<p><hr><p>
<center>
Properties of White Dwarfs
</center>
<h2><p>
White dwarfs are the endpoints of the evolution of low mass stars.  
They are interesting
objects in that they are supported by degenerate electron 
pressure and thus do not need
internal nuclear energy sources.  
White dwarfs radiate because they are born hot and
because they slowly contract releasing gravitational energy as they cool.
<p>
White dwarfs cannot be more massive than 1.4 M(sun) 
(<font color=magenta><i>Chandrasekhar Limit</i></font>, see below) and
they have radii on the order of the radius of the Earth, 
R(wd) ~ 10,000 kilometers. Comment --
this means that white dwarfs are extremely dense; densities on the order 
of 200,000
grams per cc to 100,000,000 grams per cc.  Recall that the density of lead is ~ 11 grams
per cc. A sugar cube of white dwarf material would weigh anywhere from 
400 pounds to 200 tons at the surface of the Earth!
<p><hr><p>
<center><table cell padding=10 border=10 bgcolor=purple><tr><td>
<h2><font color=yellow><center>Mass-Radius Relationship</center>
<p>
There is a well-defined relationship between the mass of a white dwarf and its
radius. The relationship is not intuitive in that
the larger the mass of the white dwarf, the smaller is its radius!!
<p>
This can be understood by noting the size of the degenerate pressure depends on 
the density of the gas in such a way that 
the pressure is greater when the density is greater. 
That is, the higher the density, the higher the pressure and 
the harder it is to compress 
the white dwarf. 
<p>
If one thinks a little about the mass-radius relationship, a plausible
scenario arises.  Because the more massive the white dwarf, the stronger is the 
tendency for gravity to force the white dwarf to contract. To counteract this
tendency, higher pressure is needed (and so higher density is needed) and the 
radius of a massive white dwarf must be small to balance gravity.
For a very massive white dwarf, the radius must be tiny.  
<i>Question:</i> Is there a limit on how small 
one can make the radius of a white dwarf to compensate for an 
increase in the mass of the white dwarf?
<I>Answer:</i> Yes, there is a limit. Performing a detailed
analysis, one can show that for a white dwarf of mass
~ 1.4 M(sun), the radius of the white dwarf must be 0 kilometers 
in order for the density to be large enough for the 
degenerate electron pressure to counter-act the force of gravity. 
Huh. Say what?? What does this mean? 
<p>
This means that there is an upper limit to the mass of a white dwarf.  The
limit is ~ 1.4 M(sun) and is referred to as the 
<font color=magenta><i>Chandrasekar Limit</i></font></td><td>
<img src="http://pages.uoregon.edu/~imamura/122/images/wd_mass_radius.jpg">
</td></tr></table></center>
<p><hr><p>
An interesting possibility for white dwarf evolution concerns white dwarfs 
which are in short orbital period (P ~ hours) binary star systems.  Such systems
are so small that the white dwarf is actually able to steal material from its
companion star.  Such binary systems are known as
<font color=magenta>cataclysmic variables</font>.
<p><hr><p>
<center><table cell padding=10 border=10 bgcolor=purple><tr><td
width=500><font color=
yellow>
<center><h2>Cataclysmic Variables</h2></center></td>
<td><img src="http://pages.uoregon.edu/~imamura/122/images/RS_Ophiuchi.jpeg">
</td></tr></table></center>
<p>
<h2>
Cataclysmic variables</a> are 
short orbital period (hours to days) binary
star systems composed of a white dwarf and a low mass main seqeunce
star, in general (sometimes the companion star is a red evolved star).
As their name implies, cataclysmic variables (CV's) are sites for
cataclysmic events.  However, the events are not so cataclysmic as to
destroy the binary star systems, usually. The events lead to rapid
increases in the luminosities of the systems.  
There are four main types of 
cataclcysmic variables (CV's), <i>Dwarf Novae, 
Recurrent Novae</i>, <i>Classical Novae</i>, and Type Ia Supernovas. 
Many CV's are 
strong sources of x-ray emission.
<p>
<center>
Mass Transfer in Cataclysmic Variables
</center>
<p>
The systems must have short
orbital periods (hours to a few days) or else the stars will be too far
apart to exchange significant amounts of mass. Let's define some
things.
</h2>
<p>
<center>
<img width=500 src="http://pages.uoregon.edu/~imamura/122/images/roche.png">
<p>
</center>
<p><hr><p>
<center><table cellpadding=10 border=10 bgcolor=purple><tr><td>
<img width=300 
src="http://pages.uoregon.edu/~imamura/122/images/close_binary.jpg">
</td>
<td><font color=yellow>
<h2>
What happens in close binary systems, depends upon the secondary (the less 
massive star). There may be
detached, semi-detached, and contact systems depending
upon whether the secondary star fills its Roche lobe.  CV's are
semi-detached systems.
The companion fills its Roche lobe and transfers
material to the white dwarf.
</td>
</tr>
</table></center>
<p><hr><p>
<center>
<h2>What Happens in Accreting White Dwarf Binaries?</h2>
</center>
<h2>
<p><p><hr><p>
<p><center><table cellpadding=10 border=10 bgcolor=purple>
<tr>
<td><h2><font color=yellow>
<Center> X_RAY BINARIES</center><p><center>Gravitational Energy</center>
</h2><h2><p>
<ul>
<li>The material accelerates as it falls onto the white dwarf.  If I
dropped some mass onto a white dwarf it would hit the surface
of the white dwarf at a speed of
around 10,000 kilometers per second. This is
a lot of kinetic energy (gained at the expense of the potential energy
of the white dwarf).
