Lecture 17: Stellar Evolution

 The End of a Star's Life

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Dramatic changes to a star's structure and output occur when the hydrogen in the star's core is consumed. To understand the behavior of the star and its core through these changes, we need to understand the differences between ordinary and degenerate gases.

Ordinary, non-degenerate gas:

    P = nkT        

( n = no. of particles/m3, k=Boltzman's Constant, T=temperature)

If the temperature of such a gas increases, the pressure increases:

Pressure = force per unit area exerted by gas particles

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momentum transfer across area A in time t =

No. of particles x average x-momentum per particle =

No. density x volume x ave x-momentum per particle

= n x A x vx x t x px = nAvxtpx

n = no. of particles/m3, A = area vx = x-velocity of particles px = x-momentum

or P = nvx px=nmvx2

Since a gas is comprised of particles moving in any of 3 directions, not necessarily towards the"wall",


where the brackets mean "average".

Recall that mv2 = 3kT, so

            P = nkT for normal matter.

A degenerate gas is one where quantum mechanical effects on the electrons dominate the behavior:

Pauli's exclusion principle: No two electrons can have exactly the same quantum mechanical state meaning that no two can have identical velocities, spins, and positions.

Heisenberg's Uncertainty Principle: It is impossible to know the position and momentum of a particle to an accuracy better than Planck's constant h for the product of the uncertainty in position and momentum:


For room temperature circumstances, this limit prevents us from knowing exactly where an electron is within an atom:

Note that the full velocity was used in the above, 10% of that might be a reasonable uncertainty so the allowed value for the position uncertainty would increase to 10-7 meters.

What do these principles mean for the behavior of a gas?

Note that the core of a star becomes a collection of positively charged ions (largely He and C nuclei) floating in a sea of electrons which have been stripped from the nuclei after H burning ceases.

Start from

            Pressure = nvxpx

as shown above and apply it and the quantum mechanical principles to a gas comprised of electrons.


So the electron gas pressure becomes


A more careful derivation that averages over a distribution of positions, momenta, and so on yields


Since we do not usually measure the number density of electrons ne but rather the mass density of the object, take advantage of the fact that a star's core (or any similar object) will be electrically neutral:


The mass density is


Look at this pressure carefully! It does not depend on temperature!

If we assume that a stellar core (or other object such as a white dwarf) is supported by electron gas pressure at some time after fusion stops, we can derive a relation between the mass of the core (or object) and its radius:

(recall the calculations of central pressure for the Sun which have been modified here to reflect the fact that P is proportional to density5/3):

"central" indicates pressure, density at the center


Equate this central pressure with that which can be provided by an electron gas and substitute for the density:

This expression for the radius illustrates the interesting property that an object supported by an electron gas is smaller if it is more massive. Note that Z/A~0.5 for most elements from He to Fe which are likely to be found in the cores of stars. This relation is also applicable to white dwarfs -- note that if a white dwarf becomes too massive, its radius approaches 0. A more complete calculation that includes the effects of electrons moving at close to the speed of light reveals that R->0 when M->1.4Msun. This limit is called the Chandrasekar limit in honor of the astrophysicist who first derived  it.

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Evolution of a Low Mass Star from the Main Sequence

  1. Core H is exhausted so energy generation stops. The gas pressure cannot resist the force of gravity so the core begins to contract. As it contracts, it heats up.
  2. Gas layer immediately above the core is heated by the core contraction. H begins to burn to He in a shell surrounding the core. The energy released in this shell causes the overlying layers to expand.
  3. The core continues to contract, aided by He "ash" falling on it from the H burning shell above.

    During these initial steps the star's luminosity changes little but the outer layers are cooler due to expansion. Because the change in surface area just about counterbalances the decline in luminosity due to decreasing T, the star moves horizontally to the right in the HR diagram and is called a subgiant.

