PART II ETP — ASTROPHYSICSsteve/bioson/overheads1.pdf · PART II ETP — ASTROPHYSICS 22 Lectures Prof. S.F. Gull & Prof. A.N. Lasenby 16 lectures this Term; astrophysics of (more

PART II ETP — ASTROPHYSICS

22 Lectures Prof. S.F. Gull & Prof. A.N. Lasenby

• 16 lectures this Term; astrophysics of (more or less) normal matter

from scales of stars to clusters of galaxies, using ordinary physics and

Newtonian gravitation theory.

• 6 lectures next Term. Prof. Lasenby will deal with the overall structure

and evolution of the universe, using proper gravity.

• Level of the course: not too much detailed mathematics; ideas needed

from previous courses will be reviewed; theoretical derivations done to

“astrophysical accuracy”. Not too much detailed astronomical factual

material. Emphasis on understanding astrophysical “test cases”.

• HANDOUT — Syllabus; books; essential astronomical facts and

jargon; orders of magnitudes; how we measure distances, velocities,

masses; basic information about astrophysical objects we will meet;

states of condensed matter. More may (or may not) follow. Please

report any errors or typos.

• NOTES — Copies of overheads available on web – hardcopy will follow.

• SUMMARY SHEETS — 1 page summary of each lecture.

• EXAMPLES — 4 sheets this Term — 2 examples per lecture — Mock

Exam paper will be issued.

• WORKED EXAMPLES — Given out at lectures 16 and 22.

• WEB PAGE — For feedback, additional pictures, movies etc.

THE OBSERVABLE UNIVERSE

• Astrophysics is the extension of laboratory physics to large-scale

structures in the universe.

• “Large” means bigger than the Earth (radius 6400 km) — the nearest

external object is our Moon (distance 400, 000 km radius 1738 km).

• The universe seems to organise itself preferentially into stars, which

are objects of size 109 m and 1030 kg.

• The nearest star is the Sun at distance 1.5 × 1011 m (8 light min).

• Sun is a fairly typical star: radius 7× 108 m; mass (M�) 2× 1030 kg.

• Next nearest stars are 30, 000 times further away at about 5 light

years 5 × 1016 m).

• Stars organise themselves into various scales, but there is another

preferential scale: galaxies (1021 m, 1042 kg).

• Other larger scales: clusters, superclusters.

• Universe originated from hot, dense state 15 billion years ago.

• Finite observable universe 2 × 1026 m containing 1012 galaxies.

• And that’s just the stuff we can see. . .

ASTROPHYSICS AND ASTRONOMY

• Universe contains wide range of exotic phenomena: stars; star

formation; supernovae; galaxies; radio galaxies; quasars; clusters;

cosmic microwave background.

• Laws of physics governing the behaviour of astrophysical objects is

exactly the same as here on Earth (we keep an open mind on this, but

would need a lot of convincing otherwise).

• Extreme conditions found in the cosmic laboratory can provide tests of

our understanding of physics.

• Astronomy is an observational science: we have to make do with the

objects Nature provides, and we can only see them from one viewpoint.

• Astrophysical timescales can be very long (e.g. dynamics of radio jets

in galaxies: 107 years). Can often only see a snapshot of a long

evolutionary process; have to infer evolution from many examples.

• Inevitable selection effects: we can only see objects that emit radiation;

we have limited dynamic range of instruments; we can only see rare,

very luminous phenomena to the greatest distances; some objects

emit anisotropically – interstellar masars, pulsars, relativistic jets.

PHYSICS NEEDED FOR ASTROPHYSICS

• We will need almost everything you have been taught – and a bit more.

• Gravity The glue of the universe — makes all objects tend to attract

and collapse. Needed throughout the course: particle orbits in globular

clusters, galaxies and clusters; Hydrostatic equilibrium of stars and

cluster gas; star formation; evolution and death of stars; formation of

galaxies and other structures from the expanding universe; dark matter.

• Fluid dynamics and plasma physics Many astrophysical

phenomena involve jets, turbulence, shocks, explosions. Hot gas in

clusters of galaxies.

• Nuclear and Statistical physics Needed to understand the

physics of normal and degenerate stars; end points of evolution of

stars; supernovae and synthesis of post-Fe elements; nucleosynthesis

in the early universe.

• Radiation mechanisms and radiative transfer This is how

we see astrophysical objects; also important for energy transport in

stars; cooling and determining the evolution of objects.

• Exotic physics (Mainly next Term) General relativity and Black

holes; dark matter; dark energy; the epoch of inflation.

