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PHGN324: Planetary motionFred Sarazin ([email protected])Physics Department, Colorado School of Mines
Planetary motionEarth, Moon, other planets…
PHGN324: Planetary motionFred Sarazin ([email protected])Physics Department, Colorado School of Mines
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
PHGN324: Planetary motionFred Sarazin ([email protected])Physics Department, Colorado School of Mines
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).
PHGN324: Planetary motionFred Sarazin ([email protected])Physics Department, Colorado School of Mines
Epicycles
The “standard model” of ancient astronomy before Copernic
PHGN324: Planetary motionFred Sarazin ([email protected])Physics Department, Colorado School of Mines
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)
PHGN324: Planetary motionFred Sarazin ([email protected])Physics Department, Colorado School of Mines
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.
PHGN324: Planetary motionFred Sarazin ([email protected])Physics Department, Colorado School of Mines
100 years of modern astronomy
Kepler’s laws predate Newton’s law ofGravity!
PHGN324: Planetary motionFred Sarazin ([email protected])Physics Department, Colorado School of Mines
• First law: the orbits of the planets are ellipses with the Sun at one focus.
Kepler’s laws of planetary motion (I)
With a: semi-major axis and e: eccentricity
(Derivation on the board – using Newton’s law of gravity)
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.
r(θ ) = a(1− e2 )1+ ecos(θ )
Earth: e=0.0167Pluto (extreme case): e=0.248
PHGN324: Planetary motionFred Sarazin ([email protected])Physics Department, Colorado School of Mines
• Second law: a line segment joining a planet and he Sun sweeps out equal areas during equal intervals of time
Kepler’s laws of planetary motion (II)
(Derivation on the board – using Newton’s law of gravity)
PHGN324: Planetary motionFred Sarazin ([email protected])Physics Department, Colorado School of Mines
• Third law: the square of the orbital period of a planet is proportional to the cube of the semi-major axis of its orbit.
Kepler’s laws of planetary motion (III)
Where:T: orbital period of the planet around the Suna: semi-major axis of the planet orbitM: mass of the star / Sunm: 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: a3 = T 2 with a in astronomical unit [AU]and T in Earth year.
(Derivation on the board – using Newton’s law of gravity)
𝑇"
𝑎$ =4𝜋"
𝐺(𝑀 +𝑚) ≈4𝜋"
𝐺𝑀
PHGN324: Planetary motionFred Sarazin ([email protected])Physics Department, Colorado School of Mines
Kepler’s laws of planetary motion – 3rd law
PHGN324: Planetary motionFred Sarazin ([email protected])Physics Department, Colorado School of Mines
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).
PHGN324: Planetary motionFred Sarazin ([email protected])Physics Department, Colorado School of Mines
Exercise
Find the altitude of the geostationary orbits using Kepler’s third law (with M⊕=5.972x1024kg and R⊕=6370 km).
PHGN324: Planetary motionFred Sarazin ([email protected])Physics Department, Colorado School of Mines
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!