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  4. The outer layers continue to expand as the H burning shell increases its rate of energy production due to increased temperature in the shell driven by the continuing collapse of the core. The temperature of the outer layers cannot fall to arbitrarily small levels, however. Photons can escape less easily from low temperature material which is more likely to absorb them. Consequently more energy begins to move via convection and the surface temperature never gets lower than ~3000° K. The star expands without cooling off so its luminosity increases and the star moves almost vertically upwards in the HR diagram. It has become a red giant.giant.tif (135804 bytes)
  5. The star's core contracts to the point that it becomes degenerate and supported by electron pressure. The gravitational field exerted by the core on the H-burning layer above is extremely strong and the H-burning layer increases in temperature so that its gas pressure can resist gravity. The increased temperature increase the rate of H-burning so the luminosity increases and the star "ascends the giant branch" quickly.
  6. At the tip of the giant branch the core has contracted to the point that its T is high enough to ignite He to C burning via the triple-alpha process . Because the core is a degenerate gas, an increase in T due to the He-burning does not cause it to expand. Instead, the energy goes into causing even higher T which in turn causes more nuclear burning. A thermonuclear runaway occurs called the He-flash. Eventually the T rises to the point where the core is no longer degenerate and the core continues to fuse He to C in a stable fashion. A star with a stable He-burning core and an H-burning shell is called a horizontal branch star.hbranch.tif (151206 bytes)
  7. Eventually the core He is exhausted. The core again contracts and He-burning begins in a shell just above the core. H-burning continues in a layer further out. The star ascends the giant branch again but now as an asymptotic giant branch star.agb.tif (140220 bytes)
  8. The star is likely to become very unstable and begin to oscillate where it expands, overcools, contracts, overheats, expands and so on. These oscillate frequently drive off much of the outer layers of the star. Eventually all of the star's material above the core is blown away forming a planetary nebula. The core remains visible as a white dwarf.

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White dwarfs are luminous not because they are generating any energy but rather because they have trapped a large amount of heat. The luminosity of a white dwarf is slowly decreasing with time as the remnant cools off. Typical cooling times for white dwarfs are a few billion years to reach the low temperatures present in space when the white dwarf would no longer be observable.

Evolution of Massive Stars

Stars with masses > ~6 MSun follow a slightly different set of steps than those outlined above because their cores are hotter. Their cores may approach or exceed the Chandrasekar limit in mass. The He-flash does not occur in these stars because the core is hot enough before it contracts all the way to a degenerate state to ignite the triple-alpha process. The formation of an H-burning shell is very similar to that in the low mass stars. The star moves back and forth in a horizontal direction in the HR diagram as the core contracts, heats up, begins burning yet another element, and the outer layers expand.highmass.tif (212138 bytes)

When a particular nuclear fuel is exhausted, the core contracts and the star moves back to the left in the diagram until another atom is ignited. highmass2.tif (307746 bytes)

The star looks like an onion with a series of shells burning heavier and heavier elements up to Fe.highmass3.tif (482744 bytes)

When a solid Fe core is achieved, the core collapses catastrophically and the star becomes a supernova.

Novae and Supernovae

The "nova" comes from the Latin word for new star. The first such objects observed were actually supernovae exploding within our galaxy and were thought to be new, temporary stars because they brightnened sufficiently to become naked-eye objects but were invisible to the naked-eye in their pre-explosion states. The light curves for novae and supernovae are superficially similar but supernovae are much more luminous and the two classes of object are produced in very different situations.

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Properties of Novae

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Detailed observations of a nova's shell can yield an estimate of the distance to the nova:

  1. Measure the radial velocity VR of the shell which gives the expansion rate in km/sec.
  2. Measure the proper motion VT of outer edge of the shell.


where VT is measured in radians/sec, and distance in km.

Example: A nova has an radial velocity of 1000 km/sec. Its outer edge moves 0.1" in a year.

- This technique assumes that the novae's shell is expanding symmetrically which appears to be a safe assumption based on the shapes of novae shells.

A Nova Model

How much mass must disappear to produce the observed 1037 joules?

This is a very modest amount of mass compared to the masses of stars so nothing really exotic is likely to be needed.

Recall that novae are binaries so

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A star's Roche lobe is that volume where its gravitational field is stronger than that of its companion.

Further support for this model comes from recurrent novae where the magnitude of the brightness increase is larger for novae with longer intervals between episodes.

Note that any mechanism which would cause accretion of H on to a white dwarf's surface could result in a nova.


The last supernova to explode in the Milky Way went off in 1604 and was observed by Kepler, Galileo and many others. Since then, many supernovae in other galaxies have been observed -- the most dramatic of these was Supernova 1987A which exploded in the Large Magellanic Cloud, a relatively close neighbor of the Milky Way.

Supernovae are completely different beasts than novae -- a SN have light curves that are similar to those of novae but a SN can have a luminosity of 1010 Lsun.

Observations of these SN in other galaxies has revealed two broad categories of SN:

Property Type I Type II
Location Near old stars Near young stars
Spectra No H, many other lines Dominated by H lines
Ejected mass 0.5 Msun 5 Msun
Total energy 5x1043 Joules 1044 Joules
MV at maximum -19 to -20 -17
Ejection velocities 10,000 km/sec 5,000 km/sec

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What Causes a Supernova explosion?

Consider the total gravitational collapse energy that could be available if an object with M=1.4Msun and r=15 km (values appropriate to a neutron star) shrank to 0 radius:

which is more than enough to power a supernova.