<p>
<li>In <i>dwarf nova</i> systems, the energy which powers the outbursts
is gravitational in nature -- it comes from the energy the material 
gained by falling onto the white dwarf. 
</p>
<li>Dwarf nova outbursts are smaller and occur much more often (every
several weeks to months) than nova outbursts.
<ul>
<p>Comments:
<p>
<li>efficiency = 1/2 mv<sup>2</sup>/mc<sup>2</sup> = 1/2 (v/c)<sup>2</sup> ~ 0.0005
<p>
<li>efficiency ~ 0.007
<p>
===> nuclear burning efficiency is higher
</ul>
<p>
<li>The release of the gravitational energy can also lead to the production of
x-ray emission near the surface of the white dwarf.  The <i>Classical
Nova</i> and <i>Dwarf Nova</i>, GK Per is also a strong source of 
<a href="gkper.gif">x-rays</a>.
</h2></td><td>
<img width=500 src="http://pages.uoregon.edu/~imamura/122/images/cv_disk.gif">
</td></tr></table><p><hr><p>
<table cellpadding=10 border=10 bgcolor=purple><tr>
<td><h2><font color=yellow><center>
Classical Novae and Type Ia SN, Standardizable Candles?</center><p>
</h2><p><h2>
The common lore is that Classical Novae and 
Type Ia SN occur in close mass-transfer binary systems composed of a 
white dwarf and a <i>normal</i> companion star, that is, 
in <i><font color=magenta>
Cataclysmic Variables</font></i> (CVs).<p>
Recall, CVs are 
short orbital period (hours to days) binary star systems composed of a white dwarf and a low mass main seqeunce
star (in general, sometimes the companion star is a red evolved star).
As their name implies, cataclysmic variables (CV's) are sites for
cataclysmic events.  However, the events are not so cataclysmic as to
destroy the binary star systems (in general).  The events lead to rapid
increases in the luminosities of the systems. Here, we consider the 
not so cataclysmic <i>Classical Novae</i> and the very cataclysmic  
<i>Type Ia Supernovas</i>.</td>
<td><img width=450 src="http://pages.uoregon.edu/~imamura/123/lecture-8/
Iaprogenitor.jpg"></td></tr></table>
</center><p><hr><p>
<center>
<h2>Energy Sources for Classical Novae and TYpe Ia Supernovas</h2>
</center>
<p>
<h2>
CVs generate energy either through nuclear burning or through
<font color=red><i>gravity</i> (accretion)</font>. 
Classical Novae and Type Ia Supernovas
power themselves through <font color=red><i>nuclear energy</i></font>.
<p>
The material which flows onto the white dwarf simply piles up on the
surface of the white dwarf. The material is rich in hydrogen since it
comes from the envelope of the companion star. This is a key point,
because white dwarfs being the ashes of nuclear burning have no nuclear
fuel left.  The companion replenishes its fuel supply.
<p>
Depending upon whether the mass flow (accretion) is high or low, different
outcomes may result.
<p><hr><p>
For slow accretion rates, the material as it piles up can lose its energy
gained as it fell onto the white dwarf. The material remains cold and reaches
high densities. The scenario is then:
<p><ul>
<li>The material accreted compresses due to the weight of the 
recently added material. The compression causes the temperature 
and pressure of the accreted material to increase but only slowly. 
After around 10,000
to 100,000 years of accretion, the conditions become right for nuclear
burning.
<li>The ignition of the nuclear burning is not gentle because of the
high densities. The ignition of the burning leads to an  explosion
(either because the material is degenerate or the ignition occurs in a
thin shell).
<li>The thermonuclear explosion causes the nuclear burning shell 
to be <a href=
"http://pages.uoregon.edu/~imamura/122/images/gkper1.gif">ejected</a> 
leading to a <i>Classical Nova</i> 
<a href=
"http://pages.uoregon.edu/~imamura/122/images/v705cas.gif">
outburst</a>.
</ul>
<p>
For the fast accretion rate, the material does not have time to lose its
energy it gained as it fell onto the white dwarf. The material thus increases
in temperature and pressure strongly as it accretes onto the white dwarf.
<P><ul>
<li>In this case when the conditions needed for the onset of nuclear burning
are reached, the material is not degenerate and the ignition is <i>gentle</i>,
there is no explosion. 
<li>The ashes of the nuclear burning then just settle onto the core of the 
white dwarf increasing its mass.
<li>This gentle increase in the mass of the white dwarf eventually causes the
white dwarf to approach the <i>Chandrasekhar Mass Limit</i>. 
For a carbon/oxygen white dwarf, like the Sun will become, this leads to 
ignition of the carbon which causes the entire white dwarf to rapidly undergo
a thermonuclar outburst which incinerates the star. This process leads to a
Type I SN.
<li><font color=magenta>Because the ignition occurs when the white dwarf 
is near the <i>Chandrasekhar Mass limit</i>, the progenitor stars for Type I
SN have nearly the same properties. This led to the suggestion that Type I SN
should all appear to be similar in appearance. This is as opposed to Type II
SN where the progenitor have widely differing properties.
</font>
<p>
<center><font color=blue>Type I SN are thus likely to be <i>Standardizable
Candles</i></font></center>
</ul>
<p><hr><p>