THE DIRECT DETECTABILITY OF GIANT EXOPLANETS IN THE OPTICAL

At optical wavelengths (∼0.4–1.0 μm), the flux from a giant exoplanet is dominated by reflected starlight, provided the planet is sufficiently old and of low enough mass that its thermal emission is negligible (see Section 5.3 for a discussion of thermal emission from young EGPs). Observationally, the wavelength- and phase-dependence of this reflection is encompassed by the geometric albedo spectrum, ${A}_{g}(\lambda )$, and the classical phase function, Φ(λ, α). Here, λ is the photon wavelength, and α is the exoplanet-centric angle between the star and the observer. In terms of these parameters, the planet/star flux ratio is given by

$$\begin{eqnarray}&&\displaystyle \frac{{F}_{p}}{{F}_{\star }}={A}_{g}(\lambda ){\left(\displaystyle \frac{{R}_{p}}{d}\right)}^{2}{\rm{\Phi }}(\lambda ,\alpha ),\end{eqnarray} \tag{ 1 }$$

where R_p is the planet radius and d is the orbital distance (see Appendix, Equation (8)). Φ(α, λ) is normalized to unity at full phase (α = 0°), which defines the geometric albedo. The planet's atmospheric composition, temperature/pressure structure, cloud/haze profiles, and scattering properties govern ${A}_{g}(\lambda )$ and Φ(λ, α). Hence, observations of optical planet/star flux ratios, as a function of wavelength and phase angle, have great potential to constrain the theory of exoplanet atmospheres (e.g., Marley et al. 1999; Sudarsky et al. 2000; Burrows et al. 2004; Cahoy et al. 2010). Furthermore, the shape and magnitude of an exoplanet's phase-folded light curve are highly sensitive to the Keplerian elements of its orbit (Dyudina et al. 2005; Sudarsky et al. 2005; Kane & Gelino 2010, 2011; Madhusudhan & Burrows 2012), providing an independent method for measuring orbital parameters that complements astrometric and radial-velocity methods.

In this work, we are interested in the general character, as opposed to detailed quantitative predictions, of the direct detectability of EGPs. We, therefore, generally assume planets with constant A_g (the exception is in Section 5.1.5, where we vary A_g with orbital distance using two simple models) and use the formalism presented in Madhusudhan & Burrows (2012) to calculate the associated single-scattering albedo (ω) and phase function for an assumed scattering mechanism. In Table 1, we have tabulated A_g as a function of ω for scalar and vector Rayleigh scattering, Lambert reflection, and isotropic scattering. Unless stated otherwise, we will hereafter refer to vector Rayleigh scattering as Rayleigh scattering.

The geometric albedo for a Lambert surface has a well-known analytic expression: ${A}_{g}(\omega )=(2/3)\;\omega$. Analytic fits to ${A}_{g}(\omega )$ for isotropic (iso) and scalar/vector (sca/vec) Rayleigh scattering are:

$$\begin{eqnarray}&&{A}_{g}^{\mathrm{iso}}=0.6896\displaystyle \frac{(1+0.4380s)(1-s)}{(1+1.2966s)(1+0.7269s)},\end{eqnarray} \tag{ 2 }$$

$$\begin{eqnarray}&&{A}_{g}^{\mathrm{sca}}=0.8190\displaystyle \frac{(1+0.1924s)(1-s)}{(1+1.5946s)(1+0.0886\omega )},\end{eqnarray} \tag{ 3 }$$

$$\begin{eqnarray}&&{A}_{g}^{\mathrm{vec}}=0.6901\displaystyle \frac{(1+0.4664s)(1-s)}{(1+1.7095s)(1-0.1345\omega )},\end{eqnarray} \tag{ 4 }$$

where $s\equiv \sqrt{1-\omega }$. These fits are accurate to within 0.5% for all ω, with the largest discrepancies corresponding to ω < 0.4² . Fits such as these are particularly useful for efficient theoretical calculations of the geometric albedo, as well as for retrieving a representative atmosphere-averaged scattering albedo from an observed geometric albedo.

It is common to assume the Lambert phase function when modeling reflected-light observations. However, this assumption neglects the albedo- and wavelength-dependence of physical phase curves, and generally predicts higher values than the more physically motivated Rayleigh phase function (Madhusudhan & Burrows 2012). Figure 1 compares the phase curves for scattering due to Rayleigh, isotropic, and Lambert scattering. We see that Rayleigh scattering phase curves are systematically lower than isotropic and Lambert phase curves for $\alpha \;\lesssim$ 120°, differing by as much as a factor of ∼2 for phase angles in the range 60° ≲ α ≲ 90° (see also, Madhusudhan & Burrows 2012). For most phase angles, increasing ω results in better agreement between these scattering processes. Rayleigh phase curves generally increase with increasing ω, whereas isotropic scattering phase curves show the opposite trend. In Section 5.1.1, we compare predictions of the direct detectability of EGPs with Rayleigh, isotopic, and Lambert phase functions.

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2.1.1. Keplerian Elements

2.1.2. Albedo Spectra from Homogeneous Atmospheres

Semi-infinite homogeneous atmospheres are characterized by an average single-scattering albedo spectrum: $\omega (\lambda )\equiv {\kappa }_{s}/({\kappa }_{s}+{\kappa }_{a})$, where ${\kappa }_{s}$ is the scattering opacity, and ${\kappa }_{a}$ is the absorptive opacity. In this section, we provide illustrative scattering albedo spectra for cool EGPs, which will aid in the interpretation of some of the results presented in this paper. Our calculations use the opacity database from Sharp & Burrows (2007) and the equilibrium chemical abundances from Burrows & Sharp (1999).