STUDYING ASTROPHYSICS

• Overall aim We will try to develop an intuitive view of the various

astrophysical phenomena we observe, and be able to apply simple

physical models to explain them quantitatively.

• Realistic goal I’d like you enjoy astrophysics, to be able to do the

problems on the question sheet and to do well in the Exam.

• Astronomical context We have to have some appreciation of the

observational data: wavebands used; resolution achieved relative to

scale of the object; nature of radiation processes; spectral and velocity

information (if any). The is the necessary astronomical legwork. You

can’t progress in astrophysical research without building telescopes,

doing surveys and studying sources in gruesome detail.

• Theoretical treatment This is very difficult. Theories we can

compute with are approximate (noone knows how to do the 2-body

problem in GR. Even numerical N -body Newtonian gravitational work

and 3-d fluid dynamics are fraught with difficulties, though can give

insight. Analytical treatments are usually worse, but can still be very

useful. We use approximate theory and imperfect simulations to try to

educate our physical intuition. We iterate round the loop of theory,

observation and simulation and eventually hope to progress.

• Method This course will consist of generally applicable theoretical

ideas, a necessary minimum of astronomical facts, illustrated by about

a dozen examples of astrophysical test cases which I will analyse in

detail. These will be interspersed throughout the course, i.e. some will

occur before you have the all relevant facts and observations are to

hand. Don’t worry; research is like that — you are thrown in the deep

end. . .

ASTRONOMICAL FACTS AND JARGON

• Angles are measured in degrees, arc minutes and arc seconds:

180/π radians = 1 degree = 60 arcmin = 3600 arcsec

• 1 AU (astronomical unit) = 1.5 × 1011 m

• 1 pc (parsec) = 1 AU subtends 1 arcsec = 180 × 3600/π AU

= 3 × 1016 m

• 1 M� (Solar mass) 2 × 1030 kg

• Redshift: z ≡ λobs − λrest

λrest

• m = apparent magnitude = −2.5 log10

(

Flux(ν)

StandardFlux(ν)

)

M = absolute magnitude = m − 5 log10

(

Distance

10 pc

)

m M

Sun −27 +5

Full Moon −13 +32

Sirius −1.5 +1.5

A0 star at 10 pc 0 0

White dwarf at 100 pc +20 +15

Galaxy at z = 1 +22 −22

Type Ia SN at z = 1 +24.5 −19.5

THE EXPANDING UNIVERSE AND HUBBLE’S “CONSTANT”

• In 1929 Edwin Hubble found that all galaxies seemed to be mov-

ing away from us, with velocity v proportional to distance D: v = H0D.

• He determined the constant H0 to be 500 km s−1 Mpc−1.

• In real units this is 1/H0 = 1.5 Gyr.

• This caused a huge problem, because the age of the Earth is 4.5 Gyr.

• Cosmologists were resourceful and invented lots of crazy theories to

account for this impossible observation.

• Hubble’s calibration had underestimated the luminosity of Cepheid

variables. When corrected, the problem disappeared.

• The value of H0 is still uncertain.: H0 = 75 km s−1 Mpc−1 is

currently popular.

EVOLUTION OF HUBBLE’S CONSTANT

• Published values of Hubble’s constant up to 1980.

THE BIG BANG

• Although the Hubble constant settled down so that the universe was

older than the Earth, people still didn’t like the idea that the universe

had a definite “beginning” in a hot, dense state 10–15 billion years ago.

• Fred Hoyle (one of the originators of the“Steady-State” theory) said:

“You might as well say it all started in a big bang!”

• The discovery of the relic 2.7 K cosmic microwave background

radiation and the clear evidence of evolution in the radio source counts

(radio source were much more common at z = 2 than they are today)

largely settled the matter, though there were persistent pockets of

resistance for many years afterwards.

• So now we call the dense, hot early universe the “Big Bang”.

• A version of the “Steady-State” theory is now popular again (for the

very early universe). . .

DETERMINATION OF THE DISTANCE SCALE

Traditional “step by step” approach: the distance ladder

• Solar system: planetary radar, tracking of spacecraft and pulsar timing

gives value of AU to a few metres.

• Nearby stars: use parallax. Hipparchos satellite measured parallaxes

to 0.001 arcsec (expect a further factor of 100 improvement soon).

Thus establish luminosity as a function of spectral type for main

sequence stars.

• Apply to more distant stars, especially clusters. Find luminosity of

bright “standard candles” (in particular Cepheid variables, for which the

absolute luminosity is well correlated with the period of oscillation).

