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月 球 探 测 PowerPoint PPT Presentation


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月 球 探 测. Gravity and Orbits(1). Basic Knowledge for Lec3. Key Ideas:. Newton Generalized Kepler ’ s 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.

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月 球 探 测

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Gravity and Orbits(1)

Basic Knowledge for Lec3


Key Ideas:

  • 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


Keplers Laws Revisited

  • 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.


Newtons Generalization

  • 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.


First Law of Orbital Motion

  • 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 Section Curves


Closed and Open Orbits

  • 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.


Circular Velocity

  • 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 ...


Escape Velocity

  • 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

Center of Mass

  • 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


Example: Earth and Sun

  • 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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