Mass, and type of Stellar Corpse

• Brown Dwarf         0.002 < M_o < 0.08            Never becomes a star
• White Dwarf            0.08 < M_o < 1.4
• Neutron Star             1.4 < M_o < 3
• Black Hole                    3 < M_o
• Note, Jupiter's mass is 0.001 M_o. (M_o = solar mass)

Mass on the main sequence is higher than final mass at end of life.  Stars like the Sun end up burning He in a shell around the core (=> thermal pulses => planetary nebula => the "bare core" becomes a white dwarf).

Main sequence stars with M < 7 M_o end up as white dwarfs:

(< 0.5 Mo)	=> He white dwarf
(0.5 - 5 Mo)	=> Carbon/Oxygen white dwarf
(5 -7 Mo)	=> Oxygen/Neon/Magnesium white dwarf

Physical Structure
The core is made of electron degenerate matter (all available electron energy states are filled, nuclei are in tightly packed "crystalline-lattice-like" state), with a very thin atmosphere of non-degenerate matter over the surface.  The pressure inside the star does not depend on temperature, in contrast to ordinary matter.

The more massive a white dwarf, the smaller its radius.  Since white dwarfs have no nuclear engine, they cool off over several billion years, until they become a black dwarf.  They never change size once they become a white dwarf, so as they cool they follow the constant radius line in the H-R diagram.

Observations

Sirius and Companion
White Dwarfs in Globular Cluster

Sirius B has luminosity L = 3 x 10^-3 L_sun, and a temperature of T ~ 29,000 K.  From its temperature and luminosity its radius can be calculated.  Sirius B is less than 1/100th of the radius of the Sun, or about the size of the Earth.

White dwarfs have spectra in three classes
 

DA	"A-star" type spectra (lines of H Balmer series)
DB	"B-star" type spectra (lines of He)
DC	"C" = continuum spectra (no lines at all)

But the interiors are all the same--degenerate He, C, or O-Ne.

Gravitational Redshift
We will learn in the next lecture that photons lose energy in "climbing out of" a gravitational well (like the surface of a white dwarf).  Like throwing a ball into the air causes the ball to lose kinetic energy (changing it into potential energy), so too, photons lose energy.  But photons have to travel at the speed of light -- they can never slow down.  Photons lose energy by going to longer wavelengths.  This is equivalent to a "red shift" and is called the gravitational redshift.  This shift is in addition to any doppler shifts caused by motions, so in order to test this prediction observationally we must use binary systems where doppler shifts are accurately known.  For Sirius B, best measurement is Dl / l ~ 0.03% compared to theoretically exact value 0.028%.

Magnetic White Dwarfs
B ~ 10⁴ T are seen in a few percent of white dwarfs, especially those in binary systems, due to enhancement as star collapses (conservation of magnetic flux, as we saw during the collapse of protostars).

Brown Dwarfs
Objects with mass in the range 0.002 < M/M_o < 0.08 never make it to the nuclear burning stage and simply cool off (like white dwarfs) as they age.  They glow like a weak star, due to the energy left over from gravitational contraction.  Below 0.002 M_o, objects are rather arbitrarily defined as a planet.

When a star's mass at the end of its life is M > 1.4 M_o, electron degeneracy is no longer enough to keep gravity at bay, and matter is crushed to force inverse b decay

p⁺ + e^- ---> n + n.

so the protons and electrons are combined to form neutrons--a neutron star.  The state of matter is a neutron degenerate gas, with same temperature-independent pressure as for electron degeneracy.  As before, objects with greater mass have smaller radii, up to ~ 3 solar masses.  Radii are typically 10-30 km.  Gravitational redshift is now up to 20%. It is hard to visualize how great the compression is from the white-dwarf-sized core to a neutron star. Here is a picture.

Pulsars

Pulsars are rotating neutron stars with very high magnetic fields, tilted with respect to the spin axis.  Pulses originate at the magnetic poes.  Pulsars spin at periods ranging from 4 s to 1.6 ms.  Here is how they sound: pulsar sounds

You can imagine the forces on the fastest pulsar.  The surface speed is greater than 1/10th the speed of light.  Its centripetal acceleration is so great that the star could rip itself apart.  We can calculate when this would happen (when the centripetal acceleration exceeds the local acceleration of gravity).  What is the smallest possible period P that a pulsar could have?  It turns out to be about 0.3 ms, so the fastest pulsar known (1.6 ms) is pretty close to the limit.
 

When a stellar corpse is between 0.08 and 1.4 solar masses, the state of matter in the star is electron degenerate matter.  Between 1.4 and 3 solar masses it is neutron degenerate.  What happens when it is greater than 3 solar masses?  At such a mass we know of no force that can support the star against the crush of gravity.  As far as we know, it collapses to a "singularity."  What really happens we do not know, but we do know that when the object gets small enough the surface gravity gets so high that even light cannot escape.  In other words, the gravitational redshift is infinite.  The object becomes a black hole.

Although the mass inside the black hole is infinitely small, we can talk about the "size" of a black hole by considering the radius at which the escape velocity (the outward velocity that an object would have at the surface of an star or planet in order to reach infinity with no velocity) is equal to the speed of light.  At this point, even light cannot escape, so everything outside this point can be considered outside the black hole, and everything inside this point is inside the black hole.  The radius of this point is called the Schwarzschild Radius, and has a value of 3 (M/M_o) km.  This point is also called the Event Horizon.  Thus, the smallest stellar corpse black hole should be about 9 km in size (since M/M_o = ~3).  Unlike the case for white dwarfs and neutron stars, the event horizon grows as mass increases.  However, the event horizon is merely the location of the point where the escape speed reaches the speed of light.  The object itself has shrunk to a singularity.

