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3 4.1. EQUILIBRIUM TEMPERATURE 79 Greenhouse e ect for terrestrial planets. For the planets Earth and Venus the ground temperature T ground is a significantly higher than T eq due to the greenhouse e ect. Figure 4.1: Energy flow diagram for the greenhouse e ect on Earth. In the greenhouse e ect (e.g. for Earth) the visual light from the sun penetrates through the atmosphere down to the surface and heats e ciently the ground. However, the thermal IR-radiation from the ground can only escape in certain spectral windows without strong molecular absorptions (H 2 O, CO 2 ), while the rest is absorbed in the atmosphere. Energy transport from the warm/hot ground to higher cold layers occurs therefore through convection and radiation until the thermal radiation can escape to space. T eq represents the temperature of the atmospheric layers from which the thermal radiation can escape. Therefore, the ground temperature is higher than T eq. The e ect is stronger on Venus because of its much thicker atmosphere (9 bar) when compared to Earth (1 bar).

5 4.2. THERMAL RADIATION FROM PLANETS 81 Thermal luminosity and flux contrast between planet and sun. The thermal luminosity L P of an irradiated planet without intrinsic energy source is given by Equations 4.1 or 4.2. This can be expressed as thermal luminosity contrast C th between the planet and the sun L P L = R2 T P eq 4 R 2 T 4 = R2 P 1 d 2 P 4 (1 A B). (4.8) For solar system planets this ratio is very small, of the order 1 9 to 1 11 (see Table 4.1). Equation 4.8 for the luminosity contrast is also valid for extra-solar systems. For hot Jupiters the ratio L P /L is much larger than for solar system planets. The flux contrast as function of wavelength is important for observational studies. For long wavelengths, in the Rayleigh-Jeans part of the Planck function of the planet, one can use equation 4.6 which yields: F P ( F ( ) max) = R2 P R 2 T eq T. (4.9) The factor for the temperature ratio between planet and sun (or star) T eq /T is of the order 1 1. Thus the flux contrast at long wavelengths max is several orders of magnitudes ( ) larger than the total luminosity contrast (see Table 4.1). On the other hand, the planet to star flux contrast at short wavelengths < max decreases rapidly to very small values because the thermal radiation of the planet drops-o exponentially. At short wavelengths the scattered light will therefore dominate.

6 82 CHAPTER 4. RADIATION FROM PLANETS 4.3 Reflection from planets Reflection by a Lambert surface. A Lambert surface is used as reference in many technical and scientific studies on reflectivities. A Lambert surface reflects all incident light and the surface brightness is the same for all viewing angles. However, for viewing directions with an angle with respect to the surface normal the apparent reflecting area and therefore also the reflected flux is reduced / cos. ThusthereflectedintensityI Lam of a flat Lambert surface per unit solid angle is I Lam ( ) =F i cos for apple <9 (4.1) where F i is the incident flux onto the considered surface. Thereby, the reflection from a Lambert surface does not depend on the direction of the irradiation. A sheet of white paper, a with screen or a white wall are close to a Lambert surface. Figure 4.2: Reflection from a Lambert surface. The factor 1/ in Equation 4.1 is the normalization factor because energy conservation requires that the reflected intensity I Lam integrated over all direction is equal to F i Z 2 Z /2 /2 I Lam ( )sin d d = Z 2 Z /2 F i cos sin d d = F i. An observer at a distance D (much larger than the linear dimension of the surface area) measures a reflected flux F Lam per unit area of F Lam ( ) = I Lam( ) D 2 = F i cos D 2 Normal retro-reflection of a Lambert disk irradiated by the sun. Based on the reflection law for a Lambert surface we can derive the normal retro-reflection (= normal irradiation and normal reflection) of solar light by a round Lambert disk with radius R disk at the distance d disk from the sun: F disk (, = ) = F i( ) D 2 = L ( ) 4 d 2 R 2 1 disk disk D 2, where F i is replaced by the explicit formula for the sunlight intercepted by the Lambert disk.