From its definition, it is clear that ω(λ) is a function of atmospheric parameters such as temperature, pressure, and composition. In Figure 2, we demonstrate the characteristic strong dependence of scattering albedos (bottom row) and the corresponding Rayleigh scattering geometric albedos (top row) upon temperature and metallicity. The models are cloud-free, homogeneous, semi-infinite, and use opacities at a pressure of 0.5 atmospheres. For the temperatures and pressures relevant to cool EGPs, we find ω(λ) is relatively insensitive to variations in atmospheric pressure. In general, higher temperatures/metallicities result in lower albedos, particularly for wavelengths longward of ∼0.65 μm.

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The spectra in Figure 2 show prominent methane features near ∼0.62, ∼0.74, ∼0.81, and ∼0.89 μm, ammonia spectral bands near ∼0.65 and ∼0.79 μm, and a broad water band at ∼0.94 μm. Although there is clearly some degeneracy between metallicity and temperature, this figure suggests optical flux ratio measurements, particularly between ∼0.5 and 0.6 μm and/or ∼0.82 and ∼0.94 μm, have great potential as temperature/composition diagnostics.

Given a single-scattering albedo spectrum, one can use the formalism presented in Madhusudhan & Burrows (2012) to calculate the phase function for an assumed scattering process as a function of phase angle and wavelength. Note that the often-assumed Lambert phase function is independent of ω(λ), and hence λ. In Figure 3, we show the phase function for Rayleigh scattering as a function of wavelength and phase angle, where we have assumed the scattering albedo spectrum shown as the black line in the bottom left panel of Figure 2. As expected, the phase curve is equal to one for all λ at α = 0°, but note the variation in Φ(λ, α) as α increases from 0° to 170°. In contrast, a Lambert surface would result in featureless, horizontal lines for all α.

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In addition to the planet/star contrast ratio, the planet/star angular separation is an essential parameter in determining the feasibility of the direct detection of reflected light from an exoplanet. For coronagraphic observations, the figure of merit is given as an IWA and OWA. An IWA (OWA) is the minimum (maximum) planet/star angular separation that can be resolved by a given instrument. Using the formalism outlined in the appendix, the projected angular separation, as a function of phase angle and time, can be obtained via the following equation:

$$\begin{eqnarray}&&\delta \phi =\displaystyle \frac{d}{{D}_{\oplus }}\sqrt{{\mathrm{cos}}^{2}(\theta +{\omega }_{p})+{\mathrm{sin}}^{2}(\theta +{\omega }_{p}){\mathrm{cos}}^{2}i},\end{eqnarray} \tag{ 5 }$$

where ${D}_{\oplus }$ is the system's distance from Earth and θ is the planet's orbital angle as measured from periastron (the true anomaly). Inspection of the above equation reveals that, as with the planet/star flux ratio, the time dependence of the planet/star angular separation will depend sensitively upon the Keplerian orbital elements.

As mentioned in Sections 2.1.1 and 2.2, observed flux ratios and separations are strong functions of the Keplerian orbital elements. Figure 4 demonstrates this dependence for circular orbits with many different inclinations (left column) and face-on orbits with many different eccentricities (right column). The top row shows planet/star flux ratios, and the bottom row shows the corresponding angular separations for a system located 10 pc from Earth. All calculations in this figure assume a Jupiter-like planet in an orbit with a = 5 AU and ${\omega }_{p}$ = 0°.

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Although it has been pointed out previously in the literature, it is worth reemphasizing that direct-imaging observations are well-suited for studying face-on systems (the opposite is true for the radial-velocity and transit methods, which are biased toward high-inclination systems). The reason for this is that an exoplanet on a circular, face-on orbit with ${F}_{p}/{F}_{\star }\gt {C}_{\mathrm{min}}$ and $\delta \phi \;\gt$ IWA will be directly detectable at all times (note we are assuming $\delta \phi \;\lt$ OWA). If, on the other hand, its inclination is ∼90°, both the planet/star flux ratio and angular separation will drop below their detection thresholds for some fraction of the orbit as the planet passes through the full range of possible phase angles. For instance, if ${C}_{\mathrm{min}}={10}^{-10}$ and $\mathrm{IWA}=0\buildrel{\prime\prime}\over{.} 3$, then a face-on orbit (flat purple lines in Figure 4) will be well above the detection thresholds at all times. However, as we increase the inclination in the left column of this figure, we see that the flux ratio dips below ${C}_{\mathrm{min}}$ for an increasingly larger fraction of the orbit centered at new phase (α = 180°). Similarly, the minimum planet/star angular separation decreases with increasing inclination, falling below the IWA for an increasingly larger fraction of the orbit centered at new phase and full phase. Although the flux ratio near full phase is largest for high-inclination orbits, the planet will still be undetectable due to the small planet/star angular separation.

In the right column of Figure 4, we show the effect of increasing the eccentricity of a face-on orbit from circular (e = 0) to highly eccentric (e = 0.8). In this case, the phase variations are due to the time dependence of the orbital distance. On one hand, a planet on a highly eccentric orbit can be an excellent direct-imaging target, as it spends a large fraction of its orbit at larger orbital distances from its parent star, and for certain sets of Keplerian elements, this can provide angular separations that are highly favorable for direct-imaging observations. This, however, comes at the cost of lower flux ratios throughout most of its orbit and very small angular separations when it is near periastron passage.

