Astronomy 224 Lecture 3

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Astronomy 224Lecture 3

• Applying Newtons work to Keplers Laws

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Keplers Laws

• Keplers Law are descriptive
• Newton expressed the underlying principles in his
  3 laws of motion and law of universal gravity
• We can derive Keplers laws from Newtons Laws

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Newtons Laws of Motion

• The velocity of an object remains constant (in
  both magnitude and velocity) unless a NET force
  acts upon the body.
• The acceleration imparted to a body is
  proportional to and in the direction of the force
  and inversely proportional to the mass of the
  body
• For every force acting on a body (in a closed
  system) there is an equal and opposite force

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Universal Law of Gravitation

• where G 6.67 10-11 N/m2kg2
• the force of gravity is attractive, obeys
  Newtons 3rd law
• describes the gravitational force between two
  point masses spherically symmetric bodies
  behave gravitationally as though the entire mass
  were concentrated at the center.

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Gravitation Example

• What is the acceleration of a person of mass m
  just above the surface of the Earth (neglecting)
  rotation ?

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Gravitation Example

• What is the orbital period of a satellite of mass
  m in a circular orbit about the Earth ?

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Geosynchronous Satellites

• These satellites remain directly above a certain
  point on the equator for communication purposes
  and thus have an orbital period of 24 hours.
  What is their altitude?

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Applying the Law of Areas
Fig 1-13
So H is the angular momentum per unit mass
Now at perihelion an aphelion, vt is the total
velocity. Substituting for a and r yields
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Substituting for the semi-minor axis
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Ex Calculate vap and vper for Mercury
So the speed of Mercury ranges from 39 km/s to 59
km/s. Note that for the Earth the ranges if only
29.3 km/s to 30.3 km/s much smaller because of
the smaller eccentricity.
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Newtons Version of Keplers 3rd Law
Two bodies of masses m1 and m2 orbiting their
stationary center of mass at distances r1 and r2.
Fig. 1.4
and since these must be equal
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Expressing one distance in terms of the masses
Substituting this into an expression for the
force of gravity
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Ex. Get the suns mass from a planets orbit

• Uranus P 84.01 years, a 19.18 AU

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Vis-Viva Equation
The Vis-Viva is a very useful expression of
Conservation of Energy. Returning to our two
masses moving around their common center of mass.

From conservation of momentum
Paralleling what we derived for a, v v1 v2
since v is the relative speed of either body with
respect to the other.
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and using
Evaluate at perihelion where
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Now since energy is conserved, the object must
have this value of total energy everywhere in its
orbit.
The vis-viva equation tells us the total orbital
speed depends only upon the separation and the
orbits semi-major axis. We can now solve for
the speed anywhere in an orbit.