7 4.3. REFLECTION FROM PLANETS 83 Geometric albedo for solar system planets. The geometric albedo A g ( )isthe spectral reflectivity of a planet at zero phase angle = (full phase) relative to the reflectivity of a Lambert disk with the same cross section as the planet A g ( )= F P ( ) F disk ( ). (4.11) Thus, the geometric albedo of a planet can be determined by measuring the magnitude of that planet at opposition (normal retro-reflection), which is then compared to the calculated reflection of a Lambert disk with the same cross section. It is convenient to express the theoretically reflected flux from a Lambert disk relative to the flux of the sun measured from Earth F ( )=L ( )/4 d 2 E (where d E = 1 AU), because this ratio is independent of wavelength: R = F disk F = d2 E Rdisk 2 D 2 d 2. (4.12) disk Opposition = occurs for the outer planets almost every year. Because the phase angles for the giant planets is never really large, < 12 for Jupiter, < 5, and less for Uranus and Neptune, one can correct for the small deviations for an ideal geometric albedo measurement. Example Jupiter: As example we calculate with Equation 4.12 the case for Jupiter for which the distance to Earth at opposition is D =5.2 1 AU, R disk = R J = 6991 km and d disk = d J =5.2 AUwith1AU= km. The ratio between the flux of a Lambert disk with a cross section equivalent to Jupiter and the solar flux is R = F disk F = or m disk m = 2.5 log R = 23.36, where the result is also given as magnitude di erence. The apparent V-band magnitude for the sun is m (V) = mag and for Jupiter at opposition about m J (V) = 2.7 mag. This yields an opposition contrast of m J m = 24.4 mag or about m =.7 mag more than expected for a Lambertian disk. The geometric albedo of Jupiter is this magnitude di erence m expressed as ratio A g = 1.4 m =.52 in good agreement with available literature values. Figure 4.3: Typical constellation for the geometric albedo measurement of Jupiter or another outer planet.

8 84 CHAPTER 4. RADIATION FROM PLANETS Table 4.2: Reflection properties of solar system planets: geometric albedos for the V-band and the IR, phase integral q and calculated spherical albedos. The last columns give the factor R 2 p/d 2 p and the flux contrast for the scattered light at quadrature phase assuming f(9 )=.3. planet A g (V ) A g (IR) q A s (V ) A s (IR) A B Rp/d 2 2 p F P /F Mercury Venus Earth Mars Jupiter Saturn Uranus Neptune Geometric albedo of a Lambert sphere. It is important to note that a Lambert sphere has a geometric albedo of A g =2/3. The surface brightness of a Lambert disk of normalized radius R = 1 is constant over the whole disk and one can write for the normal retro-reflection ( = ): I disk (r) = F i and Z 1 Z 1 I disk (r)2 r dr =2F i rdr = F i A sphere (not a disk) at zero phase angle has a surface brightness distribution with a limb darkening which behaves for the normalized radius apple r apple 1like I sph (r) =F i cos (r) = F i p 1 r 2. Figure 4.4: Schematic di erence of the geometric albedo of a Lambert disk and a Lambert sphere.

