Download Lecture 11, PPT version

Outline - Feb. 25, 2010
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Observational evidence for Black Holes (pgs. 600-601)
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Properties of Stars (Ch. 16)
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Luminosities (pgs. 519-523)
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Temperatures (pg. 524)
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Radii (pgs. 534-535)
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Masses (pgs. 528-529)
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“Spectral Type” (pgs. 525-527)
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Hertzsprung-Russel Diagram (pgs. 530-533)
Black Hole “Binary” System
A star is observed to be in orbit (by Doppler
shift of spectrum) around an “invisible” object.
Optical image of Cygnus X-1
Artist’s conception of black hole
binary system Cygnus X-1
If star is sufficiently large and is sufficiently
close to its unseen companion, matter from
the star may transfer over and build up in an
“accretion disk” around the black hole.
Gas spirals toward BH, is accelerated up to
high speeds by gravity, suffers violent
collisions and heats up (millions of degrees =
X-ray emission).
Note: stars are not strong X-ray sources
Constraining the Size of the Region that Contains the
Invisible Mass
If the X-ray light flickers (on/off) very rapidly, this places a direct constraint
on the size of the accretion disk (just outside the event horizon).
Time scale over which you observe the light to be flickering must be
smaller than the time it takes for light to travel across the accretion
disk, or you won’t notice the flickering - it will be smeared out!!
Example: sound waves (time delay of arrival of sound due to its
distance; e.g. thunder vs. lightning)
World’s Longest, Loudest Marching Band
All band members play one short, staccato note.
What do you hear?
Speed of sound = 343 m/s, so you don’t hear the back
row of the band until 10 seconds after the single
note is played
If band plays 1 staccato note every half second you
would hear continuous sound (no “quiet” or “off” time)
If band plays staccato notes more than 10 seconds
apart, then you will notice breaks in the sound
Band is 37.5 times the length
of a football field…
Time to traverse the length of the band has to be
shorter than the time between which the notes occur
in order for you to experience “off” time (same goes
for light)
Constraining the Size, II
Diameter of the emitting region has to be less than the distance light
could travel over a time equal to the time scale for flickering (t)
D < c t
If t < 1 second, D < 300,000 km (i.e., 20% of the diameter of the sun)
So, if you see flickering on a time scale less than about 4 or 5
seconds, the size of the emitting region (the accretion disk) is
smaller than a star, so the companion cannot possibly be a star!
Back to Cygnus X-1
Cygnus X-1 consists of a bright star with
mass = 18 Msun and an unseen
companion with mass = 10 Msun
Rapidly flickering of X-rays says
companion is much too small to be a star
Most theoretically conservative
conclusion: companion is a black hole
Many such X-ray binary systems exist in
our Galaxy, with black holes that have
masses between 4 Msun and 10 Msun
These black holes were formed when an
extremely massive star died in a
supernova explosion
“Supermassive” Black Holes
MBH > 106 Msun
If something (star, disk of gas) is orbiting about a black hole, the speed of
rotation should decrease with distance from the black hole:
V = (G MBH / R)1/2
If you can measure V and R, you can deduce MBH
Look for rapidly rotating disks at the very centers of big galaxies,
motions of stars near the very center of our own galaxy (Milky Way).
What do you find?
“Supermassive” Black Holes
MBH > 106 Msun
Rapidly rotating disk within only 16 light years of the center of
giant elliptical galaxy M87 gives MBH = 3 x 109 Msun
“Supermassive” Black Holes
MBH > 106 Msun
Rapidly rotating disk within only 0.64 light years of the center of
spiral galaxy NGC 4258 gives MBH = 4 x 107 Msun
“Supermassive” Black Holes
MBH > 106 Msun
Over course of about 20 years astronomers have followed the motions of
stars at the very center of the Milky Way, and have determined their orbits
with very high accuracy. From orbital speeds of stars within 0.03 light
years of the center of the Milky Way, MBH = 106 Msun
Last word on Black Holes
• Black holes really do exit
• Black holes with mass MBH < 10 Msun probably result from death
of massive star in a supernova explosion
• Probably all large galaxies (galaxies at least as big as our own)
harbor “supermassive” black holes at their centers (formation
mechanism not yet understood)
• You have nothing to fear from black holes, you just want to stay
far enough away that the maximum speed of your space ship
exceeds the local escape speed.
