Photometry Polarimetry and Spectroscopy

Having defined the phase function and albedos we are ready to derive a formula for planetary magnitudes. The flux density of the reflected light is

CA 1

We now substitute the incident flux

At the phase angle zero 0(a = 0°) = 1 and the reflected flux density is

4n A2

If we replace the object with a Lambertian surface of the same size, we get

The ratio of these flux densities is and the constant factor expressed in terms of the geometric albedo

4n n

Thus we get

The observed solar flux density at a distance of a = 1 AU from the Sun is L0

Now we have found a physical interpretation for p: the geometric albedo is the ratio of the flux densities at phase angle a = 0° reflected by a planet and a Lambertian surface of the same cross section.

The geometric albedo depends on the reflectance of the surface but also on the phase function Many rough surfaces reflect most of the incident radiation directly backward. In such a case the geometric albedo p is greater than in the case of an isotropically reflecting surface. On some surfaces p > 1, and in the most extreme case, the specular reflection, p = The geometric albedo of solar system bodies vary between 0.03-1. The geometric albedo of the Moon is p = 0.12 and the greatest value, p = 1.0, has been measured for the Saturnian moon Enceladus.


If the apparent solar magnitude at a distance of 1 AU is m0 and the apparent magnitude of the planet m we have m — m0 = -2.5 lg

a2 A2r2

R2 a


R2 Ar

If we denote

then the magnitude of a planet can be expressed as m = V(1, 0) + 5 lg TAt — 2.5lg ^(a). (7.43)

The first term V(1, 0) depends only on the size of the planet and its reflection properties. So it is a quantity intrinsic to the planet, and it is called the absolute magnitude (not to be confused with the absolute magnitude in stellar astronomy!). The second term contains the dis tance dependence and the third one the dependence on the phase angle.

If the phase angle is zero, and we set r = A = a, (7.43) becomes simply m = V(1,0). The absolute magnitude can be interpreted as the magnitude of a body if it is at a distance of 1 AU from the Earth and the Sun at a phase angle a = 0°. As will be immediately noticed, this is physically impossible because the observer would be in the very centre of the Sun. Thus V(1, 0) can never be observed.

The last term in (7.43) is the most problematic one. For many objects the phase function is not known very well. This means that from the observations, one can

Asteroid Phase Curve

Fig. 7.20. The phase curves and polarization of different types tometric and Polarimetrie phase effects, in Bottke, Binzel, of asteroids. The asteroid characteristics are discussed in more Cellino, Paolizhi (Eds.) Asteroids III, University of Arizona detail in Sect. 7.14. (From Muinonen et al., Asteroid pho- Press, Tucson.)

calculate only

which is the absolute magnitude at phase angle a. V(1,a), plotted as a function of the phase angle, is called the phase curve (Fig. 7.20). The phase curve extrapolated to a = 0° gives V(1,0).

By using (7.41) at a = 0°, the geometric albedo can be solved for in terms of observed values:

where mo = m(a = 0°). As can easily be seen, p can be greater than unity but in the real world, it is normally well below that. A typical value for p is 0.1-0.5.

The Bond albedo can be determined only if the phase function $ is known. Superior planets (and other bodies orbiting outside the orbit of the Earth) can be observed only in a limited phase angle range, and therefore $ is poorly known. The situation is somewhat better for the inferior planets. Especially in popular texts the Bond albedo is given instead of p (naturally without mentioning the exact names!). A good excuse for this is the obvious physical meaning of the former, and also the fact that the Bond albedo is normalised to [0,1].

Opposition Effect. The brightness of an atmosphere-less body increases rapidly when the phase angle approaches zero. When the phase is larger than about 10°, the changes are smaller. This rapid brightening close to the opposition is called the opposition effect. The full explanation is still in dispute. A qualitative (but only partial) explanation is that close to the opposition, no shadows are visible. When the phase angle increases, the shadows become visible and the brightness drops. An atmosphere destroys the opposition effect.

