Test Review 3

Test 3

• Test covers Chapters 9 and 11-14 (except variable stars)

• Show your work

• Don’t forget to prepare formula sheet

• Bring your calculator

• Textbook and lecture notes are not allowed

Less math, but more concepts

H-R diagram

90% of stars

The mass-luminosity relation for 192 stars in double-lined spectroscopic binary systems.

L ~ M3.5 only for main-sequence stars!

5.3

sunsun M

M

L

L

star mass (solar masses) time (years) Spectral type

60 3 million O3

30 11 million O7

10 32 million B4

3 370 million A5

1.5 3 billion F5

1 10 billion G2 (Sun)

0.1 1000's billions M7

Lifetime T ~ M/L ~ 1/Mp-1 = 1/M2.5 ; p ~ 3.5

M = 4M; 32

15.2

M

M

T

T sun

sun

Lifetime = Amount of hydrogen fuel

Rate of energy loss

T ~ 3x108 years

Estimating the Age of a Cluster

The lower on the MS

the turn-off point, the older the cluster.

5.2

1~M

T

Age of a cluster = lifetime of stars on the turnoff point

5.2

M

M

T

T sun

sun

H-R diagram

• 90% of stars are on the main sequence and obey the mass-luminosity dependence L ~ M3.5

• Stars on the main sequence generate energy due to nuclear fusion of hydrogen

• In the end of their lives stars move to the upper right corner of the H-R diagram

Cutoff at masses > 100 M and < 0.08 M

Spectral Lines of Giants

=> Absorption lines in spectra of giants and supergiants are narrower than in main sequence stars

Pressure and density in the atmospheres of giants are lower than in main sequence stars.

=> From the line widths, we can estimate the size and therefore, the luminosity of a star.

Distance estimate (spectroscopic parallax)

Luminosity Classes

Ia Bright Supergiants

Ib Supergiants

II Bright Giants

III Giants

IV Subgiants

V Main-Sequence Stars

IaIb

IIIII

IV

V

Luminosity classes

• Ia bright supergiant

• Ib Supergiant

• II bright giant

• III giant

• IV subgiant

• V main-sequence star

Example Luminosity Classes

• Our Sun: G2 star on the Main Sequence:

G2V

• Polaris: G2 star with Supergiant luminosity:

G2Ib

Luminosity

sT

24

m

Jstar theof areaunit fromFlux

Surface area of the star = 4R2

Luminosity, or total radiated power L = T4 4R2 J/s

Intensity, or radiation flux on the Earth:

)mJ/(s4

22 d

LI

R

d

It is convenient to compare with the Sun or any other star:

42

b

a

b

a

b

a

T

T

R

R

L

L

Surface temperature and color indices

K)(

103)nm(

6

T

Can be applied to any black-body emitter!

Binary Stars More than 50 % of all stars in our Milky Way

are not single stars, but belong to binaries:

Pairs or multiple systems of stars which

orbit their common center of mass.

If we can measure and understand their orbital

motion, we can

estimate the stellar masses.

Measuring diameters and masses

A

B

B

A

m

m

r

r

Estimating Stellar Masses

Recall Kepler’s 3rd Law:

Py2 = aAU

3

Valid for the Solar system: star with 1 solar mass in the center.

We find almost the same law for binary stars with masses MA and MB different

from 1 solar mass:

MA + MB = aAU

3 ____ Py

2

(MA and MB in units of solar masses)

Examples: Estimating Mass

Binary system with period of P = 32 years and separation of a = 16 AU:

MA + MB = = 4 solar masses.163____322

How to measure period and separation?

Arbitrary units:

22

324

PG

aMM BA

Visual Binaries

The ideal case:

Both stars can be seen directly, and

their separation and relative motion can be followed directly.

Spectroscopic binaries

Stars are seen as a single point

• Spectra of both stars are distinguishable

• Sometimes spectrum of only one star is seen

The Doppler EffectThe light of a moving source is blue/red shifted by

/0 = vr/c

0 = actual wavelength

emitted by the source

Wavelength change due to

Doppler effect

vr = radial velocity( along

the line of sight)

Blue Shift (to higher frequencies)

Red Shift (to lower frequencies)

vr

Shift z = (Observed wavelength - Rest wavelength)

(Rest wavelength)

Doppler effect: cVc

Vz rad

rad

;0

00

0

z

The Doppler effect: apparent change in the wavelength of radiation caused by the motion of the source

The Doppler Effect The Doppler effect allows us to

measure the source’s radial velocity.

vr

/0 = vr/c

Determining the orbital period

Eclipsing Binaries

From the light curve of Algol, we can infer that the system contains two stars of very different surface temperature, orbiting in a slightly inclined plane.

Example:

Algol in the constellation of Perseus

Estimating the Age of a Cluster

The lower on the MS

the turn-off point, the older the cluster.

