5.1 Basic Concepts and Formulae
191
Relation Between g and G
g = G M/r
2
where M is the mass of parent body and r ≥ R.
Kepler’s Laws
(i) All planets move in an elliptic path, with the sun at one focus. This is a consequence of inverse square law of gravitation and the constancy of total energy
and angular momentum.
(ii) A line drawn from the sun to the planet sweeps out equal areas in equal times.
This is a consequence of the constancy of angular momentum.
(iii) The squares of the period of rotation of planets about the sun are proportional
to the cubes of the semi-major axes of the ellipses. This is a consequence of
the inverse square law of gravitation for circular orbits.
Central force is a conservative force which acts along a line connecting the centres of particles.
If F is a central force then curl F = 0.
Areal velocity (C) and the angular momentum (J ):
J = 2mC
(5.14)
where m is the mass of the orbiting body.
Orbits of Planets and Satellites
Circular orbits:
Orbital velocity v 0 =
G M/r =
√ gr
(5.15)
Escape velocity v e =
√
2v 0 =
2G M/R =
2g 0 R
(5.16)
Time period
T = 2π
r 3 /G M
(5.17)
If the planet’s mass cannot be ignored in comparison with the sun’s mass then
(5.17) is modified as
T = 2π
r 3 /G(M + m)
(5.18)
191
Relation Between g and G
g = G M/r
2
where M is the mass of parent body and r ≥ R.
Kepler’s Laws
(i) All planets move in an elliptic path, with the sun at one focus. This is a consequence of inverse square law of gravitation and the constancy of total energy
and angular momentum.
(ii) A line drawn from the sun to the planet sweeps out equal areas in equal times.
This is a consequence of the constancy of angular momentum.
(iii) The squares of the period of rotation of planets about the sun are proportional
to the cubes of the semi-major axes of the ellipses. This is a consequence of
the inverse square law of gravitation for circular orbits.
Central force is a conservative force which acts along a line connecting the centres of particles.
If F is a central force then curl F = 0.
Areal velocity (C) and the angular momentum (J ):
J = 2mC
(5.14)
where m is the mass of the orbiting body.
Orbits of Planets and Satellites
Circular orbits:
Orbital velocity v 0 =
G M/r =
√ gr
(5.15)
Escape velocity v e =
√
2v 0 =
2G M/R =
2g 0 R
(5.16)
Time period
T = 2π
r 3 /G M
(5.17)
If the planet’s mass cannot be ignored in comparison with the sun’s mass then
(5.17) is modified as
T = 2π
r 3 /G(M + m)
(5.18)