THE DISTANCE SCALE II

• Observe Cepheids in nearby galaxies to establish distances to them

and thus obtain the absolute luminosity of still brighter objects: globular

clusters, H+ regions and whole galaxies.

• Extend these to find distances to galaxies that are sufficiently far away

that the overall expansion of the universe dominates over random

motions. This should provide the value of the Hubble constant H0.

• Beyond that the recession velocity is used as the indicator of distance,

though the true form of the distance–redshift relation remains to be

determined.

• The brightest standard candles available are Type Ia supernovae. They

arise from the ignition of a white dwarf star following accretion of matter

from its normal companion in a binary system. Ignition occurs at a

definite mass 1.4 M�, and there is usually not much absorbing material

in the way, since the lifetime of these stellar systems is very long and

they are likely to have moved away from the dense regions in which

they were formed. These standard candles have extended the distance

scale to high redshift, but calibration will continue to require refinement.

HUBBLE DIAGRAM FROM TYPE IA SUPERNOVAE

• Hubble diagram from Perlmutter & Schmidt (2003).

• A distance modulus of 40 corresponds to a distance of 1 Gpc, and

increase of 5 in the distance modulus is equivalent to an factor of 10

increase in distance. (There are lots of technical issues here.)

COSMIC ABUNDANCES OF THE ELEMENTS

• Only hydrogen and helium were formed in the Big Bang.

• Binding energy graph shows that Li, Be, B will be rare.

• C, O, Ne are formed in normal stars, post-Fe arise from supernovae.

THE FATE OF COLD MATTER IN THE UNIVERSE

• Gravity tries to concentrate matter.

• Other forces resist:

1) Coulomb force can do the job for M < 2 × 1027 kg

2) Degeneracy pressure i.e. Pauli principle for identical fermions

a) Electrons in white dwarf stars

are sufficient for 2 × 1027 kg < M < 2 × 1030 kg

b) Neutrons in neutron stars (pulsars)

manage for 2 × 1030 kg < M < 1031 kg

• Above this limit gravity must win in the end.

• Collapse is postponed by:

3) Entropy

a) Compression + opacity → heat → pressure.

b) Compression + turbulence → heat → pressure.

c) If temperature T > 107K → nuclear fusion.

ASTROPHYSICAL OBJECTS — MASS VERSUS RADIUS PLOT

• Over a very range of masses (factor of 1054) objects have a density of

order 103 kg m−3.

• Narrow range of masses (1027–1031 kg with wide range of densities.

• On larger scales see dynamical groups of stars.

THE SOLAR SYSTEM

• The Sun

- Contains 99.9% of the mass of the solar system

- Rotation period: 25 days at equator, 30 days at poles.

- The angular momentum of the rotation is only 2% of that of the

solar system (most is in the orbital motion of Jupiter).

- The differential rotation creates magnetic field, typically 10−4 T, but

0.3 T in sunspots.

- Optical photosphere has Teff = 6000 K (emits continuum).

- Chromosphere has T = 4500 K (absorption lines).

- Whiplash effect of convection cells heats the low density corona to

T = 106 K.

- Generates solar wind 400–700 km s−1, with spiral sector structure.

- Heat is generated by conversion of hydrogen to helium in the centre.

- Heat is transported outwards by radiative diffusion, except for the

outer 1%, where the decreasing density makes the fluid develop

convection cells.

- Clues to interior: solar oscillations; neutrinos.

THE SOLAR SYSTEM II

• Planetary system

- Orbits are approximately circular and lie in a plane — must have

been formed from a gaseous disc.

- Composition shows temperature gradient: inner planets have

ρ ≈ 5 × 103 kg m−3; outer gas giants have ρ ≈ 103 kg m−3

• Earth

- Differentiated: Iron/Nickel core; silicate mantle and crust.

- Heated by radioactive decay, causing geological activity.

- Age of oldest surface rock is about 3.8 Gyr.

• Mars and Venus

- Similar histories to Earth.

- Differences in atmosphere can be understood from temperature and

gravity.

• Moon and Mercury

- Geological activity ceased before the end of asteroid bombardment.

- Age of oldest Moon rock is 4.5 Gyr.

THE SOLAR SYSTEM III

• Asteroids

- Smaller bodies, mostly lying between Mars and Jupiter, sizes from

< 1 km to a few hundred km..

- Total mass < 10−3 that of the Earth.

- Interactions and collisions generate meteors which we can study

directly. Differentiated: stony; stony/iron; irons. Some stony

meteors contain “chondrites” — appear to be pre-solar material

(contain SiC which cannot form in the presence of O).

• Outer planets

- Formed from “ices” — H2O, NH3, CH4.