As has been pointed out by Kane & Gelino (2011), the interaction between the argument of periastron and inclination in $({F}_{p}/{F}_{\star })-\delta \phi$ parameter space can be quite complex. Figure 5 shows various combinations of these parameters for a Jupiter-like planet on an eccentric orbit with e = 0.6 and a = 5 AU. Each column shows planet/star flux ratios and angular separations for orbits with inclinations that vary from 0° to 80°, with the argument of periastron held constant at the value indicated in each top left corner. Note we use the convention that when α = θ = 0°, ${\omega }_{p}=i=90^\circ$. In other words, periastron passage coincides with full phase when the argument of periastron is 90° and with new phase when it is 270°.

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The detectability of planets on face-on orbits is independent of ${\omega }_{p}$, as can be seen by the constant purple curve in all panels of Figure 5. As we increase the inclination, the flux ratio can either increase or decrease, depending upon the periastron argument and phase of the orbit. For ${\omega }_{p}=90^\circ$, the flux ratio decreases with inclination for most of the orbit as the planet transitions from being permanently oriented at quarter phase when i = 0° to displaying nearly all phase angles when i = 80°. At full phase, which corresponds to periastron passage, the flux ratio achieves its maximum for this set of parameters. However, as is typically the case, this maximum flux ratio corresponds to a minimum planet/star angular separation that, in the limit of a perfectly edge-on system, drops to zero at the time of periastron passage. Interesting behavior also occurs when ${\omega }_{p}$ = 270°. For this periastron argument, periastron passage corresponds to new phase, and, in exactly the opposite sense from the ${\omega }_{p}$ = 90° case, the flux ratio increases with inclination for most of the orbit and results in a minimum at the time of periastron passage for high-inclination orbits. For orbits with i ≳ 50°, the minimum angular separation actually coincides with the minimum flux ratio, and the maximum angular separation coincides with the relatively large fraction of the orbit with a large flux ratio.

Each random orbit requires the generation of four Keplerian elements: $\{a,e,i,{\omega }_{p}\}$. To accomplish this, we assume planetary orbital orientations are isotropically distributed with respect to the observer, which leads to periastron arguments distributed uniformly in $[0,2\pi ]$ and $\mathrm{cos}i$ distributed uniformly in [−1, 1]. The semimajor axis and eccentricity distributions must be determined empirically. Most of our knowledge of these distributions has come from radial-velocity (Cumming et al. 2008; Howard et al. 2010) and transit (Batalha et al. 2013; Fressin et al. 2013) surveys that are biased toward short-period planets and have only been in operation long enough to fully describe planets with periods up to a few thousand days.

Nevertheless, we assume both the semimajor axis and eccentricity distributions derived from radial-velocity planets can be extrapolated to wider separations. For the semimajor axis distribution, we adopt a power-law, ${dN}/{da}\propto {a}^{\beta }$, where our fiducial calculations assume β = −0.61 (Cumming et al. 2008). This distribution is based on radial-velocity planets with orbital periods in the range 2–2000 days, corresponding to semimajor axes in the range 0.03–3.0 AU. Results from direct-imaging surveys have been used to set model-dependent limits on the maximum semimajor axis (${a}_{\mathrm{max}}$) to which the radial-velocity distribution can be extrapolated. Using MC simulations to examine the null results from direct-imaging searches for EGPs around 118 stars, Nielsen & Close (2010) determined ${a}_{\mathrm{max}}$ ∼ 65–234 AU, depending on the assumed planet luminosity model. Brandt et al. (2014) performed a statistical analysis of direct-imaging data from a sample of nearly 250 stars, finding ${a}_{\mathrm{max}}$ ∼ 30–100 AU, which again depends on the adopted cooling model. With these studies as our motivation, our fiducial calculations assume the semimajor axis distribution is given by a power-law with a minimum ${a}_{\mathrm{min}}=0.03$ AU and a maximum ${a}_{\mathrm{max}}=65$ AU.

For the eccentricity distribution, we use the two-parameter Beta distribution given by Kipping (2013):

$$\begin{eqnarray}&&{P}_{\beta }(e;a,b)=\displaystyle \frac{{\rm{\Gamma }}(a+b)}{{\rm{\Gamma }}(a){\rm{\Gamma }}(b)}{e}^{a-1}{(1-e)}^{b-1},\end{eqnarray} \tag{ 6 }$$

where Γ is the Gamma function. Our fiducial calculations assume $a=1.12$ and $b=3.09$, which are the long-period parameters given by Kipping (2013), who derived this distribution by regressing the known cumulative density function of orbital eccentricities from exoplanets detected from the radial-velocity method. The orbits of planets with $a\lt 0.1$ AU are expected to be rapidly circularized due to strong tidal forces. Although we do not expect the eccentricities of such planets to influence our results, we set e = 0 for any planet with $a\lt 0.1$ AU.

We find the direct detectability of EGPs is highly sensitive to the assumed semimajor axis distribution, but relatively insensitive to the eccentricity distribution. Our fiducial orbital parameter distributions are summarized in Table 2.

Table 2.  Fiducial Orbital Parameter Distributions

Paramter	Range	Distribution
a	[0.03, 65 AU]	${dN}/{da}\propto {a}^{\beta }$
e	[0,1)	${P}_{\beta }(e;a,b)$ a
$\mathrm{cos}i$	[−1, 1]	Uniform
${\omega }_{p}$	[0, 2π]	Uniform

Notes. For our fiducial semimajor axis distribution, we assume a power-law index β = −0.61 (Cumming et al. 2008). For our fiducial eccentricity distribution, we assume a = 1.12 and b = 3.09, which are the long-period parameters given in Kipping (2013). We set e = 0 for any planet with $a\lt 0.1$ AU.

