Fundamental Astronomy

calculate only
V(1,α)≡ V(1, 0) − 2.5lgΦ(α) , (7.44)
which is the absolute magnitude at phase angle α.
V(1,α), plotted as a function of the phase angle, is
called the phase curve (Fig. 7.20). The phase curve
extrapolated to α = 0 ◦ gives V(1, 0).
By using (7.41) at α = 0 ◦ , the geometric albedo can
be solved for in terms of observed values:
2 r∆
p = 10
aR
−0.4(m0 − m⊙)
, (7.45)
where m0 = m(α = 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 Φ 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 Φ 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 atmosphereless
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 radiometric
(thermal radiation) observations. Beginning from
7.8 Photometry, Polarimetry and Spectroscopy
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 α = 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
a1 = (1 − G) × 10 −0.4 H ,
a2 = G × 10 −0.4 H (7.46)
.
The phase curve can be approximated by
V(1,α)=−2.5
× log a1 exp − 3.33 tan α
0.63
(7.47)
2
+ a2 exp − 1.87 tan α
1.22
.
2
When the phase angle is α = 0◦ (7.47) becomes
V(1, 0) =−2.5log(a1 + a2)
=−2.5log 10 −0.4 H (7.48)
= H .
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
P = F⊥ − F
, (7.49)
F⊥ + F
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