Download PHY2083 ASTRONOMY

PHY2083
ASTRONOMY
Tides
THE DIMENSIONS OF THE SOLAR SYSTEM
Using Kepler’s Laws, we can construct a model of the
solar system with the correct shapes of the orbits, and
correct relative sizes.
The actual sizes cannot be obtained from Kepler’s laws
=> What is 1 AU?
1 AU = ?
Early measurements from transiting inner
(i.e. Mercury or Venus) planets + geometry
Transit times can be measured
very accurately from different
locations on the Earth
Transit of Mercury (2003) Extreme UV
λ
1 AU = ?
If we know the distance to
Venus, we can work out the
value of 1 AU.
Aim to measure the
distance to Venus when it is
closest to the Earth using
Radio Detection And
Ranging (RADAR)
0.3 AU = 45 x 106 km
=> 1 AU = ?
Why can’t we use RADAR to measure the distance to the Sun?
Kepler’s laws were derived empirically i.e. from an analysis
of Tycho’s observational data.
Newton’s Laws
+ Gravity
Isaac Newton
Kepler’s laws do not explain why the planets revolve
around the Sun and do not fly off into space.
Planet Found in Nearest Star System to Earth
M ~ 1.14 M_Earth
d ~ 4 light-years
radial velocity ~ 51cm/s
P ~ 3.2d
1 AU
0.04 AU
0.4 AU
SUMMARY
Kepler’s 3 Laws:
I Orbits are ellipses
II The rate of change of area swept by a line
connecting a body to the focus of its elliptical orbit =
constant
∝ (semi-major axis)3 and
Period ∝ 1/(total mass)
III (Period)2
Orbital velocity
∝
1/(semi-major axis)1/2
Black Hole Orbits
The solar system
Terrestrial (Earth-like)
planets
Jovian (Jupiter-like)
planets
TERRESTRIAL
JOVIAN
Basic Form
Rock
Gas/Ice/Rock
Mean orbital dist. (AU)
0.4-1.5
5.2-30
Mean surface temp. (K)
215-733
70-165
Mass (Earth-masses)
0.055-1.0
14.5-318
Equatorial radius
(Earth-radii)
0.38-1.0
3.88-11.2
Mean density kg/m3
3933-5515
687-1638
Rotation period
(equator)
23.9h--243d
9.9h-17.2h
No. of moons
0-2
13-63+
Ring systems
no
yes
Rotation axes: why?
Tidal Forces
For Kepler’s laws, we treated the stars etc. as
point masses, and assumed that they were spherically
symmetric.
Tidal Forces One side of the moon is closer to the
Earth than the opposite side. Newton’s
3rd law => same must happen on Earth
=> differential forces => non-spherical
shape
m
M
dr
r
Lunar and Solar Tides
Sun also exerts a force on Earth => 2 tidal bulges: one
pointing towards the moon, and one towards the Sun.
Sun x375 further away, but 27 million times more
massive!
Lunar and Solar Tides
• Tidal bulges not directly
aligned with the moon,
because rotation period
of Earth slightly shorter
than rotation period of
the moon.
• Frictional forces drag the
bulge axis ahead of the
Earth-Moon line.
Dissipative force =>
rotational KE being lost
=> Earth’s spin rate is
being slowed down
0.0016s / century
•
Frictional forces drag the bulge axis ahead of the
Earth-Moon line. Dissipative force => rotational
KE being lost => Earth’s spin rate is being slowed
down 0.0016s / century
• Some of this energy transferred to the moon c.f.
fairground ride (faster spin, larger force
outwards)
• => Moon is moving away from Earth at the rate
of ~3.8 cm per year
• Newly-formed Earth: 1 day ~ 5 hours
• Moon much closer to the Earth
• Braking force due to moon over 4.5bn years =>
1day = 24h
• Coral records
• Slow-down of Earth’s rotation => same side of
Earth will face the moon => travel half-way
around the world to see the moon!
“synchronous rotation” (c.f. Martian moons)
• Consequences for humans?!
• Triton giant moon of Neptune is in synchronous
rotation, but retrograde orbit => tidal bulges are
causing it to spiral inwards
• ---> Roche limits
The Big Bang Theory (excerpt)
Summary from last lecture:
Tidal forces:
• Gravitational force proportional to r
• This results in a differential force (and therefore a
-2
differential acceleration). Proportional to r-3
• For the Earth-Moon system, energy is lost through
tidal friction => rotation rate of Earth slowing down
• Conservation of angular momentum => moon is
being pushed further away from Earth
Summary cont’d:
• Synchronous orbit: orbiting body (e.g. moon) has a
period equal to the body that it is orbiting (e.g. Earth)
• This results in tidal-locking i.e. same side of body faces
the other.
• Interesting case of Triton, the giant moon of Neptune
which is in a retrograde orbit. Tidal bulges such that
Triton is slowly spiralling inwards towards Neptune.
(c.f. recent Earth-mass planet)
Summary cont’d:
Prograde vs Retrograde
Roche Limits
R
r
µ
d
At some distance, the tidal force lifting the rock from
the surface F (tidal) = F(grav.).
Remember: differential force goes as r-3 so as r
decreases, tidal forces get more severe
R
r
µ
d
Example 1: Interacting Binary Star
“Spaghettification”
Ma
gra teria
vit l di
ati rec
on
al ted a
for
ce long
ve
cto
r
Example 2: Tidal forces near a black hole
Example 3: Comet Shoemaker-Levy 9
Example 4: Saturn’s Rings
RP
2.5RP
Example 4: Saturn’s Rings
Average density of Saturn ~ 687 kg/m3
Radius (Saturn) ~ 6 x 107 m
For a typical moon density ~ 1200 kg / m3, can calculate
Roche limit.
=> Ring system lies within this limit, larger moons are
further out.