5.2

1~M

T

Age of a cluster = lifetime of stars on the turnoff point

5.2

M

M

T

T sun

sun

Jeans instability:

Thermal pressure cannot support the gas cloud against its self-gravity. The cloud collapses and fragments.

Coldest spots in the galaxy:T ~ 1-10 K

Composition:• Mainly molecular hydrogen• 1% dust

Shocks Triggering Star Formation

Globules = sites where stars are being born right now!

Trifid Nebula

Heating By Contraction

As a protostar contracts, it heats up:

Free-fall contraction→ Heating

Heating does not stop contraction because the core cools down due to radiation

Protostellar Disks and Jets – Herbig Haro Objects

Herbig Haro Object HH30

• The matter stops falling on the star• Nuclear fusion starts in the core• Planets can be formed from the remaining disk

The Source of Stellar Energy

In the sun, this happens primarily through the proton-proton (PP) chain

Recall from our discussion of the sun:

Stars produce energy by nuclear fusion of hydrogen into helium.

The CNO Cycle

In stars slightly more massive than the sun, a more powerful

energy generation mechanism than

the PP chain takes over:

The CNO Cycle.

Net result is the same: four hydrogen nuclei fuse to form one helium nucleus; 27 MeV is released.

Why p-p and CNO cycles? Why so complicated?

Because simultaneous collision of 4 protons is too improbable

Energy Transport Structure

Inner radiative, outer convective

zone

Inner convective, outer radiative

zone

CNO cycle dominant PP chain dominant

Evolution off the Main Sequence: Expansion into a Red Giant

Hydrogen in the core completely converted into He:

H burning continues in a shell around the core.

He Core + H-burning shell produce more energy than needed for pressure support

Expansion and cooling of the outer layers of the

star Red Giant

“Hydrogen burning” (i.e. fusion of H into He) ceases in the core.

Formation of degenerate core

Red Giant Evolution

4 H → He

He

H-burning shell keeps dumping He

onto the core.

He-core gets denser and hotter until the

next stage of nuclear burning can begin in

the core:

He fusion through the

“Triple-Alpha Process”

4He + 4He 8Be + 8Be + 4He 12C +

p. 192

The Fate of Our Sun and the End of Earth• Sun will expand to a

Red giant in ~ 5 billion years

• Expands to ~ Earth’s orbit radius or more

• Earth will then be incinerated!

• It will be too hot for life in 200 million years

• Sun may form a planetary nebula (but uncertain)

• Sun’s C,O core will become a white dwarf

What is left?

A stellar remnant: white dwarf, composed mainly of carbon and oxygen

Sirius B is very hot: surface temperature 25,000 KYet, it is 10,000 times fainter than Sirius A

It should be very small: R ~ 2 Rearth

Its mass M ~ 1 Msun

It should be extremely dense!

M/V ~ 106 g/cm3

V

M

All atoms are smashed and the object is supported by pressure of degenerate electrons

White dwarf should be extremely dense!

M/V ~ 106 g/cm3V

M

Strange properties of degenerate matter

• It strongly resists compression: P ~ 5/3

• Pressure does not depend on temperature

Compare with classical gas: P ~ T

Evolution of sun-like stars on H-R diagram

Chandrasekhar limit: 1.4 Msun

This is because gravitational pressure increases with mass. Electron pressure should also increase, and the only way to do it is to compress the star.

Stars > 8 solar masses

Reactions proceed faster and faster, until Fe and Ni are synthesized

The iron core of a giant star cannot sustain the pressure of gravity. It collapses inward in less than a second.

The shock wave blows away outer layers of a star, creating a SUPERNOVA EXPLOSION!

Summary of Post Main-Sequence Evolution of Stars

M > 8 Msun

M < 4 Msun

Evolution of 4 - 8 Msun stars is still uncertain.

Fusion stops at formation of C,O core.

Mass loss in stellar winds may reduce them all to < 4 Msun stars.

Red dwarfs: He burning never ignites

M < 0.4 Msun

Supernova

Fusion proceeds; formation of Fe core.

• Evolution of sun-like stars: red giant, planetary nebula, white dwarf

• Evolution of massive stars: red giant or supergiant, supernova

• Three types of compact objects – stellar remnants: white dwarfs, neutron stars, black holes. Limits on their masses. Pulsars as rotating neutron stars

• Compact objects in binary systems. Accreting X-ray binaries

Type I and II SupernovaeCore collapse of a massive star:

Type II Supernova

If an accreting White Dwarf exceeds the Chandrasekhar mass limit, it collapses,

triggering a Type Ia Supernova.

Type I: No hydrogen lines in the spectrum

Type II: Hydrogen lines in the spectrum

Energy release due to radioactive decay of 56Ni and 56Co

Stellar nucleosynthesis

• All elements up to Atomic mass ~ 250 u are synthesized!