- Massive enough to keep most of the H and He.

- Systems of moons — formation similar to planetary system as a

whole?.

- Ring systems: dust to bolder-sized particles inside the Roche limit.

• Comets

- Made from“Dirty ices”. There are probably about 1010 comets.

- Highly eccentric orbits — regular visitors must have had interaction

with planets that circularised their orbits.

- Probably originate in the Oort cloud at 105 AU

NEWTONIAN GRAVITY THEORY

m1

m2F =Gm1m2

r212

• All material bodies attract one another: force proportional to the

product of masses; inversely proportional to square of separation.

• Force on a body of mass m at position r is determined by an

acceleration field g(r): F = mg. This gravitational field is the sum of

accelerations from all bodies: g(r) =∑

i

Gmi(ri − r)

|ri − r|3

• Gravitational field is conservative — derived from potential φ(r):

g = −∇φ

.• Analogy with electrostatics: φel =q

4πε0R↔ φ = −GM

R.

• Consider volume V surrounded by surface S

• Gauss’ theorem in electrostatics:∮

dS·E =

∫

dV ∇·E =Q

ε0=

1

ε0

∫

dV ρel ⇒ ∇·E =ρel

ε0• For gravity the analogy gives

∮

dS·g =

∫

dV ∇·g = −4πGM = −4πG

∫

dV ρ

⇒ ∇·g = −4πGρ

NEWTONIAN GRAVITY THEORY

• We can use Gauss’ theorem and the full apparatus of potential theory

to solve gravitational problems.

• In terms of the potential we have Poisson’s equation:

∇2φ = 4πGρ

• We will only need cases where there is high symmetry:

- spherical distribution gr =GM(r)

r2, where M(r) is the mass

inside r;

- cylindrical distributions will not arise in this part of the course.

- symmetric slab distribution gz = 2πGM(z) (simple model of a

gravitating disc).

• In fact we can do quite complicated problems. . .

Gravitational fieldlines and equipotentials for a uniform disc.

GRAVITATIONAL FIELD AND POTENTIAL OF A SPHERICAL BODY

• Spherical body of density ρ and radius a (total mass M = 43πρa3).

• Field at radius r determined by mass M(r) inside: g(r) =GM(r)

r2.

• Outside the body we just have:

g(r) =GM

r2; φ(r) = −GM

r.

• Inside the body M(r) =Mr3

a3,

so that g(r) =GMr

a3= 4

3πGρr.

• The potential φ(r) is the integral

of g(r) with the constant of integration

chosen to match the potential at r = a:

φ(r) =GM

2a3

(

r2 − 3a2)

.

• If the matter in the sphere is at rest and has the correct pressure

distribution P (r) it can support itself against collapse.

• Hydrostatic equilibrium: balance of forces on element ∇P = ρg.

Here ρ is constant so we can integrate: P = −ρφ + constant.

• Setting P = 0 at r = a gives P (r) = 23πGρ2

(

a2 − r2)

• In terms of mass and radius the central pressure is P0 =3GM2

8πa4

DYNAMICS OF EXPANDING SHELL

• If there is no pressure (a “dust” sphere), a uniform sphere started at

rest will collapse (see examples1: Q6 — method given below works).

• The gravitational field is proportional to radius, so that the collapse has

radial velocity v(r) ∝ r, and the density remains spatially uniform.

• We call this type of motion “self-similar‘’, and we will meet it several

times (outflows; blast waves).

• We will look at the case of an expanding sphere with v(r) ∝ r, and

use it as a simple Newtonian model for the expanding universe.

• We write the radial position r(t) of a particle in the sphere as λR(t),

where 0 < λ < 1 is a time-invariant label and R(t) is the radial scale

(R(0) = a). We also have ρ(t) = ρ0

a3

R3(t).

• Then v(r, t) = λR = H0(t)r, where H0 ≡ R/R.

• With these substitutions the equation v = −g becomes (λ cancels):

R = −4πGρ0a3

3R2

• Multiply by R and integrate: 12R2 =

4πGρ0a3

3R+ constant.

• The constant of integration determines whether the expansion will

cease: if negative the sphere stops and recontracts. The critical case

is ρcrit =3H2

0

8πG.

• Not a bad model for the universe. . .

THE FRIEDMANN EQUATION

• Our expanding dust model with Newtonian gravity is actually a very

good model for the expanding universe — it has reproduced the

Friedmann equation for the evolution of the cosmological scale

parameter. It is the same equation you get from general relativity if the

pressure and cosmological constant are zero.