Notice that the radius depends only on mass.  Here are some values
 

Radius for Black Hole of a Given Mass  Object Mass Black Hole Radius  Earth 5.98 x 1027 g 0.9 cm  Sun 1.989 x 1033 g 2.9 km  5 Solar Mass Star 9.945 x 1033 g 15 km  Galactic Core 109 Solar Masses 3 x 109 km

Getting Close to a Black Hole: What do you see?

Distortion due to bending of light

• Principle of equivalence in General Relativity
• Visit to a black hole

What do you feel?

Normally we are so far from the center of a gravitating body (say the Sun or the Earth) that our local space is flat, and only large objects feel any difference in the force of gravity from one side to the other.  Because a black hole can be so tiny and yet so massive, we can get very close to it and even small objects like ourselves can feel a difference in the force of gravity between our feet and our heads, for example.  This difference in force can literally be enough to rip us apart.

Trip to a Black Hole

Say we are in a ship, in orbit at 1 AU around a block hole of mass 10 M_o.  Our orbit is normal, and is not affected by the black hole any differently than any other star of 10 M_o.  We now let you drop into the black hole with a laser and a stop watch.

For most of the trip, nothing much happens, but as you get very close to the event horizon your laser signal gets progressively redshifted, as seen by me, still on the ship.  You also feel progressively stronger tidal forces that stretch you head to toe.  At about 3000 km radius, you would be pulled apart, unfortunately.  The pieces, however, continue on through the event horizon rather uneventfully.

Back at the ship, your laser signal is redshifted, and your stop watch appears to tick more and more slowly due to time dilation.  These two things can be considered the same phenomenon, in fact.  Just as you cross the event horizon, any signal you send back takes forever to arrive (and it is red shifted to zero energy anyway).  As seen from the ship, in fact, you appear to move more and more slowly and freeze at the event horizon, never crossing.

Observational evidence
A bare black hole would be impossible to see.  We can hope to detect black holes from interactions in binary systems: Accretion disk and X-ray emission

At least half of all stars in the sky are actually multiple star systems, with two or more stars in orbit about their common center of mass.  In this lecture we will be concerned with systems that are close enough to significantly interact over their lifetimes.  As one star fills its Roche lobe, mass can be transferred from the star to the other star through the L₁ point.  The matter that is transferred goes into an accretion disk, in which the matter loses energy (through viscous forces not well understood) and spirals down onto the surface of the star. Let's take a look at the locations and importance of the Lagrangian points, especially the inner Lagrangian point, L₁. Further discussion, with pictures.

This class of outburst in brightness is thought to be due to quasi-periodic brightenings in the accretion disk around a white dwarf.  During quiescent periods, the mass loss rate from the inflated secondary star is about 10^-10 solar masses per year.  Episodically, the mass transfer from the secondary goes up by a factor of 100, to 10^-8 solar masses per year. SS Cygni is a prototype of this kind of system.

In this type of outburst, it is the matter actually falling onto the primary white dwarf that powers the explosion.  It requires a mass transfer rate of about 10^-9-10^-8 solar masses per year, which over 10⁴-10⁵ y accumulates in a layer on the surface of the white dwarf in the form of hydrogen rich electron-degenerate matter.  This layer eventually reaches a temperature of a few million degrees, and with the help of CNO from the white dwarf, initiates run-away nucear burning that causes much of this outer atmosphere to expand into a shell.  The nova outbursts come in two types, fast and slow.  Both have an equally rapid rise, but fast novae are within 2 magnitudes of maximum brightness for only a week or two, while slow novae may take nearly 100 days for a similar decline in brightness.  About 2-3 are seen in our galaxy every year, but about 30 are seen in the Andromeda Galaxy each year, so most of those occuring in our galaxy are probably obscured by intervening dust.  The speed of ejection can be measured from doppler shifts during the outburst, and later the expanding nebula can be used to determine the distance to the nova (recall that transverse velocity is related to proper motion and distance by v_t = 4.74m" d_pc).

There are two basic types of supernovae, classified according to observational characteristics of their spectra near maximum brightness.  Those that show no strong hydrogen lines are Type I, those that do show strong hydrogen lines are Type II.  One type is the result of core collapse in a giant star, while the other involves a white dwarf in a close binary system.  (Can you identify which is which?)

A recent, famous Type II supernova was Supernova 1987A.  Type II supernovae, which we talked about last time, are responsible for all of the elements heavier than iron in the universe, through nucleosynthesis.

There are two models for Type I supernovae, although our text prefers the first scenario:

A carbon-core white dwarf accretes matter from a companion until it reaches around 1.3 solar masses.

At that mass, the core suddenly reaches sufficient pressure to begin fusing C into heaver elements.

Just as in the Helium flash, the degenerate C cannot expand as temperature goes up, so there is a complete conflagration of the core until the degeneracy is explosively broken and the star releases such energy that it is completely destroyed.

The second scenario starts the same as the first, but the mass eventually exceeds the Chandrasekhar Limit of 1.4 solar masses.

Type I supernovae are subdivided into three classes, Type Ia,b, and c, depending on the presence of certain spectral lines (Si, He).  Type Ia supernovae are the "cleanest" and have remarkably constant peak brightness at a whopping -20 absolute magnitude.  Because of this, they can be used, when seen in distant galaxies, to determine the distances to these galaxies.  This is a critical observation, since it allow us to calibrate one of our distance scale measurements, the redshift of galaxies, which we will show later implies that the universe is expanding.