Properties of Stars
(Ch. 15)
What colors can you see?
What does the color tell you?
Are “bigger” stars on the image
intrinsically more luminous that the
“smaller” stars?
It’s time to put all of our tools to use!!!!
What is a star?
(Two catch phrases)
“The sun is a mass of incandescent gas.”
“A star is a self-gravitating nuclear reactor.”
Incandescent (think standard “light bulb”) gas = extremely dense, opaque
gas, emits Black Body radiation
Self-gravitating = star holds itself together by gravity (has to balance the
“pressure” that pushes outward)
Nuclear reactor = power source is nuclear fusion (E=mc2); for 90% of a
star’s lifetime it is hydrogen that is fused, like a controlled H-bomb
Luminosity vs. “Brightness”
How much light do stars emit?
Luminosity (L) = intrinsic brightness of a source of light (amount of
radiative energy emitted per second)
Units of Luminosity: Watts (W), 1 W = 1 J/s
Apparent Brightness (b) = amount of radiative energy passing
through a given area per second
Units of Brightness: Watts per square meter (W/m2)
Luminosity is INDEPENDENT of distance (d) to source
Brightness DEPENDS on distance (d) to source
Luminosity vs. Brightness
b = L / (4 d2)
Flip the equation around and you get
L = 4 b d2
If you measure b and d, you can determine L.
How would you do this for your favorite star in the sky?
Direct measurement of distance: stellar parallax
The farther is a star, the smaller is
its parallax.
If p is measured in arcseconds, the
distance to the star is
d = 1/p
where d is in units of “parsecs”
1 parsec = 1 pc = 3.26 ly
Most accurate measurements done from space (Hipparchos satellite);
stars with distances < 1,000 pc. New satellite (GAIA) to be launched in
2012 will be able to measure distances of > 10,000 pc using parallax.
Example: What is the luminosity of Betelgeuse?
L = 4 b d2
We need “b” in units of Watts/meter2 and d in units of meters, then L will
be in units of Watts (W)
Brightness of Betelgeuse is 5.19x10-11 times the brightness of the sun and
the brightness of the sun is 1.30x103 W/m2. The brightness of Betelgeuse
is
b = (5.19x10-11)(1.30x103) = 6.75x10-8 W/m2
Parallax of Betelgeuse is p = 0.0076 arcseconds, so distance to
Betelgeuse is
d = 1/p = 1/0.0076 = 131 pc = (131 pc)(3.09x1016 m/pc) = 4.05x1018 m
So, the luminosity of Betelgeuse is
L = 4 (6.75x10-8)(4.05x1018)2 = 1.39x1031 W
Luminosity of the sun is Lsun = 3.84x1026 W, so Betelgeuse is intrinsically
(1.39x1031 / 3.84x1026) = 36,000 times more luminous than the sun.
How luminous are stars intrinsically?
Huge range (a factor of 10 billion) in stellar luminosities:
10-4 Lsun to 106 Lsun
Interestingly, the most intrinsically luminous stars are very rare,
while the most intrinsically dim stars are very numerous…
Surface Temperatures of Stars
For max measured in cm
and T measured in Kelvin,
max = 0.29 / T
T = 0.29 / max
For Betelgeuse,
max = 8.53x10-5 cm, so
T = 0.29 / 8.53x10-5 = 3,400 K
Range of stellar surface temperatures is small: about 3,000 K
to about 30,000 K. Surface temperature of the sun is 5,800 K.
