Astronomy 2 - Lecture Notes

Stellar Mass, Binary Stars and Star Clusters

Quiz today:

1. What two quantities must be known about a star to determine its 
radius indirectly?
 
2.Which type star is the most common in the Galaxy? 
Why don't we see many of them in the H-R diagrams?

• Bolometric Magnitude, M_B, BC, CI, and T_eff
• Visual Binaries, Orbits and Mass Determination
• Spectrosopic Binaries and Orbits
• Eclipsing Binaries, Light Curves and Stellar Radii
• Mass - Luminosity Relationship
• Lifetime of Stars
• HR Diagram and Star Clusters

Luminosity Classes: 
(Table 17.3)
				Example of T=4000 K
Ia - Bright Super Giants	
Ib - Supergiants		L = 3000	R = 100
II - Bright Giants		
III - Giants			L = 20		R = 20
IV - Subgiants
V - Main Sequence/Dwarf stars	L = 0.1		R = 0.7

Spectrum of two B8 Stars:
Rigel (L = 58,000 Ls)
Algol (L = 100 Ls)

Note the difference in the width of the spectral lines
for the same spectral class. These stars have different
luminosity class. Algol is smaller and denser and its
lines are broadened by pressure.

Bolometeric Magnitude
	MV = absolute visual magnitude and does not include
		all the radiation from the star and corresponds to a 
		luminosity less than the total of all radiation.
	Mbol = absolute bolometric magnitude is derived from
		the luminosity of the star that includes all the 
		radiation       L ~ R2 T4
	Mbol = MV + BC	where BC = Bolometric Correction
		BC depends on the Temperature of the stars photosphere
		Main Sequence and Java Script Calculator for MB	
Examples:
		Sun   		B8 star 
	T	5780 K		15000 K
	B-V 	0.69  		-0.171
	BC 	1.79		-1.31
	MV	 4.93		-0.46
	Mbol	 4.75		-1.77
Use Mbol to calculate Radius not MV


Binary Stars:
Visual Binaries
	1827: First binary star measured, xi Ursae Majoris
	1844: Bessel saw wavy motion of Sirius (alpha Canis Majoris)
	1862: Alvin Clark saw Sirius B with his new refracting lens.

	Measured - 	separation (" arc)
			position angle (N to E)
	Instrument (visual - filar micrometer)
		   (photographic + microsope)
		   (CCD + SAO Image program)
	Plot versus time, eg. 70 Ophiuchi
		mA = 4.2 mB = 6.0  pa = 151  sep = 3.5"arc 
		Period = 88 years (same position as 1910)


	Problems: Projection onto the sky, not true orbit
		  Have to find inclination of orbit to line of sight

Astro-mechanics of Binary Stars
	Kepler's 3rd Law
	P2 =a3/(MA+MB)
		a in AU
		P in years
		M, mass in solar Masses

Centre of Mass
Note that only the sum of the masses may be determined from the period 
and the separation of the stars.
	In order to deduce the individual masses we must know the 
	center of mass of the system. This can be determined only 
	if the absolute mostion of the individual stars is measured.

	if rA and rB are the respective distance of stars A and B from
	the center of mass then:

		MA rA = MB rB	


EXAMPLE

We can only get the sum of the masses
If we did not know the inclination we could only get the mass and inclinatio combined.
But the inclination may be deduced by using Kepler's laws on the full orbit motion.

Spectroscopic Binary Stars

Each star is in motion about the other and the velocity 
produce a doppler shift in the spectrum that has a 
period equal to the period of rotation.

Double line spectrum. As one star is moving away another is 
moving towards us. This give a double line doppler shifted 
spectrum. (See figure below)



Example:



Single Line Spectroscopic systms
Many spectroscopic binary have only one star bright enough
to produce a spectrum and then we only see one line shifting
back and forth between red and blue shift - We can get the
period and velocity of one star.

Mass Luminosity Relation

After many star masses have been measured a graph can be
made of the masses versus the brightness of the stars.
This produces a mass-luminosity relation for Main Sequence
stars - but not other luminosity classes
 


The relation can be approximated by the formula:

	L/Ls = (M/Ms)n

	The value of n = 3 to 4 with an average of 3.5.

Most references use n = 3.5 but Chaisson and MacMillan use n = 3.

EXAMPLE:
	Sirius A has a luminosity about 100 Ls
	which would mean that it has a mass of (L/Ls)2/7Ms
		= 1002/7 Ms = 3.7 Ms	
	(A bit bigger than the actual value found from its binary system)

Lifetime of a Star
	The rate with which a star uses up its energy (and hence mass)
	is proportional to its luminosity so its lifetime will be 
	proportional its mass and inversely proportional to its 
	luminosity.
		lifetime ~ mass/luminosity

	But since L ~ M3.5 we can see that
		lifetime ~ M/M3.5 = 1/M2.5

	This gives the surprising result that 
	MORE MASSIVE STARS HAVE SHORTER LIFETIMES.
	
	Since the Sun is known to have a lifetime of 10 billion years
	we can form an equality for main sequence stars.

		lifetime = 1010 yr x (Ms/M)2.5

Example:
	Massive Star, M = 10 Ms
		lifetime = (10 billion yrs)/102.5 = 31 million years
   
	Two solar Mass Star
		lifetime = (10 billion yrs)/22.5 = 1.8 billion years

	Red Dwarf Star M = 0.1 Ms
		lifetime = (10 billion yrs)/(0.1)2.5 = 3.2 trillion years

We see that a small change in mass makes a dramatic change in lifetime.

Eclipsing Binaries

There can be partial or total eclipse of the stars.
The size of the stars effect the light curve of 
the stars and the light curve can be used to determine
the sizes.



If the stars are very close binary stars,
with separations comparable to the size of
the star then the effects of tidal distortion
and mutual heating can be seen in the light curve.



A very special eclipsing binaries:
	
	Visible short period: beta Persei, Algol

	Supergiant system: beta Lyrae