The Bond albedo

A determination of the planetary albedo requires a large set of individual photometric measurements. Two approaches may be followed in collecting the data. In one approach the flux from a surface element is determined by recording the intensities emanating from that element in all directions and calculating the local flux according to the definition (Chandrasekhar, 1950), 

The bidirectional reflectivity, p, of the area element da is a function of the location and the inclination of the element with respect to the observer and the Sun. The reflectivity depends, therefore, on the angles, a, 9, and e, and is defined by 

Integration of Eq. (8.6.4) over the apparent planetary disk yields the observable mean disk intensity at phase angle, 9, and azimuth angle, (/ o. 

The first integral in Eq. (8.6.9) is over the fully visible and illuminated segment, while the second integral covers the segment containing the illuminated horizon, that is, the apparent planetary limb. If 9 is between jr/2 and n, the planet appears to the observer as a crescent and only one integral needs to be evaluated, 

Ideally, one attempts to derive an analytical expression for the disk measurements and their dependence on the phase angle. Conceivably, a family of such expressions could serve as interpolation functions to cover gaps between sparsely available data points. In reality integration of Eq. (8.6.9) is far from simple. In most cases p(a, 9, (p) is not well known. Alternatively, disk measurements must be made for many values of phase and azimuth angles, 9 and po, respectively. Then one can integrate the individual disk measurements over 4jr steradians, which yields the total power reflected by the planet. 

---

In this definition, the two edges of the HZ (see Fig. 5.3), depend on the Bond albedo of the planet. A, the luminosity of the star, the planet s semi major axis, D, as well as the eccentricity, e, of the orbit, and in turn the average stellar irradiation... 

The evaluation of the phase integral, Eq. (8.6.18), is more complicated. As mentioned before, a measurement of the phase function of the outer planets, 1(9)/1(0), requires observations from a space platform. Actual measurements obtained so far have produced only a few values of 7(0). Alternatively, one may try to find p((/>, 9, a) from radiative transfer calculations and evaluate Eqs. (8.6.9) and (8.6.11) to obtain estimates for 1(9), as has been done by Pollack et al. (1986). The phase function for a Lambert sphere can be determined analytically, and the corresponding phase integral is g =. Therefore, a Lambert sphere has the Bond albedo p. 

Recent determinations of the geometric albedos, the phase integrals, and the Bond albedos for the outer planets are summarized in Table 8.6.1. References indicating the sources of the quoted values are also shown. The measurement of the Saturn albedo is complicated by the existence of the ring system, which not only casts a shadow on Saturn but also scatters additional sunlight towards that planet. Both effects vary over the Saturnian year due to the relatively large inclination of the equator towards the orbital plane ( 27°). 

As mentioned in Subsection 8.6.a, a determination of the emitted planetary power requires measurements of the spectrally integrated disk intensity over all directions, that is, over 47T steradians. Since thermal emission does not directly depend on the solar flux, calculations of the total thermal emission are simpler than those required to find the Bond albedo. It is sufficient to use the conventional latitude-longitude system. The coordinates of the direction towards the observer are 9 and (f>o, and those of the area element, da, are a and . However, to save ourselves cumbersome trigonometric transformations, we use the emission angle s and the azimuth angle t/r with reference to the subobserver point as coordinates of da. The full disk intensity at wavenumber v, which can be measured by a distant observer, is then... 

On the outer planets the internal heat is found by measuring the thermal emission and subtracting the term representing absorbed solar power. Consequently, careful measurements of the effective planetary temperatures and the Bond albedos are required. The quantities R and D in Eq. (9.4.1) are relatively well known for each planet, as is S. On Earth the internal power is small in comparison with the other terms of Eq. (9.4.1). Thus it would be difficult to find the internal heat by subtracting two almost equal quantities. The terrestrial internal power can be found much more... 

The reflected power divided by the incident power intercepted by the object gives the albedo. If the albedo is measured over narrow spectral intervals, then the spectral albedo is determined. If the spectral response of the instrument is flat over all wavelengths pertinent to solar radiation (from 0.2 m to 4 fcm, for example), the global bolometric or Bond albedo is obtained. 

For the terrestrial planets, only items 4 and 5 are of importance. For the outer planets, all items are of interest, except item 6 which plays a dominant role only on lo, a much smaller one on Europa, and possibly a very small one on Ganymede and Callisto. On Jupiter and Saturn items 1,2,3,4, and 5 contribute to the total infrared emission, the only quantity which can be measured by remote sensing techniques [left term in Eq. (9.4.1)]. On Uranus and Neptvme a metallic hydrogen core is not expected to exist therefore, the radial redistribution of helium caimot take place. However, other processes can conceivably affect the measured helium-to-hydrogen ratio as will be discussed further below. Item 4 is important for all planets. The absorbed solar radiation [first term on the right side of Eq. (9.4.1)] is found by a measurement of the planetary Bond albedo and a knowledge of the radius, the solar constant, and the heliocentric distance of the object. 

Using the same on-board calibration with the diffuse reflecting plate, a geometric albedo of 0.242 0.012 was derived. Again combining this with a Pioneer-derived phase integral led to a Bond albedo of 0.342 0.030. Because of the uncertainty of the radiometer calibration this number must be taken with the same reservations as the corresponding number for Jupiter. 

The values pertinent to the energy balance of all giant planets are summarized in Table 9.4.1. Although Uranus and Neptune are very similar in size, Bond albedo, and other parameters, they are quite different in the energy balance see also the review paper by Hubbard et al. (1995). 

---