How the Sun Shines

The Luminosities of Stars Stellar distances can be determined via parallax – the larger the distance, the smaller the parallax angle,  From geometry D(pc) = 1 /  (in parsecs) The distance then gives Absolute Luminosity L = l  r 2 (or Absolute Magnitude)

The Temperatures of Stars In order of temperature (hot to cool), the spectral sequence of stars is O-B-A-F-G-K-M. The traditional mnemonic is Oh Be A Fine Girl Kiss Me. (Recently, types L and T have been added to the cool end.)

The H-R Diagram Most stars are on the main sequence. But some are very cool, but also very bright. Since cool objects don’t emit much light, these stars must be huge. They are red giants. Some stars are faint, but very hot. These must therefore be very small – they are white dwarf stars.

The Sizes of Stars The sizes of stars can be anywhere from 0.01 R  to 1000 R  !

The Masses of Stars Stellar masses can only be determined via the application of Kepler’s and Newton’s laws, i.e., (M 1 + M 2 ) P 2 = a 3 where  M 1 and M 2 are the stellar masses (in solar units)  P is the orbital period (in years)  a is the semi-major axis of the orbit (in A.U.) This requires binary stars!

Binary Stars Half the stars in the sky are binary stars! Many stars look like they are binaries, but are really not: they are optical doubles. Fortunately, many other are true binaries, even if they’re too close together to be resolved. Binary systems are called different things, depending on how we perceive them. A true double star can be called a visual binary, a spectrum binary, a spectroscopic binary, and/or an eclipsing binary, depending on what we see.

Visual Binaries When both stars can be seen, it’s called a Visual Binary. Castor

Spectrum Binaries If the stars are too close together to be seen separately, it is possible to identify the object as a binary based on its spectrum.

Spectroscopic Binaries If the Doppler shift of a star’s absorption lines changes with time (redshift, then blueshift, then redshift, etc.), it’s a spectroscopic binary. If one star is much fainter than the other, you may not see its lines. The object is then a single- line spectroscopic binary. If both sets of lines are seen, then it’s called a double-line spectroscopic binary.

Eclipsing Binaries If two stars eclipse each other while orbiting, it’s an eclipsing binary. Eclipsing binaries are somewhat rare, since they need to be seen edge on. This system is one in which both stars undergo a total eclipse. Frequently, the eclipses are partial.

Determining Masses from Binaries The relative speeds of the stars gives you their relative masses.

Determining Masses from Binaries The relative speeds of the stars gives you their relative masses.

Determining Masses from Binaries The absolute velocities of the stars (times the period) gives you the circumference of their orbits. From that, you can derive the orbits’ semi-major axes. In other words, Circumference = v · t = 2  a (at least for circular orbits) From the semi-major axis and the period, you can derive the total mass of the system through (M 1 + M 2 ) P 2 = a 3 Since you already know the relative masses, you now know everything! The relative speeds of the stars gives you their relative masses.

Results from Binary Stars Measurements 1)All stars have masses between 0.1 M  and 60 M  2)Main sequence stars obey a mass-luminosity relation: the brighter the star, the more massive the star.

Results from Binary Stars Measurements 1)All stars have masses between 0.1 M  and 60 M  2)Main sequence stars obey a mass-luminosity relation: the brighter the star, the more massive the star. 3)The white dwarf stars are all less than 1.4 M  4)There is no pattern to the masses of red giants.

The H-R Diagram There are patterns in the HR diagram. Most stars lie on the main sequence, and obey a mass-luminosity relation. (Low mass stars are faint, high mass stars are bright.) But there are huge red giants that have all sorts of masses, and very small white dwarfs that are all less than 1.4M  Why??

Temperature, Pressure, and Energy Gas Temperature: a measure of how fast atoms are moving in random directions Gas Pressure: the “force” these atoms put on their surroundings via their collisions Energy, Temperature, and Pressure are related through the Equation of State: where one goes, the others go! Energy   Temperature   Pressure  Energy   Temperature   Pressure 

Hydrostatic Equilibrium The Sun is very massive, so it has a lot of gravity. The center of the Sun is under great pressure! In order to keep from collapsing, the gas pressure must balance the pull of gravity. This is called hydrostatic equilibrium. High pressure means high temperature: the center of the Sun is very hot: about 14,000,000° ! High temperature means the Sun produces a lot of energy (through the blackbody law). This heat must flow out!

How Does Heat Get Transported? There are 3 ways to transport heat :  Conduction: fast moving electrons hit slower moving electrons. (This does NOT happen in most stars.)

How Does Heat Get Transported? There are 3 ways to transport heat:  Conduction: fast moving electrons hit slower moving electrons. (This does NOT happen in most stars.)  Convection: hot material is mixed into cooler material. (This only happens in the outermost layers of some stars like the Sun.)

How Does Heat Get Transported? There are 3 ways to transport heat:  Conduction: fast moving electrons hit slower moving electrons. (This does NOT happen in most stars.)  Convection: hot material is mixed into cooler material. (This only happens in the outermost layers of some stars like the Sun.)  Radiation: blackbody law photons are emitted, then absorbed by cooler material Energy is transported via the random walk of photons. In the Sun, it takes energy over 10,000,000 yr to get to the surface!

Not How the Sun Does Not Shine As the energy leaks out, the central temperature of the Sun drops Lower temperature means lower gas pressure The lower gas pressure cannot hold up against gravity – the Sun shrinks The added compression puts the Sun’s center under greater pressure, so the central temperature increases The higher temperature produces higher pressure, which fights off gravity The high temperature produces blackbody photons which leak out… Gravitational contraction could keep the Sun shining for 40,000,000 yrs!

Does How the Sun Does Shine In order to keep up the gas pressure (and prevent collapse), the center of the Sun must continually replenish the energy that is lost. This is done by nuclear fusion (of hydrogen). The energy produced maintains hydrostatic equilibrium. Net Result: 4 H  1 He The sequence of fusions is called the proton-proton chain. ( very rarely) quickly

About the Proton-Proton Chain The net result of the proton-proton chain is to turn 4 hydrogen atoms into 1 helium atom. But there is a mass defect – the 4 hydrogen atoms have 0.7% more mass than the 1 helium atom (plus the other junk). Where did the missing mass go? E = m c 2 Energy!!! If the Sun had more mass, it would have more gravity, and its center would be under greater pressure. The greater the pressure, the greater the temperature, and the more violent the nuclear collisions. More fusion would occur, and more energy would be produced. This explains the main sequence! Fusion only occurs in the core, where the temperature and density are greatest. The rest of the star just sits there.

The Sun Won’t Shine Forever Stars spend over 90% of their life on the main sequence, fusing hydrogen to helium. The Sun has already lived 4.5 billion years. It will live about 5 billion more years. Stars mostly transport energy via radiation, not convection. So hydrogen from the outside of the star does not get mixed into the core. When the core’s hydrogen runs out, bad things begin to happen.

Next time -- Stellar Evolution