^aThis is the Beta distribution from Kipping (2013) (Equation (6)).

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Here, we present a parametric study of the mean observability fraction, $\langle \;{f}_{\mathrm{obs}}\rangle$, for different sets of MC experiments, each composed of $5\times {10}^{5}$ random orbits. For a given population of planets, $\langle \;{f}_{\mathrm{obs}}\rangle$ can be interpreted as a detection probability, assuming all stars have one such planet. In Figure 7, we show the distribution of ${f}_{\mathrm{obs}}$ as a function of orbital parameters for three illustrative MC experiments. Each column corresponds to an independent experiment, where we have assumed ${C}_{\mathrm{min}}={10}^{-9}$ (left column), ${C}_{\mathrm{min}}=5\times {10}^{-10}$ (middle column), and ${C}_{\mathrm{min}}={10}^{-10}$ (right column). In all three cases, we use our fiducial orbital parameter distributions, and calculate ${f}_{\mathrm{obs}}$ for direct observations of a Jupiter-like planet orbiting a star at a distance of ${D}_{\oplus }=10$ pc from Earth, using an instrument with $\mathrm{IWA}=0\buildrel{\prime\prime}\over{.} 2$, $\mathrm{OWA}=1\buildrel{\prime\prime}\over{.} 75$, and the minimum contrast specified at the top of each column.

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In a given panel, each dot represents a single orbit, and the color-scale shows the observability fraction for the indicated assumptions about the planet and direct-imaging observatory. For a fixed minimum contrast, the semimajor axis has by far the largest effect on ${f}_{\mathrm{obs}}$, creating highly clustered regions of low and high values of ${f}_{\mathrm{obs}}$. The sharp drop in observability interior to 2 AU corresponds to an angular separation of $0\buildrel{\prime\prime}\over{.} 2$, which is the assumed IWA. Note, however, that eccentric orbits with $a\lt 2$ AU are able to achieve nonzero observability fractions, since a planet on such an orbit spends a large fraction of its time near orbital distances of $a(1+e)$, making it possible for the observed angular separations to fall outside the IWA, provided the full set of Keplerian elements allows for such geometries. These results make it clear that ${f}_{\mathrm{obs}}$ is very sensitive to the assumed semimajor axis distribution. Steeper (shallower) power-law indices will result in narrower (broader) regions of high ${f}_{\mathrm{obs}}$ values, and the assumed values of ${a}_{\mathrm{max}}$ and ${a}_{\mathrm{min}}$ determine the region within which the $5\times {10}^{5}$ planets are distributed. In addition, decreasing the minimum contrast increases the range of semimajor axes that are detectable and leads to an increasing number of orbits for which ${f}_{\mathrm{obs}}\sim 1.0$ (i.e., orbits that are observable at all times). For ${C}_{\mathrm{min}}={10}^{-9}$, low- to modest-eccentricity orbits with ${f}_{\mathrm{obs}}\gt 1\%$ are confined to $2\;\mathrm{AU}\lesssim a\lesssim 10\;\mathrm{AU}$. As we decrease ${C}_{\mathrm{min}}$ from left to right in this figure, we see orbits with increasingly larger semimajor axes achieve nonzero ${f}_{\mathrm{obs}}$, reaching $a\sim 15$ AU and $a\sim 35$ AU for ${C}_{\mathrm{min}}=5\times {10}^{-10}$ and ${C}_{\mathrm{min}}={10}^{-10}$, respectively.

As mentioned previously, low-inclination orbits are beneficial to direct-imaging observations. In the top left panel, it is particularly clear that orbits with ${f}_{\mathrm{obs}}\sim 1.0$ are associated with low inclinations. However, planets with $i\gtrsim 50^\circ$ can have ${f}_{\mathrm{obs}}\sim 15\%$ out to a ∼ 10–30 AU, depending on ${C}_{\mathrm{min}}$, since phase angles near full phase become observationally accessible. There is also an interesting feature near ${\omega }_{p}$ ∼ 270°, which corresponds to orbits with periastron passage occurring at new phase. Near this periastron argument, the region of large ${f}_{\mathrm{obs}}$ extends to semimajor axes as large as $\sim 20$ AU for ${C}_{\mathrm{min}}={10}^{-10}$. For a planet with ${\omega }_{p}$ ∼ 270°, the planet/star flux ratio rises with increasing inclination throughout most of its orbit, and the largest angular separations coincide with the highest flux ratios when $i\gtrsim 50^\circ$ (see Figure 5). These factors collectively increase the semimajor axes for which a planet on such an orbit can have a large observability fraction.

For the remainder of this section, we present the mean observability fraction, $\langle \;{f}_{\mathrm{obs}}\rangle$, for many MC experiments, which are run under different sets of assumptions. In each case, our goal is to study how the observability fraction depends upon input parameters such as albedo, planetary radius, distance from Earth, minimum achievable contrast, and the semimajor axis distribution. For each experiment, we vary one parameter with respect to a fiducial model, which is comprised of the following assumptions:

• 1.  
  the planetary atmospheres are Rayleigh scattering with ${A}_{g}=0.5$ (this corresponds to $\omega \approx 0.95$ for Rayleigh scattering),
• 2.  
  the planets have the radius of Jupiter and orbit stars located 10 pc from Earth,
• 3.  
  the direct-imaging observatory has an $\mathrm{OWA}=1\buildrel{\prime\prime}\over{.} 75$ and ${C}_{\mathrm{min}}={10}^{-9}$ (similar to what is anticipated for WFIRST/AFTA), and
• 4.  
  the Keplerian orbital elements are distributed according to the distributions given in Table 2.

