Download Comparing Earth, Sun and Jupiter

Planetary Motions
• Planets are observed to move relative to the
background stars
¾ Motion is generally regular, but sometimes shows
retrograde motion that was very difficult to
explain in geocentric theories
ƒ Led to use of epicycles
¾ In a heliocentric theory retrograde motion is a
natural consequence of the inner planets orbiting
more quickly than the outer planets
• Distances to planets can be determined using simple
geometric calculations
¾ Interior: measure angle between planet and the
Sun at greatest elongation (when they are
farthest apart)
¾ Exterior: measure same angle, separated in time
by planet’s orbital period.
¾ Copernicus had excellent values for all of the
known planets, but they were less precise for
the outermost ones (Jupiter and Saturn)
because their periods are long enough that
fewer oppositions could be observed
Kepler’s Laws
• Based on very accurate orbital data obtained by Tycho
Brahe, Kepler derived three laws of orbital motion:
1. A planet orbits the Sun in an ellipse, with
the Sun at one focus
2. A line connecting a planet to the Sun sweeps
out equal areas in equal time intervals
3. For planets orbiting the Sun, P2=a3, where P
is the period (in years) and a is the average
distance from the Sun (in AU).
Kepler’s First Law
• Planets move around the Sun in an ellipse, with the Sun
at one focus
¾ The distance r between the Sun and the planet is
related to the semimajor axis length, a, and the
(
)
a 1 − e2
eccentricity, e, by: r = 1 + e cos θ , where θ is the
angle between r and the semimajor axis.
¾ The eccentricity relates the length of the
semimajor axis (a) and semiminor axis (b) :
a 2 = a 2e 2 + b 2
b
= 1 − e2
a
• The aphelion is the point in the orbit when the planet is
farthest from the Sun. Perihelion is the point of
closest approach.
Kepler’s Second Law
• Conservation of angular momentum means the quantity
r
L = mrvθ does not change with time. Here, vθ is the
component of the planet’s velocity perpendicular to the
vector r.
¾ Thus the planet moves fastest at perihelion, and
slowest at aphelion:
va
vp
=
rp
ra
=
(1 − e)
(1 + e)
¾ The angular momentum is given by
L = 2m
πa 2 1 − e 2
P
Kepler’s Third Law
•
This law relates the Period (P) and semimajor axis
length (a) for an object of mass m, orbiting a more
massive object of mass M.
4π 2 a 3
2
P =
G ( M + m)
where, G is the gravitational constant.
• For the solar system, where the mass of the Sun is
much greater than that of any planet, a very good
4π 2 a 3
2
approximation is P = GM
Sun
¾ Since the constants are the same for all the
objects in the solar system, we can write
P 2 = a 3 , if P is measured in years, and a is
measured in AU.
Exercises
1.
Calculate the aphelion and perihelion distances
for Halley’s comet, which has a semi-major axis of
17.9 AU and an eccentricity of 0.967.
From the equation for an ellipse:
(
)
a 1 − e2
r=
1 + e cos θ
we have a=17.9 AU and e=0.967. At closest approach
(perihelion), θ=0 degrees, so:
(
)
a 1 − e2
= a(1 − e)
rp =
1+ e
= 17.9 × 0.033 = 0.5907 AU
While at aphelion, θ=180 degrees:
(
)
a 1 − e2
= a(1 + e)
ra =
1− e
= 17.9 × 1.967 = 35.21AU
which is beyond the orbit of Neptune.
2.
How much faster does Earth move at
perihelion, compared with its velocity at aphelion?
The eccentricity of Earth’s orbit is e=0.0167. The
ratio of the velocities at perihelion and aphelion are
given by:
vp
va
=
(1 + e )
(1 − e )
1.0167
0.9833
= 1.034
=
So the Earth moves 3.4% faster at perihelion.
3.
The dwarf planet Eris has a small moon,
Dysnomia. This moon orbits at a distance of
about 30,000 km, with a period of about 14
days. What is the combined mass of the
Eris/Dysnomia system?
4π 2 a 3
From Kepler’s third law, P = G ( M + m) , we can solve
4π 2 a 3
22
for ( M + m) = GP 2 = 1.09 × 10 kg , about 1/7th the mass
2
of the Earth’s moon.