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]
1
Theory of Orbits
• Laws of conservation: energy and angular
momentum
• From the energy conservation:
Z
dr
q
t=
+ const
2
L2
m [E − Φ(r)] − m2 r 2
• From the angular momentum conservation:
Z
L
r 2 dr
q
+ const
ϕ=
L2
2π[E − Φ(r)] − r2
• Exact solutions:
From the first equation r = r(t)
From the second equation: r = r(ϕ)
• Let us obtain an approximate solution
2
Retrograde Motion of Planets
3
Retrograde Motion of Planets
4
Epicyclic Theory
κ (r)
a
b
d
c
r
0
*
*
*
*
Ω (r)
• Suggest r(t) = r0 + r1 (t) and r0 = const
• Condition |r1 /ro | ≪ 1 → almost circular
orbit
• Φ0 = Φ0 (r) → central potential
• In the absence of any disturbance gravity
• κ is the epicyclic frequency
• Ω is the orbital frequency
• Expect κ ∼ Ω
• Two integrals of motion:
5
∂Φ0
∂t
= 0,
∂Φ0
∂ϕ
=0
Epicyclic Theory
• Newton’s equations of motion for an
individual “test particle” of unit mass
d2 r
∂Φ0
2
= r(ϕ̇) −
2
dt
∂r
r03 (∂Φ0 /∂r)0 ∂Φ0
−
,
=
3
(r0 + r1 )
∂r
d 2
∂Φ0
(r ϕ̇) = −
= 0,
dt
∂ϕ
q
1 ∂Φ0
where Ω = r0 ∂r 0 and
(1)
(2)
r2 ϕ̇ = r02 Ω = const → angular momentum
• A Taylor series
2 3 2
∂Φ0 (r)
r1 ∂ Φ0
∂ Φ0
∂Φ0
= ∂r 0 +r1 ∂r2 + 2! ∂r3
···
∂r
0
and take into account only the first two
terms in this expansion
6
0
Epicyclic Theory
• In Eq. (1), use the expansion
(1 + r1 /r0 )−3 ≈ 1 − 3r1 /r0
•
d 2 r1
dt2
2
= −κ2 r1 , where
2 ∂ Φ0
2
0
κ = r30 ∂Φ
=
4Ω
+
1+
2
∂r 0
∂r
0
r dΩ
2Ω dr
is the epicyclic frequency
• Solution r1 = − vκ⊥ [sin(ϕ0 − κt) − sin ϕ0 ],
where v⊥ , ϕ0 are constants of integration
• This solution and Eq. (2) give
v⊥
ϕ = Ωt + 2Ω
κ κr0 [cos(ϕ0 − κt) − cos ϕ0 ]
• Conclusion The closure condition for the
orbit Ωκ = ps , where p, s are positive integers
• Conclusion The orbits are bound if
2
and
n
≤
2
(see
κ
= ...)
Φ0 ∝ − const
rn
7
Our Solar System
• Eight planets are in orbit around the Sun.
PLuto is now classified as a dwarf planet
• Two groups of planets: terrestrial (or
“rocky”)
and giant (or “gaseous”) ones WHY?
• Rings around giant planets WHY?
Tidal disruption of a former satellite?
Primordial origin?
