A1199Are We Alone?

The Search for Life in the UniverseSummer 2019

Instructor: Shami Chatterjee

Web Page: http://www.astro.cornell.edu/academics/courses/astro1199/HW 2 posted – due Wednesday 10 July

So far: Big Bang, cosmology, galaxiesNow: Stars

What is needed to form a star?

• A star typically means an object that shines because nucleosynthesis occurs in its core.

• Initial reactions for “main sequence” stars: 4H à He (CNO cycle, proto-proton chain).

• Later: He à C à … other heavier elements … à Fe.

What is needed to form a star?• Requirements:

• Collapse of a gas cloud that contains H.• Sufficient mass in protostar so that central temperature

is high enough to drive nuclear reactions.

• Collapse of gas clouds is constrained by the temperature and density of the gas cloud.

• Jeans radius and Jeans mass are measures of whether an object (a gas cloud) is susceptible to collapse.

RJ ~ sound speed x free-fall time~ (Tgas / ρ)1/2.

• Stars form with a large range of masses: Initial mass function.

Jeans Scale and Mass

r

FP

FG

Compare: Free-fall time for a cloud to collapse:

vs.Time for pressure wave to propagate:

tP = R/Cs.

If tff < tP, then the region will collapse faster than pressure can push back.

Jeans Scale RJ ~ Cs / √(Gρ)

tff =

✓2R3

GM

◆1/2

⇠ 1pG�

Gravitational stability: The case of B68 Optical Near-Infrared

Starless Bok GlobuleGravitationally stable, or at the verge of collapse.

Example• In the Milky Way there are cold dense clouds that are

actively forming stars today.• Typical temperatures are 10 K and densities 10-22 g/cm3.• Evaluating RJ ~ (kT/m)1/2 (1 / (Gρ))1/2 ~ 3 pc.• We can also calculate the Jeans mass as

MJ ~ ρ RJ3

and we get about 50 M¤.

• Interpretation: Relatively large mass regions collapse. Sub-regions inside them fragment as their temperatures fall and their densities increase.

à A wide range of stellar masses.

Initial mass function of stars = distribution of masses at birth

Populations of Stars: I, II, III

• Population I stars: stars like the Sun; later generation, higher metal content.

• Population II stars: low metallicity, older stars like those found in globular clusters.

• Population III stars: the hypothetical first stars formed from pure H and He.

Mass fractions of elements in the Sun

Globular cluster M80

Stars mostly older than the Sun.

The Pleiades

Newborn stars, “only” ~ 108 yrs old.

Stars: Birth, Life, and DeathBIRTH: Gravitational Collapse of interstellar

clouds. “Hayashi Contraction.”

LIFE: Stability on Main Sequence.Energy from nuclear reactions in stellar cores(E = mc2).

DEATH: Lack of nuclear fuel. Instability, variability, expansion (giants, supergiants). Spectacular explosions!

H-R Diagrams

H-R

diag

ram

Entrance of a star into the HR Diagram

At equilibrium core T ≈ 15´106 K.Nuclear reactions create energy Þ E = mc2.

Stellar Evolution

Interstellar Cloud è Proto-StarèHayashi Contraction è Main Sequence

è Red Giant è Variable Starèè Explosion è White Dwarf

Evolution of a star like the SunContraction/collapse of a fragment of an interstellar cloud

• Density and temperature in core rise.• Star has large radius (R) but cool temperature (T)

so it is bright (high luminosity L) but very red (infrared).• Short-lived phase.• Collapses along axis of rotation; formation of disk possible.• When the core becomes hot enough, hydrogen burning ignites.

HK Tau –young stars (5 Myr) in binary system.

ALMA imagingreveals misaligned disks.

Star formation and protoplanetary disks

HK Tau –young stars (5 Myr) in binary system.

ALMA imagingreveals disks.

Doppler shift of emission from molecular gasà Get rotation, and infer disk axis.

Disks are misaligned!

H-R

diag

ram

The Main Sequence• Stars on the “Main

Sequence” are burning hydrogen into helium in their cores.

• The mass of a star determines its location on the Main Sequence of the H-R diagram.

• Sirius A, Altair, Procyon A are more massive than the Sun. Sirius B, Proxima Cen are less massive than the Sun.

Main Sequence: the Hydrogen-burning phase of a star’s lifetime

• Different stars have different masses.

