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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

Gravity and Orbits(1)

Basic Knowledge for Lec3


Key ideas

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


Kepler s laws revisited

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.


Newton s generalization

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

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.


Conic section curves

Circle

Ellipse

Parabola

Hyperbola

Conic Section Curves


Closed and open orbits

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

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

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)


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Parabola

v = vE

Hyperbola

v>vE

Ellipse

v<vC

Ellipse

vC<v<vE

Circle

v = vC


Center of mass

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

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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