Derivation of Kepler’s 3 RD Law in its general form
When objects of comparable mass orbit each other they orbit about the common center of mass. It can be shown that for the general situation of two stars orbiting each other in elliptical orbits with an orbital period of P years and an orbital semi-major axis a , in AU, that the combined masses of the two stars is given by a
3 m
1
 m
2

Equation 1
P
2
The full derivation is quite complex and requires knowledge of geometry and calculus beyond the requirements of this course. However, if we assume that the stars are in mutual circular motion, the problem becomes more tractable and the result in equation 1 can be understood more intuitively.
Two stars in a mutual circular orbit
Orbit of less massive star
M
1
R
1
R = R
1
+ R
2
Center of Mass
R
2
Orbit of more massive star
M
2
Pertinent Equations
Balance of Gravitatio nal and Centripeta l Forces Definition of the Center of Mass
G m
1 m
2
R 2
 m
1 v
1
2
R
1
R
1
 m
1 m
2
 m
2
R
Orbital Velocity v
1

2

R
1
P
Unit
4

G
2
Change : length to AU, time to yr, mass to M
Solar

3 .
768

10 11 s
2  m
3 kg

1 .
00 yr
2 
M
Solar
AU
3

HIP29850
Star Spectral Type
Semi-major axis, AU
Orbital
Period,
Years
Combined Mass,
Solar Masses
HIP29850 A
HIP29850 B
Eta Cassiopeia
A2 V
A2 V
8.5 9
Star Spectral Type
Semi-major axis, AU
Orbital
Period,
Years
Combined Mass,
Solar Masses

Cassiopeia A

Cassiopeia B
Aldebaran (

Tauri)
G0 V
K7 V
71 480
Star Spectral Type
Semi-major axis, AU
Orbital
Period, Years
Combined
Mass, Solar
Masses

Tauri A

Tauri B
K5 III
K7 V
650 10,180
85 Pegasi
Star Spectral Type
Semi-major axis, AU
Orbital
Period,
Years
Combined
Mass, Solar
Masses
85 Pegasi A
85 Pegasi B
G5 V
K7 V
10.0 26.28
26 Draconis
Star Spectral Type
Semi-major axis, AU
Orbital
Period,
Years
Combined Mass,
Solar Masses
26 Draconis A
26 Draconis B
G0 V
G0 V
22.2 74
Mass estimates of Main Sequence Stars
Star Type
Individual Mass,
Solar Masses
A2 V
G0 V
G5 V
K7 V
What would be a likely guess as to the original spectral type of the K5 giant star? ____________