2.11 Derivation of Kepler's Laws from Newton's Laws

Pre-Lecture Reading 2.11

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Kepler's Laws (KI, KII, and KIII) can be derived from Newton's Laws using calculus (which Newton also invented).
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For the special case of an object of mass, m, in a circular orbit around an object of mass, M, where m is negligible compared to M, KIII can be derived from Newton's Laws using only algebra.
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Additionally, KIII's constant of proportionality, and why it varies from central object to central object, can be determined.

The orbital path of one world around another is elliptical (not circular), with the system's center of mass at one focus.

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This reduces to KI, with the central mass, M, at this focus, when the orbiting mass, m, is negligible compared to M.

An imaginary line connecting the center of mass to either world sweeps out equal areas of that world's ellipse in equal intervals of time.

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This reduces to KII, with the imaginary line connecting the central mass, M, to the orbiting mass, m, when m is negligible compared to M.

( 14 )

P² =

4π²

(M + m)

× a³

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This reduces to KIII, except with the constant of proportionality determined, when the orbiting mass, m, is negligible compared to the central mass, M:

( 15 )

P² ≈

4π²

× a³

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The constant of proportionality depends on the central mass, M, which explains why Kepler found different constants of proportionality for the naked-eye planets orbiting the sun, for the Galilean moons orbiting Jupiter, and for Earth's moon orbiting Earth.

( 16 )

v_c =

	1/2

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This is the speed that an object must attain to enter into a circular orbit around a mass, M, at a distance, r, from it.

( 17 )

v_esc =

2GM

	1/2

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This is the speed that an object must attain to escape the gravitational field of a mass, M, by the time that it is a distance, r, from it.
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If escaping from the surface of a spherical world of radius, R, and if the escape speed can be attained quickly,
r ≈ R.
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v_esc is only 41% greater than v_c.

Material presented in this unit is related to material presented in Lab 3 of Astronomy 101 Laboratory: Our Place in Space. In Lab 3: Galilean Revolution, we:

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Measure a moon's orbit around a planet.
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Use this information to measure the mass of the planet.
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Measure the phase and angular diameter of Venus.
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Use this information to distinguish between the geocentric and heliocentric models of the universe.