Of course, the above assumptions do not fully represent realistic physical/statistical properties of planets or any future direct-imaging mission. Nevertheless, by varying each parameter of this very simple reference model, it is possible to gain insight into the general problem of directly detecting giant exoplanets in the optical, both in the case of a blind search and observations of stars known to host giant planets.

5.1.1. Dependence on Scattering Properties

We start by varying the first of our fiducial model assumptions. Figure 8 shows the dependence of $\langle \;{f}_{\mathrm{obs}}\rangle$ upon the geometric albedo (left panel) and the single-scattering albedo (right panel), assuming Lambert (black), isotropic (red), and Rayleigh (blue) scattering atmospheres. Each curve is composed of many distinct MC experiments, and the three sets of curves in each panel correspond to the indicated IWAs. The WFIRST/AFTA coronagraph is anticipated to have $\mathrm{IWA}\sim 0\buildrel{\prime\prime}\over{.} 2$. Note this value will depend upon the angular separation and wavelength of the observation. For each scattering mechanism, there is a one-to-one correspondence between ω and A_g (see Table 1). Additionally, the phase curves for isotropic and Rayleigh scattering are direct functions of ω, and the Lambert phase curve is independent of ω (see Figure 1). Thus, given A_g or ω, it is straightforward to calculate the product ${A}_{g}\cdot {\rm{\Phi }}(\alpha )$, which is proportional to the planet/star flux ratio, for an assumed scattering or effective scattering mechanism.

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In the left panel of Figure 8, we show the dependence of $\langle \;{f}_{\mathrm{obs}}\rangle$ upon A_g. For each MC experiment, we assume A_g and interpolate Table 1 to infer the corresponding ω, which is needed to calculate the phase curves for Rayleigh and isotropic scattering. For a given A_g, the scattering phase functions are entirely responsible for the differences seen between the assumed scattering processes. Note that, since the Lambert phase function is independent of ω, the phase function is the same for each black curve. As can be seen in Figure 1, Rayleigh scattering phase curves are systematically lower than isotropic and Lambert phase curves for $\alpha \;\lesssim$ 120°. It is for this reason that $\langle \;{f}_{\mathrm{obs}}\rangle$ is always the lowest for Rayleigh scattering. For most A_g, Rayleigh scattering predicts observability fractions that are $\sim 20\%$ below the predictions of isotropic and Lambert scatting, regardless of the IWA. At the anticipated IWA for WFIRST/AFTA, $\langle \;{f}_{\mathrm{obs}}\rangle$ decreases sublinearly with A_g for each scattering process, with maximum and minimum values for Rayleigh scattering of $\sim 11\%$ for ${A}_{g}=0.65$ and $\sim 1\%$ for ${A}_{g}\lesssim 0.1$. For context, Figure 2 suggests that cool EGPs may have large geometric albedos (${A}_{g}\sim 0.6-0.8$) in the optical, particularly for wavelengths $\lesssim 0.6\mu {\rm{m}}$ and near the broad water band at ∼0.94 μm. As expected, these results suggest planetary geometric albedos can have a dramatic effect on the results of an optical direct-imaging survey.

In the right panel of Figure 8, we show the dependence of $\langle \;{f}_{\mathrm{obs}}\rangle$ upon ω. In contrast to the left panel, here we assume ω for each MC experiment and use Table 1 to infer the corresponding A_g. In this case, both A_g and Φ(α) contribute to the differences between the assumed scattering processes. We see that $\langle \;{f}_{\mathrm{obs}}\rangle$ is comparable for Rayleigh and isotropic scattering for all ω. This arises from the fact that, for a given ω, A_g for Rayleigh scattering is systematically higher than that for isotropic scattering, whereas Φ(α) for Rayleigh scattering is systematically lower, leading to roughly consistent observability fractions. For conservative scattering ($\omega =1.0$), the geometric albedos for Lambert, isotropic, and Rayleigh scattering differ by less than $\sim 13\%$, and $\langle \;{f}_{\mathrm{obs}}\rangle$ is essentially independent of the assumed scattering mechanism. However, for non-conservative scattering (ω < 1.0), the geometric albedos for isotropic and Rayleigh scattering can be less than those for Lambert scattering by as much as a factor of ∼4. As a consequence, Lambert scattering generally predicts significantly higher $\langle \;{f}_{\mathrm{obs}}\rangle$ than both isotropic and Rayleigh scattering.

As a specific example, consider ω = 0.6. For this scattering albedo, the Lambert phase function corresponds to a geometric albedo that is a factor of ∼3 larger than isotropic scattering and a factor of ∼2 larger than Rayleigh scattering. At the WFIRST/AFTA IWA of 02, this results in $\langle \;{f}_{\mathrm{obs}}\rangle$ ∼ 3% for Rayleigh and isotropic scattering and $\langle \;{f}_{\mathrm{obs}}\rangle$ ∼ 10% for Lambert scattering. Thus, when the geometric albedo is calculated self-consistently with the single-scattering albedo, the assumed scattering phase function can greatly influence the predicted yield from an optical direct-imaging survey. In particular, assuming the Lambert phase function generally results in an overestimate of the detectability of giant planets with respect to the more physically motivated Rayleigh phase function.

