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Planetary motion
Earth, Moon, other planets…
Fred Sarazin ([email protected])
Physics Department, Colorado School of Mines
PHGN324: Planetary motion
Ancient astronomy (I)
• Ancient astronomers were not able to observe parallax as evidence of the
motion of the Earth. They concluded incorrectly that the Earth must be at the
center of the Universe. In fact, the parallax effect exist, it was just too small for
the ancient astronomers to observe and measure.
An example of the parallax effect
Fred Sarazin ([email protected])
Physics Department, Colorado School of Mines
PHGN324: Planetary motion
Ancient astronomy (II)
• The issue with the retrograde motion of the planets.
• Example: every 2.14 years, Mars
passes through a retrograde loop.
Very puzzling.
• Solution proposed: epicycles (introduced
by Ptolemy around AD 140).
Fred Sarazin ([email protected])
Physics Department, Colorado School of Mines
PHGN324: Planetary motion
Epicycles
The “standard model” of ancient astronomy before Copernic
Fred Sarazin ([email protected])
Physics Department, Colorado School of Mines
PHGN324: Planetary motion
The Copernican revolution
• Heliocentric model – the Sun is at the center of the Universe. Not correct either
since the Universe has no center (see Cosmology), but solves the problem of the
very complex motions of the planets around the Earth!
• Nicolaus Copernicus (1473-1543)
Fred Sarazin ([email protected])
Physics Department, Colorado School of Mines
PHGN324: Planetary motion
The retrograde motion of the planets explained!
• Simply an effect of the relative motion of
the planets with respect to each others.
• Further improvements from Tycho Brahe
(1546-1601) using high-precision
instruments for precise astronomical
observations meticulously reported in
tables.
• Planetary motion laws developed by
Johannes Kepler (1571-1630). Planets
move around the Sun on elliptical paths
with non-uniform velocities.
Fred Sarazin ([email protected])
Physics Department, Colorado School of Mines
PHGN324: Planetary motion
100 years of modern astronomy
Kepler’s laws predate Newton’s law of
Gravity!
Fred Sarazin ([email protected])
Physics Department, Colorado School of Mines
PHGN324: Planetary motion
Kepler’s laws of planetary motion (I)
• First law: the orbits of the planets are ellipses with the Sun at one focus.
Note: in reality, the orbits of the planets in the solar
system are virtually indistinguishable from circles, but
their centers can be significantly offset from the Sun.
a(1− e 2 )
r(θ ) =
1+ ecos(θ )
With a: semi-major axis
and e: eccentricity
Earth: e=0.0167
Pluto (extreme case): e=0.248
(Derivation on the board – using Newton’s law of gravity)
Fred Sarazin ([email protected])
Physics Department, Colorado School of Mines
PHGN324: Planetary motion
Kepler’s laws of planetary motion (II)
• Second law: a line segment joining a planet and he Sun sweeps out equal areas
during equal intervals of time
(Derivation on the board – using Newton’s law of gravity)
Fred Sarazin ([email protected])
Physics Department, Colorado School of Mines
PHGN324: Planetary motion
Kepler’s laws of planetary motion (III)
• Third law: the square of the orbital period of a planet is proportional to the cube
of the semi-major axis of its orbit.
𝑇"
4𝜋 "
4𝜋 "
=
≈
$
𝑎
𝐺(𝑀 + 𝑚) 𝐺𝑀
Where:
T: orbital period of the planet around the Sun
a: semi-major axis of the planet orbit
M: mass of the star / Sun
m: mass of the planet (m<<M in solar system)
G: Newton’s gravitational constant (G=6.67x10-11m3 kg-1 s-2 in SI units)
CAUTION: make sure to use consistent units throughout!
Note for Earth:
3
a =T
2
with a in astronomical unit [AU]
and T in Earth year.
(Derivation on the board – using Newton’s law of gravity)
Fred Sarazin ([email protected])
Physics Department, Colorado School of Mines
PHGN324: Planetary motion
Kepler’s laws of planetary motion – 3rd law
Fred Sarazin ([email protected])
Physics Department, Colorado School of Mines
PHGN324: Planetary motion
Exercise
The orbital period of Io, one of the four Galilean
moons of Jupiter is 1.77 days.
The semi-major axis of its orbit is 4.22x105 km.
Assuming the mass of Io is insignificant
compared to the mass of Jupiter. Calculate the
mass of Jupiter. Compare it to the mass of the
Earth (M⊕=5.972x1024kg).
Fred Sarazin ([email protected])
Physics Department, Colorado School of Mines
PHGN324: Planetary motion
Exercise
Find the altitude of the geostationary orbits
using Kepler’s third law (with M⊕=5.972x1024kg
and R⊕=6370 km).
Fred Sarazin ([email protected])
Physics Department, Colorado School of Mines
PHGN324: Planetary motion
Orbital motion
• Kepler’s first law describe the relative orbit of two
objects (with mass).
• In the case of a satellite to be launched into space,
the mass m of the satellite is obviously much
smaller than the mass of the Earth M⊕.
• Hence, the center of gravity of the system is
located at the Earth center, and the relative motion
really describe the trajectory of the satellite.
• Recall the Kepler’s first law depends on initial
conditions. This will define the kind of orbit the
satellite will be on. If the initial velocity is high
enough (above the escape velocity), then the
satellite will no longer be a satellite since it will be
on an open trajectory escaping the Earth
gravitational pull!
Fred Sarazin ([email protected])
Physics Department, Colorado School of Mines
PHGN324: Planetary motion