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月 球 探 测

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Gravity and Orbits(1)

Basic Knowledge for Lec3

- Newton Generalized Keplers laws to apply to any two bodies orbiting each other
- Orbits are conic sections with the center-of-mass of the two bodies at the focus.
- Second Law: angular momentum conservation.
- Generalized Third Law that depends on the masses of the two bodies.

- Triumphs of Newtonian Gravity

- First Law:
- Planets orbit on ellipses with the Sun at one focus.

- Second Law:
- Planet sweeps out equal areas in equal times.

- Third Law:
- Period squared is proportional to the size of the semi-major axis cubed.
- P2=a3, with P in years and a in AUs.

- Newton showed that Keplers Laws can be derived from first principles:
- Three Laws of Motion
- Law of Universal Gravitation.

- Newton generalized the laws to apply to any two bodies moving under the influence of their mutual gravitation.
- Moon orbiting the Earth.
- Two stars orbiting each other.

- The shape of an orbit is a conic section with the center of mass at one focus.
- Conic Sections:
- Curves found by cutting a cone with a plane.
- Circles, Ellipses, Parabolas, and Hyperbolas

- Center of Mass is at the Focus:
- The Earth does not orbit the Sun, the two orbit each other about their mutual Center of Mass.

Circle

Ellipse

Parabola

Hyperbola

- Conic curves come in two families:
- Closed Curves:
- Ellipses
- Circles: special case of an ellipse with e=0
- Orbits are bound and objects orbit perpetually.

- Open Curves:
- Hyperbolas
- Parabolas: special case of a hyperbola
- Orbits are unbound and objects escape.

- Velocity needed to sustain a circular orbit at a given radius from a massive body:

v vC , the orbit is an ellipse smaller than the circular orbit.

v vC , the orbit is an ellipse larger than the circular orbit.

Go a lot faster, and ...

- Minimum velocity to have a parabolic orbit starting at a given distance from a body:

- For the Earths surface:
- vC = 7.9 km/sec (28,400 km/hr)
- vE = 11.2 km/sec (40,300 km/hr)

Parabola

v = vE

Hyperbola

v>vE

Ellipse

v<vC

Ellipse

vC<v<vE

Circle

v = vC

a1

a2

M2

M1

a

- Two objects orbit about their center of mass:
- Balance point between the two masses

Semi-Major axis: a = a1 + a2

Relative positions: a2 / a1= M1 / M2

- Msun = 21030 kg
- Mearth = 61024 kg
- asun + aearth = 1 AU = 1.5108 km
- asun/aearth = Mearth/Msun = 3106
- asun = 450 km
- The radius of the Sun is 700,000 km.
- The C-of-M is deep inside the Sun.

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