Download Great Migrations & other natural history tales

Lecture 02, ASTB21
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
Finally, we measure periods P and orbit sizes a (semimajor 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(?)
Dust
Dust absent
around star
(~30 AU)
10000 x more comets
& asteroids than our
solar system now
comet
Beta Pic
Variable
absorption
line due to
comet’s
head
~wavelength
Absorption line variability in the b Pic spectrum shows that
comets of solar abundance of ‘metals’ sometimes evaporate near the star
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?
EQUILIBRIUM COOLING SEQUENCE
Chemical unity
of nature… and it’s all
thanks to
stellar nucleosynthesis
and mixing in ISM!
What minerals will
precipitate from a
solar-composition,
cooling gas? Mainly
Mg/Fe-rich silicates
+water ice. Planets
are made of those
T(K)
Silicates
silicates
ices
Silicates with different crystallinity
have been found in all of these
objects. They are like those found on Earth,
regarding chemical composition and
apearence.
Source: P. Kalas
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?
N-body
dynamics
Stellar
Astrophysics
Dynamics of a star:
Hydrodynamics and hydrostatics
Gen.Rel.
Radiation transfer
High ener.
physics
Thermodynamics of gas
atmospheres
Nuclear
physics
Astronomy:observations
of stars
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 = forces
where 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<m/r^2> -(1/<rho>) dP/dr, and replacing the
average gas density <rho> with estimate M/V = M/((4/3) pi R^3)
and <m/r^2> 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.).