5.1.2. Dependence on Planet Radius

The planet/star flux ratio is proportional to the square of the planet radius. Therefore, we expect $\langle \;{f}_{\mathrm{obs}}\rangle$ to be sensitive to small variations in planet size, which we have heretofore assumed to be that of Jupiter. Figure 9 shows $\langle \;{f}_{\mathrm{obs}}\rangle$ as a function of planet radius, assuming observations made with a coronagraph that has $\mathrm{IWA}=0\buildrel{\prime\prime}\over{.} 1,0\buildrel{\prime\prime}\over{.} 2,\mathrm{and}\ 0\buildrel{\prime\prime}\over{.} 3$. All assumptions other than planet radius are as listed at the end of Section 5.1. We note that gas giants are expected to have $0.8{R}_{\mathrm{Jup}}\lesssim {R}_{p}\lesssim 1.2{R}_{\mathrm{Jup}}$, but we show a wider range of radii for illustrative purposes. Within the range of expected radii for gas giants, an increase in radius by a factor of ∼1.5 leads to an increase in $\langle \;{f}_{\mathrm{obs}}\rangle$ by a factor of ∼1.5–2.5, depending on the IWA. We find that smaller IWAs are less sensitive to variations in planet radius. This effect likely arises from the ${d}^{-2}$ dependence of the planet/star flux ratio, which—because of the increasing number of planets at smaller orbital distances that are accessible to smaller IWAs—reduces the weight carried by the ${R}_{p}^{2}$ dependence of the flux ratio.

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5.1.3. Dependence on Distance and Minimum Contrast

The fiducial model listed at the end of Section 5.1 assumes direct observations of planet/star systems located a distance ${D}_{\oplus }=10$ pc from Earth with a minimum achievable contrast ${C}_{\mathrm{min}}={10}^{-9}$. It is obvious that the direct detectability of EGPs will vary strongly with both of these parameters. In Figure 10, we investigate this variation in the context of the mean observability fraction. We show $\langle \;{f}_{\mathrm{obs}}\rangle$ for planet/star systems located 0.01–50 pc from Earth, assuming minimum achievable contrasts of 10⁻⁹ (dashed lines) and 10⁻¹⁰ (solid lines). The calculations assume a coronagraphic instrument with an $\mathrm{IWA}=0\buildrel{\prime\prime}\over{.} 1,0\buildrel{\prime\prime}\over{.} 2,\mathrm{and}\ 0\buildrel{\prime\prime}\over{.} 3$. All other parameters are as listed at the end of Section 5.1.

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WFIRST/AFTA is anticipated to have $\mathrm{IWA}\;\sim 0\buildrel{\prime\prime}\over{.} 2$ and ${C}_{\mathrm{min}}\;\sim {10}^{-9}$. In this case, Jupiter-like planets located 10 pc from Earth spend on average ∼10% of their time in observable configurations, whereas the same planets located 30 pc from Earth spend on average ∼0.1% of their time in observable configurations. If instead a minimum contrast of 10⁻¹⁰ can be achieved, then these fractions become ∼30% and ∼16%, respectively. At small distances from Earth, we see $\langle \;{f}_{\mathrm{obs}}\rangle$ increases with distance. This trend is the result of planets at sufficiently wide orbital distances to spend some fraction of their orbit outside the OWA, which we have assumed to be 175. Assuming a uniform distribution of stars and $\mathrm{IWA}=0\buildrel{\prime\prime}\over{.} 2$, Jupiter-like planets orbiting stars within 10, 30, and 50 pc from Earth have volume-averaged observability fractions of ∼12%, 3%, and 0.5% for ${C}_{\mathrm{min}}\;\sim {10}^{-9}$ and ∼28%, 20%, and 13% for ${C}_{\mathrm{min}}\;\sim {10}^{-10}$.

5.1.4. Dependence on the SemiMajor Axis Distribution

Our fiducial model assumes the semimajor axis distribution derived from radial-velocity planets can be extrapolated to wider separations ($a\gt 3$ AU). However, the validity of this assumption is highly uncertain, and, as evidenced by Figure 7, the semimajor axis distribution has a large influence on the distribution of observability fractions. In this section, we explore the dependence of $\langle \;{f}_{\mathrm{obs}}\rangle$ upon the semimajor axis distribution by varying each of its components: the minimum semimajor axis (${a}_{\mathrm{min}}$), the maximum semimajor axis (${a}_{\mathrm{max}}$), and the power-law index of the distribution (β).

Our fiducial model assumes ${a}_{\mathrm{min}}=0.03$ AU (note we set e = 0 for any planet with $a\lt 0.1$ AU), which has been inferred for radial-velocity planets (Cumming et al. 2008). In Figure 11, we show $\langle \;{f}_{\mathrm{obs}}\rangle$ as a function of ${a}_{\mathrm{min}}$, assuming the indicated IWAs. At a given IWA, the mean observability increases with increasing ${a}_{\mathrm{min}}$ until ${a}_{\mathrm{min}}/{D}_{\oplus }\sim \mathrm{IWA}$, where ${D}_{\oplus }=10$ pc in our fiducial model. Since all of the MC experiments use the same number of planets (5 × 10⁵), increasing the minimum semimajor axis forces more planets to exist at larger orbital separations, leading to larger $\langle \;{f}_{\mathrm{obs}}\rangle$ when ${a}_{\mathrm{min}}/{D}_{\oplus }\lesssim \mathrm{IWA}$ and smaller $\langle \;{f}_{\mathrm{obs}}\rangle$ when ${a}_{\mathrm{min}}/{D}_{\oplus }\gtrsim \mathrm{IWA}$. At the anticipated IWA of WFIRST/AFTA, the mean observability varies by ∼1% for ${a}_{\mathrm{min}}$ in the range 0.03–2.0 AU.

