Lecture 16: Stellar Masses
Astronomy 101/103
Terry Herter, Cornell University
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Lecture
Topics
  • Binary Stars
  • Stellar Masses
  • Summary of Stellar Properties
  • Implications

 

Types
of
Binaries

  • Visual Binary
    • Stars are separated in a telescope.
  • Spectroscopic Binary
    • See two sets of spectral lines Doppler shifted due to orbital motion.
  • Eclipsing Binary (rare)
    • Stars cross in front of one another.

Visual
and
Spectroscopic
Binaries



Our simulation of a spectroscopic binary demonstrates how the radial velocity varies with various oribital parameters. Follow the link and take a look.


Eclipsing
Binary

 



Note: The vertical (apparent magnitude) axis is wrong. It should read 7 on the top and 8 on the bottom.

 

You can link to our eclipsing binary star simulation to see how this works.

Masses
of
Binary Stars
  • Newton's laws allow us to determine the total mass in a binary system.
  • For star of mass MA and MB, the total mass is related to the period, P, and the average distance between the stars, a.


Keplers
Laws of
Planetary Motion
  1. Law of Ellipses
  2. Law of Areas
  3. Harmonic Law
  • Apply all laws to binary stars


Law 1:
1. Law of Ellipses

Orbits are ellipses


Law 2:

2. Law of Areas

Equal areas in equal time.



Orbital speeds


Law 3:

3. Harmonic Law

  • P = period of revolution in years
  • a = mean separation in AU
  • MA and MB = mass in solar masses

Example:

  • If a visual binary has a period of 32 years and an average separation of 16 AU then


Newton =>
Kepler !
  • Force = mass x acceleration
  • Acceleration is change in velocity with time.
  • Force in Newtons.


Circular
Motion


Newton's
Gravity


Putting it all
Together

The velocity and the orbital period, P, are related. The circumference is 2*pi*r, so


Orbital
Elements
 

- Apparent Orbit - as seen on the sky
- Real Orbit

Orbital Elements include:

  • Size of major axis
  • Eccentricity of Orbit
  • Period of Revolution
  • Orbit orientation angle
  • Inclination of Orbit


Notes on
Binary
Masses
  • Don't mistake the mass for magnitude!
  • P is measured in years, a in AU and mass in solar masses in this formula.

  • We can only get the total mass.

 


Main
Sequence
Stellar
Properties

Spectral
Class
Mass
(Msun)
L
(Lsun)
Temp.
(K)
Radius
(Rsun)
O5
40
400,000
40,000
13
B0
15
13,000
28,000
4.9
A0
3.5
80
10,000
3.0
F0
1.7
6.4
7,500
1.5
G0
1.1
1.4
6,000
1.1
K0
.08
.46
5,000
0.9
M0
0.5
0.08
3,500
0.8

 


Stars
Die!
  • The fuel in stars is proportional to the mass, M.
  • It is found that the luminosity of stars on the main-sequence varies with mass as:


Stellar
Lifetimes
  • Assuming stars "consume" the same fraction of their mass (M), the lifetime, T, is given by:

 
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