1. Lecture 4 Stellar masses

2. Spectroscopy Obtaining a spectrum of a star allows you to measure: 1.Chemical composition 2.Distance (via spectral parallax) 3.Effective temperature 4.Radial velocity 5.Magnetic field strength

3. Doppler shifts Doppler shifts of the spectral lines yield the radial (i.e. toward the observer) velocity of the star

4. 1.Typical stars in the solar neighbourhood have velocities ~30 km/s. What is the size of their Doppler shift at 500 nm? Doppler shifts: examples

5. The Zeeman effect In the presence of an external magnetic field (which defines a preferred spatial direction) the orbital energy depends on the field strength and on the quantum number m l mlml  +1 0  0

6. Example: the Zeeman effect Pulsars are rapidly spinning neutron stars which beam light in opposite directions. They have huge magnetic fields of 10 4 – 10 8 Tesla. How large is the Zeeman splitting?

7. Kepler’s Laws Johannes Kepler derived the following 3 empirical laws, based on Tycho Brahe’s careful observations of planetary positions (astrometry). 1.A planet orbits the Sun in an ellipse, with the Sun at one focus 2.A line connecting a planet to the Sun sweeps out equal areas in equal time intervals 3.P 2 =a 3, where P is the period and a is the average distance from the Sun.

8. What is an ellipse? Definition: An ellipse is a closed curve defined by the locus of all points such that the sum of the distances from the two foci is a constant: Ellipticity: Relates the semi-major (a) and semi-minor (b) axes: Equation of an ellipse: Substituting and rearranging we get:

9. Ellipses Ellipticity: Relates the semi-major (a) and semi-minor (b) axes: Equation of an ellipse:

10. Centre of mass Where we have defined the reduced mass: More generally, it is the centre of mass that is at one focus of the ellipse For the Earth-Sun system, how far is the Sun from the centre of mass?

11. Energy and Angular momentum The two-body problem may be treated as a one-body problem with reduced mass  orbiting a fixed mass M=m 1 +m 2

12. Kepler’s Second Law 2. A line connecting a planet to the Sun sweeps out equal areas in equal time intervals This is just a consequence of angular momentum conservation.

13. Since L is constant, Example: how much faster does Earth move at perihelion compared with aphelion? Recall e=0.0167 Angular momentum conservation i.e. 3.4% faster (aphelion=perihilion)

14. Break

15. Kepler’s First Law The radius r connecting two bodies describes an ellipse, with eccentricity and semimajor axis related to the energy and angular momentum Now, since: the mass m 1 also moves in an ellipse with semi-major axis a 1 and the same eccentricity, e, and period P.

16. Examples Two stars are separated by 3 A.U. One star is three times more massive than the other. Plot their orbits for e=0.

17. Orbital angular momentum We know the angular momentum is constant; but what is its value? Since L is constant, we can take A and t at any time, or over any time interval.

18. Example: the Sun-Jupiter system What is the angular momentum of the Sun-Jupiter system, where a=5.2, e=0.048, P=11.86 yr ?

19. Derivation of Generalized KIII From and conservation of energy, we can derive