9 4.3. REFLECTION FROM PLANETS 85 The angle is the angle of incidence with respect to the surface normal which depends on the radial distance r =sin measured from the center of the illuminated hemisphere (apparent disk). The sub-solar point reflects like a disk (surface brightness F i / ) butthe irradiation of the more an more inclined surface towards the limb results in a reduced back-scattering because the strongest scattering occurs along the surface normal. Integration for a fully illuminated Lambert sphere yields: Z 1 Z 1 I sph (r)2 r dr =2F i r p (1 r 2 1 r 2 ) 3/2 dr =2F i 3 1 = 2 3 F i. A Lambert sphere reflects only 2/3 of the light for phase angle = when compared to a Lambert disk because a substantial fraction of light is scattered into direction > /2 what does not occur for an illuminated disk. Lambert disk and Lambert sphere scatter both all light and have a Bond albedo (or spherical albedo) of A B = 1 but the angular distribution of the scattered light is di erent. It is not surprising that the solar system planets have geometric albedos A g <.7 when considering the case of the perfectly reflecting Lambert sphere. Averaged over all wavelengths the A g should be smaller (about 2/3) than the Bond albedo A B. This is roughly the case for Venus and Mars (see Table 4.2). For Earth and the giant planets the situation is di erent. The geometric albedo in the visual is higher than the Bond albedo A g (V) >A B. This indicates that the geometric albedo must be low at other wavelengths, what is the case for the IR wavelength regime because of molecular absorption by H 2 O for Earth and CH 4 for the giant planets. Spherical albedo and Bond albedo. The spherical albedo A s ( ) gives the reflection in all direction and not only the normal retro-reflection as measured for the geometric albedo A g ( ). The spherical albedo is required for an accurate derivation of the Bond albedo A B. A B, which is used for energy budget calculations, is the flux weighted wavelength average of the spherical albedo: R 1 F i ( )A s ( ) d A B = R 1. (4.13) F i ( )d With a scattering model of a planet it is easy to calculate the geometric albedo and spherical albedo. Observationally, one needs to know the scattering in all direction, what is a very di cult to achieve. For example, the reflection f( ) of Earth for a phase angle = 9 will be di erent if mainly the white polar regions are seen from a polar direction when compared to the dark oceans as seen from equatorial directions. One simple way to address the problem of the reflection into di erent directions is the phase integral q defined by Z F ref ( ) q =2 sin d, F ref ( = ) where F ref ( ) is a rotationally symmetric phase angle dependence of the reflected radiation normalized to the geometric albedo F ref ( = ) = A g. With this definition the phase integral, geometric albedo, and spherical albedo are related by A s = A g q. (4.14)

10 86 CHAPTER 4. RADIATION FROM PLANETS It should be noted that this approach is only a first order approximation which is formally only correct for rotationally symmetric reflection from planets. The phase integral q for special cases is: q = 1 for a Lambert disk, q =3/2 for a Lambert sphere, q = 4 for (a theoretical) isotropically scattering body. Some values of the phase integral for solar system planets are given in Table 4.2. Reflectivity phase curves. The phase angle dependence of the reflected radiation from a planet is important for the analysis of observations. In general, planets are not observed at phase angle =. For example, the inner planets, Mercury and Venus, are behind the sun for =, and extra-solar planets are behind their star. With direct imaging of extra-solar planets only data in the range 3 < <15 can probably be obtained in the near future. For this reason one needs to study the reflectivity phase curves F ref ( ) or the phase dependence of the reflection normalized to the geometric albedo: f( ) = F ref( ) F ref ( = ). (4.15) Phase curve for a Lambert sphere. The phase curve for a Lambert sphere can be derived analytically by integrating the cos reflection law of the visible part of the illuminated sphere as function of phase angle. The solution is: f( ) = 1 (sin +( ) cos ). (4.16) Flux contrast for reflecting extra-solar planets. Equation 4.12 is also valid for a very distant observer outside of the solar system or for the observations of extra-solar planets from Earth. In this case the distance of the observer to the central star d star (which was d E for an Earth-based observer looking at a solar system planet) and the distance from the planet to the observer D are equal and very large d star = D 1 AU. Thus the contrast of a reflecting planet C ref with respect to its illuminating star is C ref = F P F star = A g ( )f( ) R2 P d 2 P, (4.17) where A g ( ) is the geometric albedo and f( ) a normalized phase function as described by Equation 4.15 which takes into account that the reflected light depends on the angle star - planet - observer. Table 4.2 gives the factors RP 2 /d2 P for the solar system planets and also estimates for the contrast of the reflected light for a phase angle = 9.

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