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As mentioned in Section 4.1, recent model-dependent limits on ${a}_{\mathrm{max}}$ fall in the range ∼30–100 AU (Brandt et al. 2014). In Figure 12, we test the sensitivity of $\langle \;{f}_{\mathrm{obs}}\rangle$ to a somewhat larger range of ${a}_{\mathrm{max}}$. For a fixed number of planets, smaller ${a}_{\mathrm{max}}$ values yield larger $\langle \;{f}_{\mathrm{obs}}\rangle$ values, which is sensible given that more planets will exist at smaller, but still detectable, orbital distances. As ${a}_{\mathrm{max}}$ increases, $\langle \;{f}_{\mathrm{obs}}\rangle$ slowly decreases as a result of the increasing number of planets at very large orbital distances, which are too faint to be detected and, in the case of nearly face-on orbits, fall outside the OWA. Over the range ${a}_{\mathrm{max}}\sim 30-100$ AU, $\langle \;{f}_{\mathrm{obs}}\rangle$ decreases by a factor of ∼1.6 for each IWA.

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The power-law index of the semimajor axis distribution of radial-velocity planets with periods in the range 2–2000 days has been constrained to be β = −0.61 ± 0.15 (Cumming et al. 2008). Our fiducial model assumes this index can be extrapolated to planets at much larger orbital separations. In Figure 13, we test the sensitivity of $\langle \;{f}_{\mathrm{obs}}\rangle$ to variations in the power-law index. As we steepen β toward more negative values, an increasing number of planets begin to pile up at small orbital separations, which leads to larger $\langle \;{f}_{\mathrm{obs}}\rangle$ values. At the anticipated IWA of WFIRST/AFTA, β = −0.61 and β = −0.85, which are the estimated indices from Cumming et al. (2008) and Brandt et al. (2014), produce $\langle \;{f}_{\mathrm{obs}}\rangle$ values that are nearly identical. Over the full range of β, the difference between the minimum and maximum $\langle \;{f}_{\mathrm{obs}}\rangle$ values is only ∼3%.

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5.1.5. Geometric Albedo as a Function of Distance

Thus far, we have assumed the geometric albedo is constant throughout the orbit of each planet. However, the atmospheric temperature of a planet varies strongly with orbital distance, and such variations determine whether water or ammonia clouds will form. We, therefore, expect the albedo spectrum to vary significantly with orbital distance (Sudarsky et al. 2005). We also expect the albedo spectrum to depend sensitively upon the planet's mass, age, and parent star. In this section, we compare our fiducial assumption of ${A}_{g}=0.5\;=$ constant with two simple models that account for the distance dependence of the albedo spectrum.

The first model is an interpolation of the models shown in Figure 9 of Sudarsky et al. (2005), which depicts the geometric albedo spectra of a Jupiter-mass, 5 Gyr planet orbiting a G2 V star at distances in the range 0.2–15 AU. In Figure 14, we show the predictions of this model for the distance dependence of the geometric albedo for various wavelengths, including wavelengths associated with methane (0.74 and 0.89 μm), ammonia (0.65 μm), and water (0.94 μm). We see the geometric albedo, and hence the planet/star flux ratio, is a strong, non-monotonic function of both distance and wavelength.

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Using this model for A_g, we carried out MC experiments for observations at the wavelengths shown in Figure 14, keeping all other parameters as listed at the end of Section 5.1. For orbital distances $d\gt 15$ AU, we set A_g equal to its value at 15 AU; this does not significantly influence our results, since planets at these separations contribute very little to $\langle \;{f}_{\mathrm{obs}}\rangle$ (see Figure 7). In Figure 15, we show $\langle \;{f}_{\mathrm{obs}}\rangle$ as a function of IWA assuming the "Sudarsky et al." albedo model (solid lines) and our fiducial assumption of ${A}_{g}=0.5\;=$ constant (solid black line). As can be seen in the first column of Figure 7, most planets with nonzero ${f}_{\mathrm{obs}}$ values have semimajor axes in the range 2–10 AU, so it is the variability of A_g in this region that has the strongest effect on $\langle \;{f}_{\mathrm{obs}}\rangle$. In the wavelength range we have considered, this model suggests the optimal observability fractions occur for λ ∼ 0.5–0.6 μm, which Figure 15 indicates is the effective bandpass of our fiducial assumption ${A}_{g}=0.5$. At the IWA of WFIRST/AFTA, direct observations in the wavelength range λ ∼ 0.5–0.6 μm of a Jupiter-like planet located 10 pc from Earth will, on average, be possible for ∼10% of its orbital period. We note that the observed reflected fluxes shortward of ∼0.6 μm from Jupiter and Saturn are less than what this model would predict, which is made manifest by the reddish hues of these solar-system planets (Karkoschka 1994; Burrows 2014).

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For the second model of the distance dependence of the albedo spectrum, we calculate the equilibrium temperature for each planet as a function of orbital distance in a self-consistent fashion with the albedo spectra. This is accomplished as follows. For each planet in an MC experiment, we assume the planetary atmosphere is homogeneous and Rayleigh scattering. In addition, solar metallicity and a pressure of 0.5 bar are assumed (albedo spectra for such a planet with T = 200 K are shown as black lines in the left column of Figure 2). At a given phase of the planet's orbit, we assume an initial Bond albedo and calculate the equilibrium temperature. Assuming this to be the temperature of the atmosphere, we generate (scattering, geometric, and spherical) albedo spectra with the methods described in Section 2.1.2. We then integrate the spherical albedo spectrum to obtain the Bond albedo and calculate a new equilibrium temperature (${T}_{\mathrm{eq}}$). We iterate this process until we have achieved self-consistency between the equilibrium temperature and albedo spectra. This procedure is performed at every phase of the planet's orbit, resulting in a geometric albedo spectrum and phase function that varies with orbital distance. The results of MC experiments assuming this "${T}_{\mathrm{eq}}$ model" are shown as dashed-lines in Figure 15. We see that this model is roughly consistent with the Sudarsky et al. model for λ < 0.7 μm, but its predictions are very different at longer wavelengths. This model also predicts that the wavelength range λ ∼ 0.5–0.6 μm offers the optimal observability fractions.

