Next convert ergs cm$^{-1}$ s$^{-1}$ cm$^{-2}$ to photons s$^{-1}$.
Energy per photon per wavelength: $$E\,[ergs] = \frac{hc}{\lambda[cm]}$$ Number of photons per wavelength: $$n_{\gamma}\left[\frac{\gamma}{cm\,s\,cm^2}\right] = \frac{F_{\lambda}(\lambda)\left[\frac{ergs}{cm\,s\,cm^2}\right]}{E\,[ergs]}$$ Then the total flux in the filter is the sum over all wavelengths of the flux times the filter transmission curve: $$\mathrm{Total \; flux} \;[\gamma \;s^{-1} cm^{-2}] = \sum(F_\lambda(\lambda)\; [\gamma \;cm^{-1} s^{-1} cm^{-2}] \times R(\lambda) \times \delta\lambda \;[cm] )$$ where $R(\lambda)$ is the filter transmission curve as a function of wavelength, and $\delta\lambda$ is the interval the spectrum is sampled in.

Now multiply by the telescope collecting area: $$\mathrm{Total \; flux} \;[\gamma \;s^{-1}] = \mathrm{Total \; flux} \;[\gamma \;s^{-1} cm^{-2}] \times \pi r^2$$ where $r$ is the radius of the primary mirror.
Done. Total flux in a filter in photons per second.

#### Noise in the atmospheric speckle limited regime

For a derivation of this and discussion of speckles in AO+coronagraph ground-based systems, see Males et al. 2021.
The noise in a single resolution element (1 $\lambda$/D) located at the planet's separation and position angle from the star ($\overrightarrow{r_p}$) can be written as: $$\sigma^2 = \underbrace{I_* \Delta t}_\text{Star poisson noise} \left[\underbrace{I_c + I_{as} + I_{qs}}_\text{Poisson noise from star halo at planet location} + \underbrace{I_*[\tau_{as}(I_{as}^2 + 2[I_c I_{as} + I_{as} I_{qs}])}_\text{Atm speckles} + \underbrace{\tau_{qs}(I_{qs}^2 + 2I_c I_{qs})]}_\text{Quasistatic speckles} \right]\;\; + \\ \underbrace{I_p \Delta t}_\text{Planet poisson noise} \;\; + \underbrace{I_{\rm{sky}}\Delta t N_{\rm{pix}}(\lambda)}_\text{Sky background poisson noise} \;\; + \underbrace{\left(RN \frac{\Delta t}{t_{\rm{exp}}}\right)^2}_\text{Read noise} \;\; + \underbrace{I_{dc}\Delta t N_{\rm{pix}}(\lambda)}_\text{dark current}$$ where:
• $I_*$ is the peak star intensity without a coronagraph in photons/sec
• $I_c$ is the fractional contribution of intensity from residual diffraction from coronagraph,
• $I_{as}$ is the contribution from atmospheric speckles,
• $I_{qs}$ is contribution from speckles caused by instrument imperfections ("quasi-static" speckles),
• $\tau_{as}$ is the average lifetime of atmospheric speckles,
• $\tau_{qs}$ is the average lifetime of quasi-static speckles,
• $I_p$ is the planet intensity in photons/sec,
• $I_{sky}$ is the average sky background count rate,
• RN is the read noise,
• $I_{dc}$ is the dark current count rate,
• $\Delta t$ is the observation time,
• $t_{exp}$ is the exposure time of a single frame.
• $N_{pix}$ is the number of pixels within the area of a circle of a 1 $\lambda$/D radius,
• with $A_{\rm{\lambda/D}} \rm{[mas]} = \pi r^2, r = 0.5\lambda/D,\; \lambda/D \rm{[mas]} = 0.2063 \frac{\lambda [\mu m]}{D [\rm{m}]} \times 10^{-3}$ and A$_{pix}$ = pixel side length [mas$^2$], then $N_{pix} = A_{\rm{\lambda/D}} \rm{[mas]} / A_{pix} \rm{[mas]}$.
This is Males et al. 2021 Eqn 7 plus the typical noise terms.
If you assume, as in Males+ 2021, that you have a perfectly functioning coronagraph and instrument such that $I_{qs}$ and $I_{c}$ terms are negligible compared to the atmospheric speckle terms, we are then in the speckle-noise limited regime. Additionally, for the purposes of these calculations I will assume that the sky, read noise, and dark current are all negligible compared to the speckles. So: $$\sigma^2 = I_* \Delta t \left[I_{as} + {I_*\tau_{as}I_{as}^2}\right]\;\; + I_p \Delta t$$ and S/N becomes: $$S/N \approx \frac{I_p \Delta t}{\sqrt{I_* \Delta t \left[I_{as} + {I_*\tau_{as}I_{as}^2}\right]\;\; + I_p \Delta t}}$$ and then: $$\Delta t = \left(\frac{S/N}{I_p}\right)^2 \left[I_* \left(I_{as} + {I_*\tau_{as}I_{as}^2}\right) + I_p \right]$$