Download AN INTRODUCTION TO ASTRONOMY Dr. Uri Griv Department of Physics, Ben-Gurion University

BEN-GURION UNIVERSITY
AN INTRODUCTION TO ASTRONOMY
Dr. Uri Griv
Department of Physics, Ben-Gurion University
Tel.: 08-6428226 Email: [email protected]
Orbits and Inclinations
2
Orbits and Inclinations
• The planetary orbits all lie nearly in the
same plane and take place with the same
sense of revolution
• The spin of the Sun is also nearly in the
plane of the ecliptic
• The spins of the planets are also nearly in
the same plane
• The orbital shapes of the planets are nearly
circular, i.e. of low eccentricity
• The satellite systems of the major planets
mimic the solar system in the above
properties
30
15
10
5
2
4
6
n
8
10
(c)
r (× 100000 km)
10
(b)
r (× 100000 km)
r (× 100000 km)
r (AU)
20
0
6
12
(a)
8
6
4
(d)
4
2
2
2
4
2
n
4
n
3
6
2
4
n
Origin of the Moon
• Two mechanisms of Moon’s formation:
• I. Gravitational instability of a gas-dust disk
2
2
2
0
0
0
−2
−2
−2
−2
0
t=0.0
2
−2
0
t=0.5
2
2
2
2
0
0
0
−2
−2
−2
−2
0
t=1.5
2
−2
0
t=2.0
2
−2
0
t=1.0
2
−2
0
t=2.5
2
• II. Earth–Comet megaimpact → Collision
an early Earth (≈ 4 × 109 years ago) and a
body (a small planet) of mass
(0.15 − 0.25) × M⊕ , where M⊕ is the
Earth’s mass
4
Origin of the Moon:
A Disk Instability or a Megaimpact?
Peculiar Composition and Unconventional
Dynamical Features of the Moon
• The fraction of iron in the Moon is 10%
• This is lower by a factor of 3 than the iron
content of the Earth and Venus, lower by a
factor of 2.5 than that of Mars
• The lunar orbital momentum attains 4/5 of
the total Earth–Moon momentum
• The relative distance R between the planet
and its satellite is longest for the Moon
(R ≈ 60R⊕ )
5
Origin of the Moon: a Megaimpact?
• Collision with the Earth of a body having
the mass M ≈ 0.2M⊕ and the velocity
V ≈ 15 km/s
• We can consider the inelastic impact
• A noninertial coordinate system with its
origin at the Earth’s center
• Conservation of momentum:
M V = (M⊕ + M )∆V⊕
• From this equation (for M ≈ 0.2M⊕ ):
V
1
∆V⊕ =
≈ V
M⊕ /M + 1 6
(1)
• If V ≈ 15 km/s then ∆V⊕ /V⊕ ≈ 1/12 ≪ 1
for the Earth’s orbital velocity V⊕ ≈ 30
km/s
6
• We use this small parameter ∆V⊕ /V⊕ in
calculating the eccentricity e of the Earth’s
orbit after the impact
• Mechanics yields at an arbitrary time:
s
2EL2
where
e= 1+
2
M⊕ α
E=
M⊕ vr2
2
M⊕ vϕ2 α
+
− ,
2
r
L = M⊕ vϕ r
α = GM⊙ M⊕ and M⊙ is the Sun’s mass
• We now show that (for striker parameters
V ≈ 15 km/s and M ≈ 0.2M⊕ ) the
eccentricity of the Earth’s orbit after the
impact turns out to be larger than the
observed value by a factor of 5 or 10
7
(2)
(3)
• After the impact:
E = E0 + ∆E ,
vr = ∆vr ,
L = L0 + ∆L (4)
vϕ = vϕ0 + ∆vϕ
(5)
where the subscripts “0” refer to the
quantities prior to the collision
• Using conditions of equilibrium in the
circular orbit
GM⊙
2
vr0 = 0 ,
= vϕ0
r
we have from (3)
2
M⊕ vϕ0
, L0 = M⊕ vϕ0 r
E0 = −
2
• Assuming e0 = 0, we have from (2)
2E0 L20
= −1
2
M⊕ α
8
(6)
(7)
• For an impact directed along the Earth’s
orbit, one derives from (3)
M⊕ (∆V⊕ )2
∆E = M⊕ vϕ0 ∆V⊕ +
(8)
2
∆L = M⊕ r∆V⊕
(9)
• Substituting (4)–(5) into (2) with allowance
for (7)–(9) and restricting ourselves to terms
of second-order smallness with respect to
the small parameter ∆V⊕ /V⊕ ≈ ∆V⊕ /vϕ0
(all terms of first-order smallness are
canceled), we obtain
2∆V⊕
1
∆e ≈
≈ ≈ 0.17
V⊕
6
which exceeds the observed eccentricity
(≈ 0.017) by an order of magnitude!
9
(10)
• For the impact along the radial direction
M⊕ (∆V⊕ )2
∆E =
, ∆L = 0
2
and similar calculations yield for the
eccentricity the value
∆V⊕
∆e =
V⊕
(11)
(12)
which is twice as small as (10)
• In the case of such an impact direction, the
eccentricity of the Earth’s orbit must exceed
the observed value by a factor of 5
• Finally, is the impact theory correct?
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