Lecture 4Lecture 4

Stellar masses

SpectroscopySpectroscopyObtaining a spectrum of a star allows you to

measure:1. Chemical composition2. Distance (via spectral parallax)3. Effective temperature4. Radial velocity5. Magnetic field strength

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

1 if

zzcv

z

r

restrest

restobs

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: examplesDoppler shifts: examples

The Zeeman effectThe Zeeman effectIn 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 ml

ml +10 -1

0

cmeB e4

cmeB e4

emeB

4

Example: the Zeeman effectExample: the Zeeman effectPulsars are rapidly spinning neutron stars which beam light in opposite directions. They have huge magnetic fields of 104 – 108 Tesla. How large is the Zeeman splitting?

Kepler’s LawsKepler’s LawsJohannes 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. P2=a3, where P is the period and a is the average distance from the Sun.

32 aP

EllipsesEllipses

Ellipticity: Relates the semi-major (a) and semi-minor (b) axes: 21 eab

Equation of an ellipse:

cos1

1 2

eear

Centre of massCentre of mass

rm

r

rm

r

rrr

22

11

12

21

21

mmmm

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?

Energy and Angular momentumEnergy and Angular momentum

2

21.. vEK

vrL

The two-body problem may be treated as a one-body problem with reduced mass orbiting a fixed mass M=m1+m2

rMGEP

..

Kepler’s Second LawKepler’s Second Law2. 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.zrvprLˆ

Since L is constant,

Example: how much faster does Earth move at perihelion compared with aphelion? Recall e=0.0167

Angular momentum conservationAngular momentum conservation

ee

rr

vv

vrvr

LL

a

p

p

a

ppaa

pa

11

034.19833.00167.1

11

ee

vv

a

p

i.e. 3.4% faster

(aphelion=perihilion)

BreakBreak

Kepler’s First LawKepler’s First LawThe radius r connecting two bodies

describes an ellipse, with eccentricity and semimajor axis related to the energy and angular momentum

Now, since: the mass m1 also moves in an ellipse with semi-major axis a1 and the same eccentricity, e, and period P.

1

2

1

2

2

1

aa

rr

mm

rm

rrm

r

22

11 ,

cos1

1 2

eear

am

a1

1

21

2

11

2121

11

1

aa

mma

mmmmama

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

Orbital angular momentumOrbital angular momentumWe know the angular momentum is constant; but what is its value?

zrvprLˆ

dtdAL 2

Since L is constant, we can take A and t at any time, or over any time interval.

Pea

PA

L ellipse

22 12

2

Example: the Sun-Jupiter systemExample: the Sun-Jupiter system

PeaL

22 12

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

Derivation of Generalized KIIIDerivation of Generalized KIII

GMaP

322 4

FromPeaL

22 12

322

22 21

MGELe

EGMa

2

and conservation of energy, we can derive