The shape of the phase curve depends on the geometric albedo. It is possible to estimate the geometric albedo if the phase curve is known. This requires at least a few observations at different phase angles. Most critical is the range 0°-10°. A known phase curve can be used to determine the diameter of the body, e. g. the size of an asteroid. Apparent diameters of asteroids are so small that for ground based observations one has to use indirect methods, like polarimetric or radio-metric (thermal radiation) observations. Beginning from the 1990's, imaging made during spacecraft fly-bys and with the Hubble Space Telescope have given also direct measures of the diameter and shape of asteroids.

Magnitudes of Asteroids. When the phase angle is greater than a few degrees, the magnitude of an asteroid depends almost linearly on the phase angle. Earlier this linear part was extrapolated to a = 0° to estimate the opposition magnitude of an asteroid. Due to the opposition effect the actual opposition magnitude can be considerably brighter.

In 1985 the IAU adopted the semi-empirical HG system where the magnitude of an asteroid is described by two constants H and G. Let a! = (1 - G) x 10-04 H , a2 = G x 10-04 H . The phase curve can be approximated by V(1, a) = -2.5

x log

a1 exp i - 3.33 (ton-J j (7.47) + a2 exp i - 1.87 (ton Oj a \ 0.63 2

a \ 1-22 v 2 When the phase angle is a = 0° (7.47) becomes V(1, 0) = -2.5log(ai + a2)

The constant H is thus the absolute magnitude and G describes the shape of the phase curve. If G is great, the phase curve is steeper and the brightness is decreasing rapidly with the phase angle. For very gentle slopes G can be negative. H and G can be determined with a least squares fit to the phase observations.

Polarimetric Observations. The light reflected by the bodies of the solar system is usually polarized. The amount of polarization depends on the reflecting material and also on the geometry: polarization is a function of the phase angle. The degree of polarization P is defined as

where F± is the flux density of radiation, perpendicular to a fixed plane, and F is the flux density parallel to the plane. In solar system studies, polarization is usually referred to the plane defined by the Earth, the Sun, and the object. According to (7.49), P can be positive or negative; thus the terms "positive" and "negative" polarization are used.

The degree of polarization as a function of the phase angle depends on the surface structure and the atmosphere. The degree of polarization of the light reflected by the surface of an atmosphereless body is positive when the phase angle is greater than about 20°. Closer to opposition, polarization is negative. When light is reflected from an atmosphere, the degree of polarization as a function of the phase angle is more complicated. By combining observations with a theory of radiative transfer, one can compute atmosphere models. For example, the composition of Venus' atmosphere could be studied before any probes were sent to the planet.

Planetary Spectroscopy. The photometric and polari-metric observations discussed above were monochromatic. However, the studies of the atmosphere of Venus also used spectral information. Broadband UBV photometry or polarimetry is the simplest example of spectrophotometry (spectropolarimetry). The term spectrophotometry usually means observations made with several narrowband filters. Naturally, solar sys

Wavelength [nm]

480 520 560 600 640 680 720

Wavelength [nm]

480 520 560 600 640 680 720


u i-M





Fig. 7.21. Spectra of the Moon and the giant planets. Strong absorption bands can be seen in the spectra of Uranus and Neptune. (Lowell Observatory Bulletin 42 (1909))






Fig. 7.21. Spectra of the Moon and the giant planets. Strong absorption bands can be seen in the spectra of Uranus and Neptune. (Lowell Observatory Bulletin 42 (1909))

tem objects are also observed by means of "classical" spectroscopy.

Spectrophotometry and polarimetry give information at discrete wavelengths only. In practise, the number of points of the spectrum (or the number of filters available) is often limited to 20-30. This means that no details can be seen in the spectra. On the other hand, in ordinary spectroscopy, the limiting magnitude is smaller, although the situation is rapidly improving with the new generation detectors, such as the CCD camera.

The spectrum observed is the spectrum of the Sun. Generally, the planetary contribution is relatively small, and these differences can be seen when the solar spectrum is subtracted. The Uranian spectrum is a typical example (Fig. 7.21). There are strong absorption bands in the near-infrared. Laboratory measurements have shown that these are due to methane. A portion of the red light is also absorbed, causing the greenish colour of the planet. The general techniques of spectral observations are discussed in the context of stellar spectroscopy in Chap. 8.

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