Given assumptions about the orbital parameter distributions and occurrence rates (f_p) of a population of planets, the mean observability fractions presented in this work can be translated into the probability that a blind search for such planets will yield a detection. Statistical studies of planets discovered by the radial-velocity and microlensing methods suggest that the frequency of giant planets with $a\lt 20$ AU falls in the range ${f}_{p}$ ∼ 10–20% (Cumming et al. 2008; Cassan et al. 2012). If we assume ${f}_{p}=15\%$, then calculations based on our fiducial model assumptions, which are listed at the end of Section 5.1, suggest that the probability of detecting a Jupiter-like planet in a blind search around a star located 10 pc from Earth is $p={f}_{p}\cdot \langle {f}_{\mathrm{obs}}\rangle \sim 1.5\%$, where we have assumed $\mathrm{IWA}=0\buildrel{\prime\prime}\over{.} 2$. If a minimum contrast of 10⁻¹⁰ is assumed, then this probability increases to p ∼ 4.2%. If the star is located 20 pc from Earth, we find p ∼ 0.4% for ${C}_{\mathrm{min}}={10}^{-9}$ and p = 3.2% for ${C}_{\mathrm{min}}={10}^{-10}$. The scaling of p with parameters such as the IWA, planet radius, distance from Earth, and atmospheric scattering properties can be inferred from the figures presented in this paper. In all but the most optimistic configurations, the probability for detection in a blind search carried out with WFIRST/AFTA's baseline coronagraphic capabilities is low (<5%).

By the time direct detection in the optical becomes feasible, long-term radial-velocity campaigns will have likely discovered many wide-separation EGPs, for which a subset of Keplerian elements will be known. Further, it has been estimated that Gaia will discover and determine robust orbits for ∼1500 planets with periods ≲6 years within 50 pc from Earth (Perryman et al. 2014), including ∼100 giant planets around known M dwarf host stars within 30 pc (Sozzetti et al. 2014). Direct-imaging surveys that leverage this information can improve their detection probabilities with respect to blind searches, and the tools we have developed for this work can be used to determine both the most promising systems to target and when to observe them. It is important to point out that, as can be inferred from Figures 4–6, the window of time that an EGP is directly detectable can be very brief and sparsely distributed throughout its orbit, suggesting that previous orbital parameter constraints must be very precise if they are to be used to optimize direct-imaging surveys. For a detailed study of the impact of uncertain radial-velocity orbital parameters on direct-imaging and spectroscopy of radial-velocity exoplanets, see Brown (2015).

Throughout this work, we have focused on cool, relatively old giant planets, for which the flux from thermal emission is negligible compared to the flux from reflected starlight. The thermal emission from a giant planet is age, mass, and metallicity dependent. Furthermore, the particular formation mechanism of giant planets significantly influences the cooling of the planet (e.g., Burrows et al. 1997; Marley et al. 2007; Fortney et al. 2008; Spiegel & Burrows 2012). Regardless of the assumed cooling model, it is reasonable to expect the thermal emission contribution to the optical planet/star flux ratio will increase with planet mass and decrease with planet age.

In Figure 16, we show the planet/star flux ratio (top panel) and associated Rayleigh scattering geometric albedo spectrum (bottom panel) for a representative seven Jupiter-mass planet orbiting a Sun-like star at an orbital distance of 5 AU, where we have included the effect of blackbody thermal emission from the planet. We use the evolutionary calculations of Burrows et al. (1997) to model the evolution of the age, radius, and temperature of the planet, and the albedo spectrum was calculated with the opacity database from Sharp & Burrows (2007) and equilibrium chemical abundances from Burrows & Sharp (1999). We assume solar metallically and an atmospheric pressure of 0.5 bar. Both the planet and star are assumed to be uniformly luminous blackbodies. That is, we assume the thermal emission component of the contrast ratio is given by ${R}_{p}^{2}{B}_{\lambda }({T}_{p})/{R}_{\star }^{2}{B}_{\lambda }({T}_{\star })$, where T_p is the effective temperature of the planet and ${B}_{\lambda }(T)$ is the Planck function.

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In this simple model, the thermal emission from the planet noticeably contributes to the optical planet/star flux ratio only for ages <80 Myr. Nonetheless, we see it briefly dominates the flux ratio at wavelengths ≳0.5 μm when the planet is a mere few tens of millions of years old, resulting in a planet that transitions from red to blue with increasing age and decreasing effective temperature. The age range over which thermal emission is important will be larger (smaller) for giant planets more (less) massive than seven Jupiter masses. The decrease in flux at short wavelengths (∼0.4–0.5 μm) for ages ≳100 Myr is due primarily to the contraction of the planet's radius, which decreases from ∼$1.4{R}_{\mathrm{Jup}}$ to ∼$1{R}_{\mathrm{Jup}}$ throughout its evolution. In contrast, the geometric albedo spectrum is approximately constant over this same wavelength and age interval.