Preliminaries from Ch.1 of the green book(#1): 1. Reasons for assuming spherical symmetry of a star [F_centrif / F_grav ~ 1e-5] 2. Reasons for defining r and m as two equivalent choices of an independent variable in a spherical star [concentration in the center] 3. What do we observe? Distance to the star: d [pc] from geometry Flux on Earth = L/(4 pi d^2) ==> L (luminosity of a star) From spectrum, T_eff (that blackbody would emit), e.g. T_eff=5780 K for the sun Stephan-Boltzmann law ==> R (stellar radius) e.g. 0.7mln km in rare instances, can measure R directly Lecture 02, ASTB21

Finally, we measure periods P and orbit sizes a (semi- major axis) in binary stars ==> masses M from Kepler’s law 4. What are the observed L(M) functions? L ~ M^3 (high mass), M^5 (low mass) [fig1.6] 5. Reasons for assuming uniform composition described by mass fractions X, Y, Z of: H, He, and the “metals” ( X+Y+Z=1 ) Q1: Is chemistry similar everywhere in the star? Q2: Is chemistry similar everywhere in the galaxy?

IMPORTANT DIGRESSION: On the similarities of chemical composition of most pop. I stars Observations show that many stars are surrounded by dust and sometimes detectable gas, in the form of the so-called debris disks or replenished dust disks, originally called Vega-type disks. The Sun has a zodiacal light disk, which is a week manifestation of the same phenomenon. Beta Pictoris (or b Pic, beta Pic) is the most prominent one. It’s a disk around a nearby star of spectral type A5V, 1.75 times more massive than the sun, and only 20 Myr old. The disk is seen almost edge-on, and extends to >1000 AU from the star. It is made of solid bodies of different sizes: dust, sand, pebbles,…, comets, planets(?)

Beta Pictoris, I-band

Beta Pictoris B Pic sky(?) x more comets & asteroids than our solar system now Dust Dust absent around star (~30 AU)

Absorption line variability in the b Pic spectrum shows that comets of solar abundance of ‘metals’ sometimes evaporate near the star Beta Pic comet Variable absorption line due to comet’s head ~wavelength

A rock is a rock is a rock… But which one is from the Earth? Mars? Beta Pic?? It’s hard to tell from spectroscopy, or even close up! Why?

What minerals will precipitate from a solar-composition, cooling gas? Mainly Mg/Fe-rich silicates +water ice. Planets are made of those Silicates silicates ices T(K) Chemical unity of nature… and it’s all thanks to stellar nucleosynthesis and mixing in ISM! EQUILIBRIUM COOLING SEQUENCE

Source: P. Kalas Silicates with different crystallinity have been found in all of these objects. They are like those found on Earth, regarding chemical composition and apearence.

? Does this mean we are going to see the same minerals and the same (H_2 O) rivers in other worlds? …despite the substantial disagreement in types of orbits and mass of planets.

Marcy and Butler (2003)

Lecture L2 agenda: What’s inside a star?

Gen.Rel. Radiation transfer Thermodynamics of gas Stellar Astrophysics Astronomy:observations of stars Dynamics of a star: Hydrodynamics and hydrostatics Nuclear physics High ener. physics atmospheres N-body dynamics

ODEs (ordinary diff. Eqs.) OF 1-D HYDRODYNAMICS : (1) Continuity equation: dm = 4 pi rho r^2 dr m = m( r ) mass inside radius r, dm = mass of a layer. (3) Equations of motion (momentum) [Newton’s 2nd law] Dv/Dt = forceswhere D/Dt = d/dt + v grad (1) energy equation Du/Dt = … where u is internal energy /dm (1) constitutive relations, EOS: P=P(rho,T) In general, these are 6 equations for 6 unknowns: rho, T, P, v_x, v_y, v_z (or similar velocity coordinates) [unless gas consists of many species… then more eqs. & var’s are needed, but a similar equality holds.] Q: Does that imply that these ODEs always have solutions?

ENERGY EQUATION heat absorption + mechanical work => internal energy change EQUATION OF MOTION acceleration * dm = gravity + pressure force

The virial theorem: total internal en. = -(1/2) * tot. grav.energy or total energy = (1/2) * tot. grav.energy

How astrophysical estimates are made: The trick is to get an order-of-magnitude estimates (i.e., approximation to with a factor of 10) from the equations governing the structure of a star. These equations are typically ODEs (ordinary differential eqs.) that contain terms like dP/dr This derivative can be estimated as dP/dr ~ [P(R) - P(0)]/(R - 0) Notice that this would be an exact expression for the derivative if pressure P were falling from a large P(0) in the center to P(R)~0 at the surface as a strait line section (linearly). Linear approximation is not necessarily accurate, but it needs not to be here. Plugging dP/dr ~ -P(0)/R into the hydrostatic radial force equation 0 = -G -(1/ ) dP/dr, and replacing the average gas density with estimate M/V = M/((4/3) pi R^3) and with M/R^2, we finally arrive at P(0) ~ GM^2/R^4, which gives 450 mln atm for the center of the sun.

(for M ~ 2e33 g = 3e30 kg = 1 M_sun and R~ 0.7e6 km) Summary of estimates of: central pressure ~ 0.5e9 atm average temperature ~ 4e6 K (hot!)

I would like to remind you that this is a fast-moving course. Reading ahead from Prialnik’s book is essential. Only then will you be able to focus during the lecture, not on notation or copying strange-looking things, but on pointers to which things in the book are most important, comments widening the scope of the text etc. And you’ll have many relevant questions concepts not clear yet from the book or lecture. So, please do yourself a big favor and BEFORE every lecture read the new chapter from the textbook once. (If you don’t have any clue about what to read, or have questions than please contact the TA or myself during our office hours.). Home assignements will be distrbuted every week to every other week. Happy reading, and looking forward to your questions!