Download Lecture 17 Review

Lecture 17 Review
Most stars lie on the Main sequence of an H&R diagram including the Sun, Sirius,
Procyon, Spica, and Proxima Centauri. This figure is a plot of logL versus logT.
The main sequence is explained by the stable burning of hydrogen, the most efficient,
lowest temperature energy producer. The radiation produced keeps the star in
hydrostatic equilibrium with the Helmholtz gravitational contraction forces.
Remember from Stefan-Boltzmann that
L = σ AT4 = 4 πσ R 2 T4 , thus log L = log( 4 πσ R 2 ) + 4 log T
This says that lines of constant radius are straight lines on this H&R diagram. These
are shown. Note that over about b of this plot, radius changes very slowly for stars
on the main sequence, i.e. increased mass contributes to increased density with little
change in radius.
While the Main Sequence represents stable hydrogen burning for a variety of different
mass stars, there are stars off the main sequence:
White Dwarfs:
T • 2 Tu, hotter than the Sun
R • RSun /100
L • LSun /100
Giants:
T • TSun , same temperature as the Sun
R • 10RSun
L • 100 LSun
Super Giants:
T • TSun /2, cooler than the Sun
R • 1000 RSun
L • 10,000 LSun
The Russell-Vogt Theorem (1926) states that the equilibrium structure of a main
sequence star is uniquely determined by
1)
mass
1)
chemical composition
This implies that all 1 MSun stars with the same chemical composition occupy the same
spot on the H&R diagram with the same temperature, radius, and luminosity. This is a
way to distinguish the Sun from white dwarfs and certain giants.
More important, this gives us a way to measure distances beyond 100 pc, the limit of
parallax measurements.
1)
Measure temperature and spectra to determine where on the H&R plot a star
belongs.
2)
This determines the absolute magnitude and luminosity.
3)
Use 1/r2 dependence and apparent magnitude to determine distance from the Sun.
This is called the method of spectrographic parallax and is good for measurements out
to the edge of the Milky Way. The method is unreliable where the interstellar medium
dims the star in an unmeasurable way.
Example:
1)
Spica has an apparent brightness of mV = 1 and a spectrographic temperature of
20,000 K.
2)
From the H&R plot you can determine that L = 2300 Lu and M = -4.
3)
Correct for 1/r2 using the following figure and you find that Spica is 84 pc from
the Sun.
Stars form from large clouds of gas. The process usually involves the formation of
clusters of stars, many bound gravitationally together. Binary stars revolve around
each other. Time and position measurements tell us the masses of the binary pair.
From this sample of stars we find that there is a linear relationship between luminosity
and mass to some power. This is the mass-luminosity relation and leads to the
following figure showing masses along the main sequence. Masses of the giants and
dwarfs can be determined in the same way.
How stars evolve and die depends totally on mass. You might guess that a more
massive star would live longer because it has more fuel to burn, but no.
Hydrogen burning time =
energy available
mass
≅
rate of burning lu min osity ≅ mass burning
time
For stars more massive than the Sun the mass-luminosity relationship says that
L ~ M3 . 8
M
⇒ time ~ 3 .8 = M −2 .8
M
2.8
⇒
t
t Sun
M 
=  Sun 
 M 
2 .8
1
 1
For M = 10 Mu t =   t Sun =
⋅ 10 billion years ≅ 16 million years
 10 
630
This implies that big stars have a much shorter life than the Sun. Also, if you see a big
star, it is relatively young. Thus, in the night sky you see a mixture of old dim stars
and bright young stars.
Relative ages of stars can be inferred from an H&R plot of stars in an open cluster.
These stars are presumed to be formed from a common gas cloud and all to be roughly
the same age. This plot of stars in the Pleiades is an example.
Where do stars come from? Are they still being formed? The evidence for current star
formation is the presence of much gas and dust in space. This is called the interstellar
medium and where the densities are large the gas and dust formed clouds called
nebula.
Emission nebula - 100 to 104 solar masses of dust and gas, very low density, found
near hot stars. Ultraviolet radiation emitted from the stars is absorbed by the gas,
ionizing the hydrogen which emits photons characteristic of hydrogen. These are
called HII regions.
Dark nebula - Dust grains block light. The nebula appears dark in front of a bright
region.
Reflection nebula - Small dust grains at low concentration scatter light. Like the
scattering of sunlight in Earth’s sky, the process is most efficient for blue light;
therefore, the nebula have a blue cast.
Interstellar reddening - As light from stars passes through the interstellar medium blue
light is slowly scattered out of the path, leaving the resultant light with a reddish color.
Thus, remote clusters appear dimmer and redder than expected from their distance and
age.
Interstellar extinction - When enough gas and dust is in the way, far away objects
cannot be seen because light is scattered out of the line of sight. Thus, we cannot see
the galactic center with visible light.
That these clouds are the source of stars follows from several observations:
1)
The solar system is only 4.6 billion years old while the Milky Way (with 100
billion stars) is believed to be around 12 billion years old and the Big Bang is on
the order of 14 billion years old.
2)
Open star clusters like the Pleiades had some hot blue stars but no giants. This
implies that the cluster is only about 50 million years old. Very young in Big
Bang terms.
3)
Short-lived massive stars, 20 to 40 solar masses , live only a few million years
but we see them.
The process for star formation is thought to involve a large cloud of gas, say 1035 kg
or 105 solar masses, cold enough so that gravity can overcome random thermal
motion. The cloud contracts gravitationally. Fragmentation and sub-fragmentation
produce clusters of stars, some individual stars, others binaries, some other larger
groupings. These may disperse with time due to random motion.
Clusters
1)
Population II clusters - globular clusters - are old stars in groups of 104 to l05
stars. They form a halo about the galaxy formed when the galaxy was a huge
sphere of gas, before gravity and angular momentum had collapsed the gas into a
disk. Their paths are typically very elliptical, passing through the center of the
galaxy and then out again. They are very old, with many stars off the main
sequence. These stars contain few heavy elements.
2)
Population I clusters - open clusters - are relatively new stars in groups of
approximately 103 stars. The are 100,00 to 100 million years old with mainly
main sequence stars. The Pleiades is an example. Their spectra are metal rich.
Stellar Evolution
1)
As gas collapses towards a single star, it heats up. When the cloud has the
dimensions of the current solar system is may have a temperature of 2000 K,
highly luminous, but radiating mostly in the infrared. This is called a protostar.
2)
As the contracts, it surface area diminishes so it becomes less luminous.
3)
As the temperature increases the cloud becomes more opaque, thus, trapping the
radiation inside the cloud and increasing the temperature within.
4)
At some point the core temperature reaches the ignition point and the protostar
becomes a star. The star is still hidden from view by much remaining gas.
5)
Magnetic fields sweep the remaining dust away, sometimes huge amounts, to
reveal the new star.
The question is, what if the mass is greater than about 50 solar masses? If the forming
star is too large, the gas cloud condenses quite fast, is unstable, gets very hot, and
either explodes or fragments into smaller clouds which form individual stars.
A second question is, can the mass of the gas be too small. The answer is yes. If the
mass of the cloud is too small it heats up from gravitational contraction, but never gets
hot enough to ignite. The gas ball then reaches some equilibrium and cools off.
Jupiter is a case in point. It still radiates more energy than it absorbs from the Sun, but
the source of the radiation is not thermonuclear processes. This type of gas giant is
called a brown dwarf. The process is shown below.