• By appropriate scalings of time and radius it can be put in the form

R2 =1

R− k, where distinct cases are distinguished by k taking one

of the three values −1, 0, +1.

• The equations can be solved in closed form for t as a function of R:

k = −1 t =√

R(1 + R) − sinh−1(√

R)

k = 0 t = 23R3/2

k = +1 t = −√

R(1 − R) + sin−1(√

R)

• You will meet these cosmological models again next Term.

HOW COMPRESSIBLE ARE ATOMS?

• Recall Bohr model for hydrogen atom:

Forces:e2

4πε0r2= meω

2r

⇒ ω2 =e2

4πε0mer3

Quantum: h = meωr2 ⇒ ω =h

mer2

• Hence radius of orbit a0 is given by a0 =4πε0h

2

e2me

= 5 × 10−11 m.

• Energy is (using Virial theorem):

E = − 12

e4

4πε0a0

= − e4me

32π2ε20h2

= −13.6 eV

.

• Atoms contain a large amount of

energy and resist compression

very strongly.

• We can estimate the pressure that

atoms can resist as approximately

equal to the energy density ∼ E/(2a0)3 ∼ 2 × 1012 Pa.

• This estimate is an upper limit: if exceeded the electrons will certainly

not be bound to the protons.

HOW COMPRESSIBLE ARE ATOMS? II

• We can make a simpler model of an atom: ignore the proton and

suppose the electron is in a box of side πa, so that its momentum is

p =h

aand the kinetic energy is

p2

2me

=h2

2mea2.

• Now remember the proton and find the total energy

E(a) =h2

2mea2− e2

4πε0a

E(a)

a

• Treat a as a variable parameter

and locate the minimum energy

as a is varied:

amin =4πε0h

2

e2me

= a0.

• The minimum energy itself is Emin = − e4me

32π2ε20h2

as before.

• I admit I fiddled that — you can also invoke the uncertainty principle if

you like. . .

• The general point is that, as you try to compress an atom, the kinetic

energy will increase and eventually the total energy will be positive. At

this point the electrons are not bound to individual protons and the

hydrogen becomes metallic (predicted in 1935).

• According to our simple model this occurs when a = 12a0; i.e. when

compressed by a factor of 8.

GRAVITY: THE ATOM-CRUSHER

• The gravitational energy of neighbouring atoms in hydrogen is about

Gm2p

2a0

∼ 10−35eV — a factor of 1036 times weaker than the

electrostatic energy.

• But gravity is always attractive and, as you add more atoms,

gravitational effects become more important. . .

• Suppose we have a body containing N atoms.

• The electrostatic forces on an atom are no

stronger than before — the electric field is

shielded on scales larger than a0.

• An atom will feel the gravitational effect of all the other atoms in the

body, but we have to allow for the increased separation.

• For an incompressible body the average separation is ∼ a0N1/3.

• The gravitational energy of the atom is nowNGm2

p

2a0N1/3; i.e. ∝ N2/3

• Although gravitation is 1036 times weaker than electromagnetism at

the atom level, it will dominate when the number of atoms exceeds

(1036)3/2 = 1054.

• This is a mass of1054mp ≈ 1027 kg — about the size of Jupiter.

PROPERTIES OF SOLID HYDROGEN

• Density of solid hydrogen is low: the molecular volume is 22.7 cm3.

• As solids go it is rather weak – it compresses by a factor of two in

volume under a pressure of “only” 2 × 104 atmospheres (3 GPa).

• Resistivity and molecular volume have now been measured up to

pressures of 300 GPa, which is about the central pressure of Jupiter.

• The resistivity drops dramatically and hydrogen becomes “metallic”

when compressed by a factor of 9.

VIRIAL THEOREM

• Balance between gravitational potential energy and kinetic energy in

systems bound by gravity and supported against collapse by internal

motions.

• System of masses mi, positions ri and velocities vi

• The quantity G ≡∑

i

(mivi)·ri is called the virial.

dG

dt=

∑

i

mi

(

v2i + vi·ri

)

= 2T −∑

i

Fi·ri

∑

i Fi·ri =∑

i

∑

j 6=i

Gmimjri·(rj − ri)

|ri − rj |3

=∑

i

∑

j<i

Gmimj(ri·(rj − ri) + rj ·(ri − rj))

|ri − rj |3

=∑

i

∑

j<i

Gmimj

|ri − rj |= −V

• If the motions are bounded limτ→∞

1

τ

∫ τ

0

dt G = 0.

• Hence Virial Theorem:

〈Kinetic energy〉 = − 12〈Gravitational potential energy〉