Detectability in reflected and emitted light

A simple calculation to define the parameters of the problem is the expected separation between the planet and star, in angular terms. For a planet located at a distanceafrom its host star, and the host star a distancedfrom the Earth, the expected angular separation is:

θ=a d (1.32) = 100mas a 1AU d 10pc −1 (1.33)

where the implicit definition of the parsec is used to generate the second expression. For context, 100 mas is not a large angular separation; it is about the width of a human hair viewed from 100 meters away.

To understand how bright planets are, and in particular how bright they are compared to stars, we recognize that planets generate their own light from internal heat, but also reflect light from their parent stars. Both the reflected and emitted light may be approximated by blackbody spectra, at the temperature of the star and planet, respectively. The blackbody law is given by

B(λ, T) = 2hc

2

λ5_exp[hc/(λk

BT)]−1

where the units of B are in specific intensity, that is, ergs/s/cm2_{/cm/sr. The law describes the}

amount of power emitted (ergs/s), per unit wavelength (cm), per unit surface area (cm2_{), per unit}

solid angle (sr) by a surface at temperatureT at wavelength λ. In this formula, c is the speed of light,his Planck’s constant, andkBis Boltmann’s constant. At a distancedfrom a star of uniform

brightness, we have F = Z BcosθdΩ =πB(λ, T) _R ∗ d 2 (1.35)

and hence the contrast between a planet and star is

Fp F∗ = R 2 p B(λ, Tp) R2 ∗ B(λ, T∗) (1.36)

Putting some numbers into this expression, we find that for a planet like Earth, the contrast at a wavelength of 10 microns is about 10−7_{, about the same as Jupiter (the smaller radius is offset by}

the higher temperature of 275K vs 150K for Jupiter). At 1 micron, both are more than 20 orders of magnitude fainter. It is clear detecting planetary thermal emission at optical and near-infrared wavelengths is not feasible for solar system analogs.

The previous expressions made no explicit use of the planet-star separation distance. There is an implicit dependence, as the effective temperature of the planet will depend on the distance from the star, to some degree, though internal sources of heat and residual heat from formation are important caveats to this statement. For reflected light, the distance to the star plays an explicit role. As stated before, the stellar flux density at a distancedfrom the star is

F(λ, d) =πB(λ, T∗) _R ∗ d 2 (1.37)

The factor ofπcomes from integrating the solid angle along the line of sight. The planet will reflect some of this light, and the amount reflected depends on the radius of the planet, Rp, its distance

from the stara, and the reflectivity of the atmosphere or surface, called the albedoA(λ):

Fp(λ, d) =πB(λ, T∗) _R ∗ a 2 A(λ)Φ(α) _R p a 2 (1.38)

There is also a phase function Φ(α) which depends on the inclination of the orbit, and the portion of the orbit the planet is in. This is exactly analogous to the phases of the moon, and runs from 0 (such as when the planet transits a star) to 1. That shows that the planet to star flux ratio is

Figure 1.11: The flux of light from a sun-like star compared to various planets in the solar system, and a hypothetical hot Jupiter. All these are blackbody curves; real planets and stars have more complicated spectra, but the approximations are quite accurate for the continuum levels. Note that the shorter, visible wavelengths are all reflected light and track the spectrum of the star, while the longer infrared wavelengths are thermal emission and are somewhat larger. The contrast between the Earth and the Sun is ten orders of magnitude in visible light. This figure is taken from Seager & Deming (2010). Fp F∗ =A(λ)Φ(α) _R p a 2 (1.39)

where we note that the wavelength dependence cancels out, except for the albedo. For general numbers of the Earth and Jupiter, and assuming an albedo of 0.3, we find reflected light contrasts of about 5x10−10_{and 2x10}−9_{. The wavelength dependent reflected and emitted contrasts are presented}

in Figure 1.11.

resolution. We now give some context as to the challenges associated with these requirements. For angular resolution, the diffraction limit of a 2m class telescope operating at a visible wavelength of 500 nm is7 _{50 mas. Of course, looking at nearby stars will help offset this, but there are only}

250 star systems closer than 10 parsecs.8 _{Large telescopes are hence preferred to improve angular}

resolution, but on the ground, they need adaptive optics systems to sharpen the starlight and reduce the scattering effects of the turbulent atmosphere. In terms of contrast, even with a perfect telescope, diffraction will cause the image of the star to be surrounded by “Airy” rings; the first Airy ring is at about 1.5 resolution elements (λ/D) from the core, and is reduced in intensity by a factor of 100. This is still thousands to millions of times brighter than any putative planet at that location. This drives the need to develop diffraction control systems, such as coronagraphs or interferometers. Even with such systems, optical errors in the telescope and instruments can destroy most of their performance, driving the need to develop sophisticated focal plane wavefront control systems and post-processing algorithms to deal with this scattered light. Finally, with all this extra optical complexity, the throughput of the system has to remain high. A 2 m space telescope will receive about 100 million photons per second from a V=7.5 magnitude star; for a Jupiter-type planetary companion, this reduces to about 1 photon per second. If the throughput of the system is low, exposure times could stretch into days to accumulate enough signal, even in the best cases.