Celestial Mechanics

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Celestial Mechanics
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Planetary motions

• The planets move relative to the background
  stars.
• Sometimes they show complex retrograde motions

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Epicycles

• Epicycles were introduced to explain the
  non-uniform velocities of planets, in a
  geocentric, circular-orbit theory

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Retrograde motion

• Retrograde motion is a natural outcome of the
  heliocentric model
• Inner planets orbit more quickly than outer
  planets, and so overtake them

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Distances to Interior planets

• Venus and Mercury follow the Sun around the
  ecliptic means their orbits are smaller than
  Earths
• At greatest elongation a line between the Sun and
  planet is perpendicular to a line between Earth
  and planet.

• E.g. for Venus, q46 degrees, so the distance
  from Venus to the Sun is 0.72 times the Earth-Sun
  distance

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Distances to exterior planets

• Exterior planets can be found anywhere in the
  zodiacal belt
• The true orbital period of the planet (sidereal
  period) tells how long it takes the planet to
  return to point P.
• Observe the angles PES(initially) and PE?S (one
  superior planet period later).
• The angle ESE is known from the Earths orbital
  period vs. the planets. And the triangles can be
  solved.

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Sidereal and synodic periods
It is easy to observe the synodic period this is
the time between successive oppostions (when the
Earth, Sun and planet are aligned).

• But how do we know when a planet has completed
  one sidereal period (i.e. is in the same position
  relative to the background stars?

The angular velocity of a circular orbit is
360/P. The synodic rate is the rate of the
planet relative to the Earth. So
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Tycho Brahe

• Brahe (1546-1601) believed in a geocentric
  Universe the Sun and moon go around the Earth
  (but the other planets go around the Sun)
• However, he also believed that this theory could
  be tested by making sufficiently accurate
  observations
• At time this was a revolutionary approach
  different from the idea that phenomena could be
  understood through philosophical discourse alone
• Arguably the first application of the scientific
  method

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Tycho Brahes observations
wall quadrant

• Made very accurate, naked eye observations of
  planetary motion
• Used devices for measuring angles and positions
• To measure time, he used the planetary motions
  themselves. Clocks were rare and the pendulum
  clock had not been invented

sextant
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Keplers Laws

• Johannes Kepler derived the following 3 empirical
  laws, based on Tycho Brahes careful observations
  of planetary positions (astrometry).
• A planet orbits the Sun in an ellipse, with the
  Sun at one focus (supporting the Copernican
  heliocentric model and disproving Brahes
  hypothesis)
• A line connecting a planet to the Sun sweeps out
  equal areas in equal time intervals
• P2a3, where P is the period and a is the average
  distance from the Sun.

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Break
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What is an ellipse?
Definition An ellipse is a closed curve defined
by the locus of all points such that the sum of
the distances from the two foci is a constant
Ellipticity Relates the semi-major (a) and
semi-minor (b) axes
Equation of an ellipse
Substituting and rearranging we get
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Ellipses

• Calculate the aphelion and perihelion distances
  for Halleys comet, which has a semi-major axis
  of 17.9 AU and an eccentricity of 0.967.

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Keplers Second Law

• 2. A line connecting a planet to the Sun sweeps
  out equal areas in equal time intervals
• This is just a consequence of angular momentum
  conservation.

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Angular momentum conservation

• Since L is constant,

(aphelionperihilion)
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Angular momentum conservation

• How much faster does Earth move at perihelion
  compared with aphelion? The eccentricity is
  e0.0167

i.e. 3.4 faster
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Orbital angular momentum

• We know the angular momentum is constant but
  what is its value?

Since L is constant, we can take A and t at any
time, or over any time interval.
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Keplers Third Law
The general form of Keplers third law can be
derived from Newtons laws.
Example 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?