Radii of Stars
Since stars are a good approximation to being black bodies, we
know that the relationship between Luminosity, Temperature, and
Radius is
L = 4 R2  T4
where  = 5.67x10-8 W / (m2 K4) is the “Stefan-Boltzmann” constant
Rearrange the equation and you get: R = (L / 4 T4)1/2
Range of radii is about a factor of 50,000: 0.01 Rsun (“white
dwarf”) to 500 Rsun (“supergiant” star = Betelgeuse)
Stellar Masses
Binary Stars
As one object orbits around
another, the lines in its spectrum
will be shifted back and forth.
When the object is coming
towards you, the lines will be
blueshifted.
When the object is going away
from you, the lines will be
redshifted.
When the object is moving tangentially with respect to
your line of sight, there is no Doppler shift and you see
the “zero velocity” line pattern. The curved magenta
line above shows you how one particular black
absorption line sweeps up and down the spectrum due
to orbital motion.
Stellar Masses
Binary Stars
To measure the masses of
the stars in a binary system,
we need to see the lines of
both stars.
When one star is moving
away from us, the other will
be coming towards us.
The star with the smaller
mass will have the larger,
faster orbit (like the planets
orbiting the sun).
Stellar Masses
Binary Stars
Both stars orbit on ellipses, and they share a
common focus (the “center of mass”).
The more massive star has the smaller orbit.
The relative amount of the Doppler shift of the
two sets of lines tells us the ratio of the stellar
masses.
If both sets of lines shift by the same amount,
the stars have the same mass.
If one set of lines shifts twice as far as the other
set, the big star is twice as massive as the small
star.
If one set of lines shifts three times as far as
the other set, the big star is three times as
massive as the small star, etc.
Get (M1 + M2) from Newton’s form of Kepler’s 3rd law and M1 / M2 from the relative
Doppler shift. After a little algebra you get M1 and M2 separately (see Mathematical
Insight 15.4 on pg. 529)
Stellar Masses
Masses of stars range from about 0.08 Msun to 150 Msun (about a factor of 1,800).
Lower limit set by how much mass you need to start H-fusion going, and upper
limit set by pressure-gravity balance.
Just because a star has a radius that is bigger than the sun doesn’t
necessarily mean that it is more massive than the sun!
Just because a star is more luminous than the sun doesn’t necessarily mean
that it is more massive than the sun!
Just because a star is hotter than the sun, doesn’t necessarily mean that it is
more massive than the sun!
It turns out that this is due to the fact that the radius, temperature and
luminosity of stars evolve over time…
Patterns to the Stars
Stellar Spectra
Depending upon the surface temperature of the star, you see different absorption lines.
The hottest stars show strong Helium lines, stars with T = 10,000 K show the strongest
Hydrogen lines, and the very coolest stars show strong lines due to molecules (like
titanium oxide).
This is really is a temperature effect, it is not reflective of different chemical
composition for the different stars!
Spectral Type
(Astronomers like to classify things / put them in bins)
The letters O, B, A, F, G, K, M
are called the “spectral type”
of the star and describe the
appearance of the spectrum
(i.e., strong helium lines but
weak hydrogen lines, strong
hydrogen lines but no helium
lines).
The spectral type classifications are historical and come from a time
when we didn’t know that the different spectra were due to different
stellar temperatures. The notation persists today, though!
Time-honored mnemonic: “Oh Be A Fine Girl/Guy, Kiss Me”
Hertzprung-Russel (H-R) Diagram for Stars
Take a huge random sample of stars
and plot up their luminosity (vertical)
and their temperature / spectral type
(horizontal, with T increasing to the
LEFT).
Remarkably, you don’t get a random
plot at all!
Roughly 90% of all stars fall on “the
Main Sequence”. These are stars that
produce energy by fusion of hydrogen
(E = mc2).
Any star that is not on the Main
Sequence is getting close to the end of
its life.