Stars Measuring Stars For any nearby star, we can pretty easily measure the brightness How much power arrives at Earth from the star By a variety of methods, we can often determine the distance Combining this, we can figure out the luminosity L How bright the star really is We can get a good estimate of the surface temperature T from the spectrum Bluer stars are hotter; redder stars are cooler We can also get a good idea of the composition from the spectrum ¾ hydrogen, ¼ helium, 1-3% other stuff Most stars are too small to resolve No direct measurement of radius R Indirectly deduce radius from formula

“Oh Be A Fine Girl, Kiss Me.” Spectral Type The following are all equivalent information: The surface temperature of a star The color of the star The spectral type of the star From hottest to coldest, OBAFGKM Subdivided 0-9, with 0 the hottest Sun is a G2 star The spectral type is easy to determine Why I hate astronomers “Oh Be A Fine Girl, Kiss Me.”

Spectral Type

Intrinsic Properties of Stars To describe stars, we want to talk about intrinsic properties Luminosity Composition Temperature Composition is almost always the same Mass is difficult to measure Radius can be deduced from Luminosity and Temperature Radius Mass Temperature and Luminosity

The Hertzsprung-Russell Diagram A plot of temperature vs. luminosity Hot on left, cold on right Luminous at top, dim at bottom Stars fall into categories: The Main Sequence contains about 90% of the bright stars The Giants are rare but very bright The Supergiants are very rare but extremely bright The White Dwarfs are not uncommon but very dim

Main Sequence Stars Main Sequence stars have different sizes, masses, and luminosities But spectral class determines everything else This diagram shows correct relative sizes and approximate colors of stars But not correct relative luminosities

Main Sequence Stars Stars in the main sequence are similar to the Sun At their core, they are fusing hydrogen to make helium Other stars are doing different things Pre-main sequence stars are not yet hot enough to fuse hydrogen Red giants have run out of hydrogen at their core, and are fusing it in a small shell surrounding the core Horizontal branch stars are fusing helium at their core to carbon and oxygen White dwarfs are burnt-out carbon/oxygen We will focus on main sequence stars And will talk a little about pre-main sequence stars And I have time, will talk about white dwarfs and neutron stars

Stars Are in Balance Stars spend almost all their lives in a steady state Pressure vs. gravity Gravity wants them to shrink Pressure wants them to expand If a star gets out of gravitational balance, it adjusts to compensate If gravity is winning, it will shrink until pressure rises to compensate If pressure is winning, it will expand until pressure drops Luminosity vs. nuclear energy generation Luminosity causes them to lose power Nuclear fusion replaces it If a star gets out of energy equilibrium, it adjusts to compensate If too little power, shrinks and heats up to increase power If too much power, expands and cools to decrease power

Building a Simple Model for Stars We would like to build an order of magnitude model for main sequence stars Approximation 1: They are spherical Pretty good approximation for realistic stars Approximation 2: The nuclear energy generation rate is infinitely sensitive to temperature At 10 million K, nuclear energy generation is very slow At 20 million K, nuclear energy generation is very fast Therefore, all stars have central temperature Tc ~ 15 million K Approximation 3: We can approximate all derivatives as differences Terrible approximation Correct way to do it is with differential equations Approximation 4: All factors of 2, , etc., are too small to worry about

Sources of Pressure There are three sources of pressure in stars Degeneracy pressure caused by the Pauli Exclusion principle Not important for most main sequence stars Ideal gas pressure You probably learned a version of this formula in high school chemistry n is the number of independent particles per unit volume Because the core is so hot, atoms are completely ionized When computing n, add together nuclei and all electrons This is the main source of pressure in the Sun Radiation pressure is from the electromagnetic radiation This is more important at higher temperatures Major source of pressure in high mass stars

Radiation Pressure Imagine a region filled with black-body radiation with a mirror on one side If the mirror weren’t there, energy would be flowing out at a rate Consider a single photon bouncing off Each photon that attempts to leave has momentum By bouncing off the mirror, it transfers momentum Collectively, the photons produce a pressure This works out to

Radius-Temperature-Mass Relation (1) A star is a sphere of mass M and radius R Divide it in half Both pieces have mass ½M ~ M Separated by a distance R The gravitational force between these two pieces is The area separating these two halves is R2 ~ R2 The pressure is force over area This must be matched by the ideal gas pressure Use central temperature for T The number density is the number of particles divided by the volume The volume of a sphere of radius R is about R3 The number of particles is the mass M divided by the mass per particle  Equate this to the pressure found above Solve for R R R

Radius-Temperature-Mass Relation (2) The star is made primarily of 1H with mass 1u But each atom has proton + electron = 2 particles Therefore So, how well did we do? Try the formula out for the Sun Far better than we deserved to do! R

Pre Main-Sequence Stars Note that for fixed M, the product RTc will be constant Pre-main sequence stars are not producing energy in their interiors They lose energy over time They will consequently shrink But this relationship will apply at all times Since M is constant, Tc must rise as R decreases Eventually, the star will reach a temperature hot enough for fusion to start 10 – 20 million K At this point, the star becomes a main sequence star

Radiative Transport (1) For the Sun, the energy mostly flows via radiation Consider a region of slowly changing temperature There are many electrons which can scatter or absorb photons It is opaque The photons scatter and/or are absorbed many times Because of these many scatterings, the photons will be almost thermal They will typically go a short distance l before being scattered or absorbed Consider an imaginary barrier perpendicular to the temperature gradient The light moving right came from a distance l to the left The light moving left came from a distance l to the right The difference between these is the net flux

Radiative Transport (2) The distance l is very small compared to the scale of the star, so we can approximate this as a derivative: We therefore have The mean free path l was how far it goes between scatterings The more stuff there is to scatter from, the smaller it will be Therefore, l is inversely proportional to the density

Mass-Luminosity Relationship Radiation moves out of the star because it is hotter at the center than the outside The temperature at the center is roughly constant Tc, and by comparison, temperature on the surface is 0 Multiply by the area of the star to get luminosity A = R2  R2 Density is mass over volume Volume proportional to R3 Radius is proportional to mass Central temperature is roughly constant By comparing to the Sun, we therefore have

Mass-Temperature Relationship The luminosity is also related to the surface temperature and radius And radius is proportional to mass Equating these two expressions, we have And therefore have Surface temperature rises slowly as a function of mass Luminosity rises quickly as a function of mass The different stars on the main sequence correspond to different masses

The Main Sequence High mass stars are hot and very luminous Intermediate mass stars are in between Low mass stars are cool and very dim

A Dose of Reality How did I lie to thee? Let me count the ways The nuclear generation rate is not infinitely sensitive to temperature Higher mass must have slightly higher central temperature This changes a little For 0.43 – 2 Msun, the relationship is more like For high mass stars, the pp-chain is not the most important fusion There is another method called the CNO cycle This is very sensitive temperature So our approximations work a little better For very high mass stars, radiation pressure dominates Entire arguments fall apart For very low mass stars, convection throughout the star, not radiative transport