Newton's law of gravitation, orbital mechanics, and Kepler's laws
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1
โ
m2โ
โ
where G=6.67ร10โ11 Nยทmยฒ/kgยฒ is gravitational constant.
Vector form:F12โ=โGr2m1โm2โโr^12โ
Gravitational Potential Energy
Choosing U=0 at r=โ:
U(r)=โGrm1โm2โโ
Work to bring masses from infinity to separation r:
W=โซโrโFdrโฒ=โGrm1โm2โโ
Gravitational Field
gโ=โr2GMโr^
Gravitational potential:ฮฆ=โrGMโ
gโ=โโฮฆ
Escape Velocity
Minimum velocity to escape gravitational field:
Set total energy = 0:
21โmvesc2โโGRMmโ=0
vescโ=R2GMโโ
For Earth: vescโโ11.2 km/s
Circular Orbits
For circular orbit of radius r:
Centripetal force = Gravitational force:
rmv2โ=Gr2Mmโ
Orbital velocity:v=rGMโโ
Orbital period:T=v2ฯrโ=2ฯGMr3โโ
Orbital energy:E=KE+PE=21โmv2โGrMmโ
E=โ2rGMmโ
(Negative: bound orbit)
Note: โฃEโฃ=KE (virial theorem)
Kepler's Laws
First Law (Law of Ellipses):
Planets move in elliptical orbits with the Sun at one focus.
Second Law (Law of Equal Areas):
Line from Sun to planet sweeps equal areas in equal times.
This follows from angular momentum conservation:
dtdAโ=2mLโ=constant