• The time a star spends on the Main Sequence depends on its mass.

• A more massive star converts all its H into He quicker than a less massive star!

• A more massive star has a shorter “Main Sequence lifetime” than a less massive star.

L µ M4

on the M.S.

Stars don’t shine forever• The “fuel” in stars is proportional to the mass, M.• The luminosity of stars on the main-sequence varies

with mass as: Luminosity µ (Mass)4

Assuming all stars “consume” the same fraction of their mass (M), the lifetime is given by:

Lifetime µ =Amount of fuel Rate of using fuel

Star’s mass, M*Star’s luminosity, L*

M*L*

MassMass4

1Mass3

==µtLifetime of star

High mass stars have SHORTER lifetimes!

Stellar Evolution

H-R

diag

ram

Globular Clusters: Older Populations

Virial Theorem and Stellar TemperaturesThe virial theorem says that in a stable object the internal and gravitational energy are balanced:

2 x KE + PE = 0.Example: a planet of mass m orbiting a star of mass M

The KE and PE are:

• So the VT is satisfied. The same is true for any stable object that is held together by gravity.

mv2

r=

GMm

r2

KE =1

2mv2 =

GMm

2rPE = �GMm

r

VT and Stars• In a star, the kinetic energy is thermal (possibly

combined with convection and turbulence, which we ignore here).

• The gravitational potential energy is (uniform density):

• The thermal energy is (uniform temperature):

• Using the VT we can solve for temperature:

PE = �3GM2

5R

KE = N

⌧1

2mv2

�=

3

2

M

mkT

T =1

5

GMm

kR

k = Boltzmann’s constantm = particle mass (e.g. mass of a proton)M = Mass of star; R = its radius.

Internal Temperatures of Stars

• Use

to estimate T ~ 4.6 x 106 K for the Sun.

• This is an average temperature but is comparable to what is needed to drive nuclear reactions.

T =1

5

GMm

kR

G = 6.67 x 10-8 cgsM¤ = 2 x 1033 gR = 7 x 1010 cmm = mH = 1.67 x 10-24

k = 1.38 x 10-16 cgs

Kelvin-Helmholtz Contraction Time

• The VT says that KE and PE are balanced. In order for a star like the Sun to contract, it must lose energy. The VT further implies that while ½ of the PE goes into an increase in KE, the other half must be radiated away.

• The measured luminosity of the Sun is L ~ 4 x 1033 erg s-1. If this luminosity were solely due to radiation of GPE, the lifetime of the Sun would be only about 3x1014 s, or ~ 10 Myr (the K-H contraction time).

• What gives? Either the solar system is very young or there is another source of energy, i.e. nuclear reactions.

Solar Interior

• Radiative zone:– Energy is

transported by electromagnetic radiation.

• Convection zone:– Energy carried by

convection.

The Core of the Sun

• The core of the sun is the place where nuclear fusion reactions power the sun.

• Approximate T ~ 15´106 K. • The sun has been “burning” for 5 billion years

and theoretically should continue burning for another 4 to 5 billion years.

• Should the core stop burning, the star’s luminous life would be at an end.

The Proton-Proton Chain Reaction

• Three steps complete this fusion reaction:

• Net effect reaction: 4p è 4He + energy

• The release of energy is about 0.007 times the rest mass of the input hydrogen.

The CNO Cycle

• Six steps complete this fusion reaction:

The CNO cycle requires higher temperatures than the proton-proton chain because C and N nuclei have larger positive charge that the proton needs to push against.

This requires higher thermal velocities for the protons.

Mass-Luminosity RelationMain Sequence Stars

• Nuclear reactions in hotter stars are faster, and T = T(M), so luminosities scale strongly with mass.

• A simple approach gives L ~ M3. More detailed analysis - get scaling laws:

The Milky Way

0

25

50

75

100

O-M F-M O B A F G K M B-F

Supergiant(I & II)

Red Giant(III)

Main Sequence (V) WhiteDwarf

Luminosity Class and Spectral Type

Percentage of Galactic LuminosityPercentage in Number Percentage of Galactic Stellar Mass

75% of the Milky Way’s luminosity

arise from the rarest stars.

K & M stars account for ¾’s of the stars in the

galaxy but contribute less than 5% of its luminosity.