8
Planet Parameters
Mercury 0.4 solid 0 N 0.06 0.4
Venus
0.7 solid 0 N 0.8 0.9
Earth
1.0 solid 1 N 1.0 1.0
Mars
1.5 solid 2 N 0.1 0.5
Jupiter 5.2 gas 16 Y 318 11
Saturn 9.5 gas 21 Y 95 9.5
Uranus 19.2 gas 18 Y 15 4
Neptune 30.1 gas 8 Y 17 3.9
where R⊙−⊕ ≈ 1 AU ≈ 1.5 × 1013 cm and
R⊕ ≈ 6 400 km
Belts
Asteroids 2.7 solid
Kuiper belt 46 solid
— — — —
— — — —
9
Mercury
• orbit: 57 910 000 km (0.38 AU) from Sun
• diameter: 4 880 km
• mass: 3.3 × 1023 kg
10
Venus
• orbit: 108 200 000 km (0.72 AU) from Sun
• diameter: 12 103 km
• mass: 4.9 × 1024 kg
11
Earth
• 49 600 000 km (1.00 AU) from Sun
• diameter: 12 756 km
• mass: 5.97 × 1024 kg
12
Earth
13
Mars
• 227 940 000 km (1.52 AU) from Sun
• diameter: 6 792 km
• mass: 6.4 × 1023 kg
14
Mars
• Mars’ surface
15
Asteroids (or Minor Planets)
• orbit (Ceres): (2.7 AU) from Sun
• diameter (Ceres): 933 km
• mass: 1022 kg Number: 26 (D > 200 km
• number: ∼ 1 000 000 (10 > D > 1) km
16
Asteroids (or Minor Planets)
17
Jupiter
• orbit: 778 330 000 km (5.20 AU) from Sun
• diameter: 142 984 km (equatorial)
• mass: 1.9 × 1027 kg
18
Saturn
• orbit: 1 429 400 000 km (9.54 AU) from Sun
• diameter: 120 536 km (equatorial)
• mass: 5.68 × 1026 kg
19
Saturn and Its Rings
20
Saturn
• Saturn’s shadow and moons
21
Saturn
• Saturn’s moon Enceladus – Artist’s view
22
Saturn
• Saturn’s moon Tethys
23
Saturn
• Saturn’s moon Rhea
24
Uranus
• orbit: 2 870 990 000 km (19.218 AU) from
Sun
• diameter: 51 118 km (equatorial)
• mass: 8.68 × 1025 kg
• William Herschel (March 13, 1781)
25
Uranus
• Uranus’ separated and narrow rings
26
Uranus’ Moons
plays, the distant ringed world Uranus was last visited in 1986 by the Voyager 2 spacecraft.
Astronomy Picture of the Day
Tomorrow’s picture: x-ray galaxy
Discover the cosmos! Each day a different image or photograph of our fascinating universe is featured,
along with a brief explanation written by a professional astronomer.
< | Archive | Index | Search | Calendar | Glossary | Education | About APOD | >
2003 January 15
Authors & editors: Robert Nemiroff (MTU) & Jerry Bonnell (USRA)
NASA Technical Rep.: Jay Norris. Specific rights apply.
A service of: LHEA at NASA / GSFC
& NASA SEU Edu. Forum
& Michigan Tech. U.
Ringed Planet Uranus
Credit: E. Lellouch, T. Encrenaz (Obs. Paris), J. Cuby , A. Jaunsen (ESO-Chile), VLT Antu, ESO
Explanation: Yes it does look like Saturn, but Saturn is only one of four giant ringed planets in our
Solar System. And while Saturn has the brightest rings, this system of rings and moons actually belongs
to planet Uranus, imaged here in near-infrared light by the Antu telescope at the ESO Paranal
Observatory in Chile. Since gas giant Uranus’ methane-laced atmosphere absorbs sunlight at
near-infrared wavelengths the planet appears substantially darkened, improving the contrast between the
otherwise relatively bright planet and the normally faint rings. In fact, the narrow Uranian rings are all
but impossible to see in visible light with earthbound telescopes and were discovered only in 1977 as
careful astronomers noticed the then unknown rings blocking light from background stars. The rings are
thought to be younger than 100 million years and may be formed of debris from the collision of a small
moon with a passing comet or asteroid-like object. With moons named for characters in Shakespeare’s
27
Uranus’ Moons
• New Uranus’ small satellites
28
Neptune
• orbit: 4 504 000 000 km (30.06 AU) from
Sun
• diameter: 49 532 km (equatorial)
• mass: 1.02 × 1026 kg
• Galle and d’Arrest (Sept 23, 1846)
29
Kuiper Belt (Discovered in 1992)
• orbit (Pluto): 5 913 520 000 km (39.5 AU)
from the Sun
• diameter: 2 274 km
• mass: 1.27 × 1022 kg
• number: ∼ 100 000
30