The Mass-Luminosity
Relationship
and Stellar Lifetimes
Gee, since we’re already
examining how things are related…
Masses of Stars
• Just like we plotted color (temperature)
against luminosity…(the HR diagram!)
• If the mass of the stars is plotted against
their luminosities, the graph reveals
another relationship.
• (We’ll cover just how the masses of stars
are determined in a couple of days.)
http://www-astronomy.mps.ohio-state.edu/~pogge/Ast162/Unit2/Images/mlrel.gif
The Equation
• A star has a luminosity that is proportional
to its mass to the 4th power. In other
words, the brightness of a star increases
much faster than its mass does.
• In equation form, the mass-luminosity
relationship is…
Lstar / Lsun
=
(Mstar / Msun)4
• The equation is only true for stars that are
bigger than 0.43 Msun.
• If the star is very small, smaller than 0.43
Msun, then the equation is modified:
Lstar / Lsun = (Mstar / Msun)2.3
Slight change
This doesn’t seem right…
• Our intuition would seem to say that since
big stars have a lot more fuel to consume,
they should last a lot longer than smaller
stars.
• It doesn’t work this way, however. If the
luminosity of a star increases with the 4th
power of the mass, that means that the
star is producing energy and using its fuel
at the same faster rate.
Here’s an example
• A star of 2 solar masses has twice the H
fuel available, but uses it 24 or 16 times
faster.
• Mathematically, 21/24 = 2/16 = 1/8. A star
of 2 solar masses would last only 1/8th as
long as a 1 solar mass star.
• This is like a bonfire burning all its fuel and
being out in an hour, where a small
campfire still has hot embers the next
morning!
Burns out quickly!
Still burning slowly
the next morning…
So How Long Do Stars Last?
• We estimate that a 1-solar-mass star like
our sun will live a total of about 10 billion
or 1 x 1010 years.
• Larger stars will live
(1 / Mstar)3 x 1010 years
• This equation is good for all stars 0.43
Msun and above.
What about very small stars?
• For small stars, their lifetimes are given by
(1 / Mstar)1.3 x 1010 years
A couple of examples
• How long would a 10-solar-mass star live?
(1 / 10)3 x 1010 = 1/1000 x 1010 =
1 x 107 or 10 million years.
• The smallest a star can be is about 0.08
solar masses. How long would a star like
this live?
(1 / 0.08)1.3 x 1010 = 26.7 x 1010 =
267 billion years
• This means that the universe isn’t yet old
enough for small red dwarf stars to have
begun dying. Some red dwarfs may have
been around since the universe began!
• The very largest a star can be (we think) is
about 100 solar masses.
• How luminous would a star like this be?
Lstar = 1004 = 100 million Lsun
• How long would a star this size live?
(1/100)3 x 1010 = 10,000 years
• Huge stars may actually live a little longer
than this due to other factors, but you get
the point!
Time to do the Homework!