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!