• S-processes: “slow” synthesis of elements up to iron

• R-processes (r = rapid): rapid neutron capture during SN explosion; all elements heavier than iron

The Remnant of SN 1987A

Ring due to SN ejecta catching up with pre-SN stellar wind; also observable in X-rays.

Synchrotron Emission and Cosmic-Ray Acceleration

The shocks of supernova remnants

accelerate protons and electrons to extremely

high, relativistic energies.

“Cosmic Rays”

In magnetic fields, these relativistic

electrons emit

Synchrotron Radiation.

Crab nebula: the remnants of supernova 1054

Fate of the collapsed core

• White dwarf if the remnant is below the Chandrasekhar limit 1.4 Msun after mass loss

• Neutron star if the core mass is less than ~ 3 solar masses

• Black hole otherwise

Deaths of stars

Formation of Neutron StarsCompact objects more massive than the

Chandrasekhar Limit (1.4 Msun) collapse further.

Density and T become so high that electrons and protons combine to form stable neutrons throughout the object:

p + e- n + e

Neutron Star

Properties of Neutron Stars

Typical size: R ~ 10 km

Mass: M ~ 1.4 – 3 Msun

Density: ~ 1014 g/cm3

Piece of neutron star matter of the size of a sugar cube has a mass of ~ 100 million tons!!!

• Neutron stars should rotate extremely fast due to conservation of the angular momentum in the collapse

• They should have huge magnetic field due to conservation of the magnetic flux in the collapse

2211 RMVRMV

12

1

1

2 R

R

V

V

The enigma of pulsarsPulse repetition: from a few to 0.03 secondsPulse duration: ~ 0.001 sPeriod extremely stable: it increases by less than 1 sec in a million years

What could it be???

Only star rotation can be so stable. However: Centrifugal acceleration < gravitational acceleration

km50~3/1

222

GM

RR

GMR

It must be a neutron star!!

General idea of a pulsar emission

Exact mechanism of pulsar radiation is still unknown!

The Crab Pulsar

Remnant of a supernova observed in A.D. 1054

Pulsar wind + jets

Only a small fraction of neutron stars are pulsars

• Their beams may not sweep over the Earth

• Rotation of old neutron stars slows down and the pulsar mechanism turns off (why?)

Pulsar Periods

Over time, pulsars lose energy and angular momentum

=> Pulsar rotation is gradually slowing down.

Binary PulsarsSome pulsars form binaries with other neutron stars (or black

holes).

Radial velocities resulting from the orbital motion lengthen the pulsar period when the pulsar is moving away from Earth...

…and shorten the pulsar period when it is approaching

Earth.

Orbital period becomes shorter: stars lose energy to gravitational radiation

Taylor and Hulse, Nobel prize 1993

Pulsar PSR1913+16: two neutron stars in a binary system

2

2

c

GMRs

Schwarzschild radius: event horizon for a nonrotating body

To make a black hole from a body of mass M, one needs to squeeze it below its Schwarzschild’s radius

Rs

Gravitational collapse: the body squeezes below its event horizon

Newton’s theory is a weak-gravity limit of a more general theory: General Relativity

Even in the weak gravity of the Earth and the Sun, there are measurable deviations from Newtonian mechanics and gravitation law!

• Advance of Mercury’s perihelion

• Bending of light by the Sun’s gravity

General Relativity predicts new effects, completely absent in the Newton’s theory: black holes, event horizon, gravitational waves.

Einstein’s idea:

Bending of light

Space-time gets curved by masses. Objects traveling in curved space-time have their paths deflected, as if a force has acted on them.

Main idea:

“Curvature” of time means that the time flows with a different rate in different points in space

"Matter tells spacetime how to bend and spacetime returns the complement by telling matter how to move."

John Wheeler

Low density star

High density star

How to observe a stellar remnant if it does not emit radiation?

• Isolated black hole or neutron star has almost no chance to be seen

• Gravitational lensing is possible but very improbable• Isolated neutron star can be detected as a pulsar, or if it is very

close and hot• Isolated white dwarf can be seen if it is close enough and hot• Good news: most stars are in binary systems

– We can detect radiation from matter accreting onto a compact object. Remember, however, this is only an indirect indicator of a black hole

– We can determine the mass of an unseen companion. If it is much larger than 3 Msun – this is likely a BH. If it is between 1.4 and 3 Msun – this is likely a neutron star.

;2

3

21 P

aMM

a – in AUP – in yearsM1+M2 – in solar masses

Binary systems

If we can calculate the total mass and measure the mass of a normal star independently, we can find the mass of an unseen companion

Observing Black HolesNo light can escape a black hole

=> Black holes can not be observed directly.

If an invisible compact object is part of a binary, we can estimate its mass from the orbital period and

radial velocity.

Mass > 3 Msun

=> Black hole!

Candidates for Black Hole

Compact object with > 3 Msun must be a

black hole!