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.

S/N Ratios for Atmospheric Speckle Limited Observations

Get photons per sec from a model spectrum