Fundamental Astronomy

468
Answers to Exercises
6.2 a = 1.4581 AU, v ≈ 23.6kms −1 .
6.3 The period must equal the sidereal rotation period
of the Earth. r = 42,339 km = 6.64 R⊕. Areas
within 8.6 ◦ from the poles cannot be seen from geostationary
satellites. The hidden area is 1.1% of the
total surface area.
6.4 ρ = 3π/(GP 2 (α/2) 3 ) ≈ 1400 kg m −3 .
6.5 M = 90 ◦ , E = 90.96 ◦ , f = 91.91 ◦ .
6.6 The orbit is hyperbolic, a = 3.55 × 10 7 AU,
e = 1 + 3.97 × 10 −16 , rp = 2.1 km. The comet will hit
the Sun.
6.7 The orbital elements of the Earth calculated from
Table C.12 are a = 1.0000, e = 0.0167, i = 0.0004◦ ,
Ω =−11.13◦ , ϖ = 102.9◦ , L = 219.5◦ . The geocentric
radius vector of the Sun in the ecliptic coordinates
is
⎛ ⎞
0.7583
⎜ ⎟
r = ⎝0.6673⎠
.
0.0
The corresponding equatorial radius vector is
⎛ ⎞
0.7583
⎜ ⎟
r = ⎝0.6089⎠
,
0.2640
which gives α ≈ 2 h 35 min 3 s, δ ≈ 15.19 ◦ . The exact
direction is α = 2h34min53s,δ = 15.17 ◦ .
Chapter 7
7.1 Assuming the orbits are circular, the greatest
elongation is arcsin(a/1AU). For Mercury this is 23 ◦
and for Venus 46 ◦ . The elongation of a superior planet
can be anything up to 180 ◦ . The sky revolves about 15 ◦
per hour, and thus corresponding times for Mercury and
Venus are 1 h 30 min and 3 h 5 min, respectively. In opposition
Mars is visible the whole night. These values,
however, depend on the actual declinations of the planets.
7.2 a) 8.7 ◦ . b) The Earth must be 90 ◦ from the ascending
node of Venus, which is the situation about
13 days before vernal and autumnal equinoxes, around
March 8 and September 10.
7.3 Psid = 11.9a, a = 5.20 AU, d = 144,000 km.
Obviously the planet is Jupiter.
7.4 a) Hint: If there is a synodic period P there must
be integers p and q such that (n2 − n1)P = 2πp and
(n3 − n1)P = 2πq. Sometimes one can see claims that
the configuration of the whole planetary system will recur
after a certain period. Such claims are obviously
nonsense. b) 7.06 d.
7.5 a) If the radii of the orbits are a1 and a2, the
angular velocity of the retrograde motion is
dλ
dt =
√
GM
√
a1a2( √ a1 + √ a2) .
b) In six days Pluto moves about 0.128 ◦ corresponding
to 4 mm. For a main belt asteroid the displacement is
almost 4 cm.
7.6 If the orbital velocity of the planet is v the
deviation in radians is
α = v
1 GM⊙
= .
c c a
This is greatest for Mercury, α = 0.00016 rad = 33 ′′ .
This planetary aberration must be taken into account
when computing accurate ephemerides. The deviation
is largest when the planet is in conjunction or opposition
and moves almost perpendicularly to the line of
sight.
7.7 p = 0.11, q = 2, and A = 0.2. In reality the
Moon reflects most of the light directly backwards (opposition
effect), and thus q and A are much smaller.
7.8 ∆m = 0.9. The surface brightness remains constant.
7.9 The absolute magnitude is V(1, 0) = 23.
a) m = 18.7. b) m = 14.2. At least a 15 cm telescope is
needed to detect the asteroid even one day before the
collision.