Type I and Type II result from quite different pre-cursors:

Type II result from the evolution of massive stars.

Type I result from the explosion of white dwarfs close to the Chandrasekar limit that accrete enough material to be pushed over the limit.

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In a Type II, a massive star reaches the point where the core is solid Fe. The core shrinks because no further energy generation can occur. The core becomes so hot that the gamma-ray photons can penetrate to the Fe nuclei and cause them to disintegrate. So much energy is absorbed in a short period that the core collapses suddenly and conditions are reached where electrons and protons merge to form neutrons. Neutrinos are released in this conversion and escape immediately. The overlying calling material bounces off the neutron core and send out a strong shock wave that blows most of the rest of the star into space. After about a day, large quantities of visible light begin escaping and the explosion becomes visible.

Supernova Remnants

Edwin Hubble deduced in 1928 that the Crab Nebula is the remnant of a supernova explosion in the Milky Way observed and recorded by Chinese and Native Americans. He did this by measuring the expansion velocity of the gas in the nebula and realizing that this implied an explosion ~900 yrs earlier.      CrabC.jpg (56058 bytes)

These remnants are frequently circular in shape and have enhanced quantities of heavy elements as would be expected for a material from a supernova. The surrounding interstellar medium also gets swept up and excited by the fast moving gas from the explosion. The gas is often hot enough to radiate even at x-ray wavelengths.

Pulsars, Neutrons Stars, and Black Holes

Recall that these are the two more massive types of stellar remnants:

Summary of endpoints of stellar evolution
Type Core Mass


Main Sequence Mass MSun Source of pressure

White Dwarf




Neutron star




Black hole




Neutron Stars



Rotation rates of Neutron Stars

Neutron stars should be expected to be fast rotators due to conservation of angular momentum.

Consider an object which starts out roughly the diameter and rotation rate of the Sun and ends up with a radius of 15km:

Can a neutron star rotate this rapidly?

For a particle to remain on the surface without flying off, the rotational speed must be less than the circular speed:


Let's look at P in terms of the density of the object:


So a neutron star could have a period as short as conservation of angular momentum predicts although that would be close to the limit.

Do neutron stars exist?

-- yes, based on the properties of pulsars.

Pulsars were discovered in 1967 by radio astronomers who detected some sources whose output varied in a regular fashion with a periods of only a few seconds or less. Careful timing revealed that some of the periods were lengthening but slowly. None were observed to be speeding up.

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A year later, the pulsar at the center of the Crab nebula was discovered and pulses were seen at a broad range of wavelengths at a period of .033 secs. Later, another pulsar with a very short period was found at the center of the Vela supernova remnant confirming the pulsar as the remains of a supernova explosion.

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The rotational rate calculations above and the existence of two pulsars associate with SN remnants has led to the association of pulsars with neutron stars. A pulsar is thus a rotating neutron star where beams of radiation sweep by our line of sight and produce the observed pulses.

Production of the pulses

--- pulsed radiation is synchrotron radiation emitted by electrons moving in a strong magnetic field

--- neutron stars are likely to have very strong magnetic fields at their surfaces because the magnetic field will be conserved in the collapse and will also strengthen as 1/R2

--- exact mechanism is unclear but probably electrons are removed from the neutron star's surface by an electric field generated by the rotating magnetic field. Electrons flow into the magnetosphere surrounding the neutron star and are accelerated by the rotating magnetic field. The "beam" may represent the pole of the magnetic field where electrons are accelerated and therefore along which the synchrotron emission will be seen

--- 'pulse' thus becomes the result of the misalignment between the neutron star's rotational and magnetic axes

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--- the observed slowing down of some neutron stars becomes the result of the energy lost in the synchrotron emission

Unanswered questions:

Are all neutron stars pulsars initially?

How many neutron stars have we missed in our surveys simply because their magnetic axes do not point towards us?

Some Special Pulsars

The Binary Pulsar -- PSR 1913+16 has a 59 msec period which varies in a cyclic fashion with a period of 7.75hrs. The period increase and decreases in a fashion analogous to a Doppler shift due to the pulsar being in a binary star. Data accumulated since 1974 has revealed that

1) The sum of the masses in the system is 2.8 Msun with the pulsar having a mass of 1.441 Msun (confirming its neutron star character) while the companion is a white dwarf of mass 1.3874 Msun.

2) Their separation is only 700,000km and shrinking

3) The orbital shrinkage matches the energy lost expected from gravitational wave emission in such a system

4) All of the characteristics of the system match predictions from general relativity

The two astronomers who discovered the binary pulsar and used it to test general relativity received the Nobel prize in physics in 1993.

Black Holes

If a remnant is left with too much mass to be supported by the pressure of a degenerate neutron gas, gravity will force it to be a black hole. The "radius" of a black hole can be defined as the radius at which the escape velocity equals the speed of light, the Schwarzschild radius. Because no information can escape from within this radius, the sphere with this radius is called the event horizon. Note that the actual size of the compact object is less than the Schwarzschild radius but we cannot see anything inside this radius.

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A black hole's behavior can be described completely from just three properties:

  1.         mass,
  2.         electrical charge, and
  3.         angular momentum.

Charge is usually ignored because a black hole would be expected to attract whatever opposite charge would be needed to make it electrically neutral. Since a black hole's parent star is likely to have been rotating, the black hole will also rotate. The rotation leads to some curious effects because space-time can get dragged along with the black hole and will cause asymmetries in the behavior of light passing along one side as compared to the other side of the back hole.

Can we prove that a black hole exists?

-- a topic of much debate a few years ago because many wanted to avoid the possibility of a singularity in space such as might exist at the center of a black hole

-- several strategies exist for proving that a black hole is present and they all rely on checking for strong gravitational fields

Gravitational lensing could reveal the presence of a black hole but this has yet to be applied in practice.

How can we be confidant that black exist?

In part because Einstein's Theory of General Relativity, a theory describing gravity and its interactions with light, has been largely confirmed:

Deflection of star light:

Stars observed near the limb of the Sun appear offset relative to their positions when the Sun is elsewhere in the sky.

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The amount of the deflection has been confirmed to match Einstein's predictions.

Orbit of Mercury:

Mercury lies close enough to the Sun that the general relativistic effects are noticeable. The perihelion of its orbit is moving 41" per century, all explainable by general relativity.

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Gravitational Redshift

To climb out of the deep gravitational potential well near a mass concentration, light must lose energy (a particle with mass loses kinetic energy and slows down). Light will shift to a longer (redder) wavelength:


where M=object's Mass and R=distance from object's center at which light is emitted.

For the white dwarf companion to Sirius,

M=1.05Msun and R=.0073Rsun so


Although black holes could have been predicted in one sense by Newton, they have been studied more now as a result of general relativity. A black hole can be thought of as a region where the curvature of space is so large that the path of a photon will bend back on itself. Equivalently, we can consider what happens when an object's escape velocity equals that of light:


But do not think of this as a radius that you could measure!

Search strategies for black holes have centered on x-ray sources because either material from a companion star or swept up material from the interstellar medium can form an accretion disk. Gas will not fall straight into a black hole but because of angular momentum, will orbit the black hole. Viscosity in the gas causes friction which makes the gas heat up and the faster it is orbiting, the hotter it gets. Gas approaching the Schwarzschild radius can have temperatures of more than 10 million degrees and hence will be strong x-ray sources.

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Consider an object with T=107°K and an x-ray luminosity of 1030 watts. What radius would it have?


This size is typical for a neutron star or an accretion disk around a black hole. How much mass must fall onto a compact object to produce the luminosity assuming 100% conversion of gravitational potential energy into observable luminosity and assuming that the object has at least as much mass as the Sun?


This accretion rate could easily happen in a binary system so the efficiency of converting gravitational energy into luminosity could be very small and still explain observed x-ray sources. However, this doesn't prove that a black hole must exist.

One of the strongest cases for the existence of a black hole comes from Cyg X-1 which is a strong x-ray source. In visible light, a blue supergiant,HDE226868, with M~20Msun is seen and cannot be the source of the x-rays as it is not hot enough.

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Spectral lines in the blue supergiant vary in radial velocity with a period of 5.6 days but no lines other than those associated with the blue supergiant are seen. No eclipses are seen in the x-ray flux although a slight variation in the output of the supergiant is detectable. This leads to an estimate of the inclination of the system of 60° but less than 90°. These facts lead to a total mass of ~35Msun for the system and therefore 15Msun for the unseen companion. Several other x-ray sources consisting of a visible star which is a spectroscopic binary but with an invisible companion exist and have masses indicating that they are black holes. The rapidity with which the x-ray flux varies also supports the existence of compact objects in these cases.

Can a Black Hole Lose Mass?

Yes, by means of quantum mechanical effects and the uncertainty principle. An alternate formulation of Heisenburg's Uncertainty Principle states that


so that a large uncertainty in energy can occur but only for a very short period of time. If a fluctuation in energy occurs and this lead to the production of a positron-electron pair briefly, no law of physics will be violated. Imagine that some energy (equivalent to mass) just inside an event horizon produces such a particle pair. If, during the interval , one of these particles is drawn into the black hole while the other "tunnels" through the event horizon, the black hole will end up with a net mass slightly smaller than it began with. However, we do not have any observable evidence of a black hole with shrinking mass!