Stellar structure and evolution Pierre Hily-Blant 2017-18 April - - PowerPoint PPT Presentation

Stellar structure and evolution Pierre Hily-Blant

2017-18

April 29, 2018

IPAG pierre.hily-blant@univ-grenoble-alpes.fr, OSUG-D/306

9 Star formation 9.1. Introduction

9– Star formation

Introduction Collapse of spherical gaseous configurations Stability of isothermal self-gravitating spheres Numerical simulations Formation of the protostar Current view of star formation Initial mass function Open questions

9 Star formation 9.1. Introduction

3 On-going star formation: evidences

• Star formation is an on-going process in the MW
• Various evidences
• We see star forming: young stellar objects (YSOs) ∼ 1 Myr
• Open clusters and associations remain visible despite

differential rotation which would bring them apart by ∼ 10 kpc in 10 Gyr);

• We see massive stars on the main sequence: lifetime on the

MS is τMS ∼ 1010M/L yr, and L ∝ Mα with α = 3 − 4.5. For M = 10 M⊙, τMS = 3Myr ≪ age of the Galaxy;

9 Star formation 9.1. Introduction

4 Star formation

All stars form in molecular clouds (here: Taurus molecular cloud)

9 Star formation 9.1. Introduction

5 Molecular gas in the Milky Way

• S. Ori

• FIG. 2.–Velocity-integrated CO map of the Milky Way. The angular resolution is 9´ over most
• f the map, including the entire Galactic plane, but is lower (15´ or 30´) in some regions out
• f the plane (see Fig. 1 & Table 1). The sensitivity varies somewhat from region to region,

Dame et al. (2001)

9 Star formation 9.1. Introduction

6 Molecular clouds distribution

Koda+09

M51 at 160 pc resolution; Molecular clouds are distributed along

9 Star formation 9.1. Introduction

7 Milky Way: Overview

• Mass of galactic disk + bulge: 6 ×1010 M⊙
• Mass is dominated by dark matter (90%)
• Baryonic mass is essentially in the form of gas
• Dust is 1% in mass
• 1011 stars in the MW

9 Star formation 9.1. Introduction

8 Molecular gas in the Milky Way

Stars form in molecular clouds

9 Star formation 9.1. Introduction

9 Star formation

• Overall: good understanding
• Outstanding issues remain: Star formation is one of the main
• pen question in astrophysics
• What determines the rate of star formation ?
• How are the kinetic energy and angular momentum removed

from the collapsing cloud ?

• What determines the initial mass function ?
• How did the first stars formed in the Universe ?

9 Star formation 9.1. Introduction

10 Observations of star formation

9 Star formation 9.1. Introduction

11 Prestellar core

Pre-stellar cores (here Barnard 68)

9 Star formation 9.1. Introduction

12 Protostar

• Class 0 protostar:

L1527

• Located in the Taurus

cloud (140 pc)

• Very early stage of

collapse

• Age ∼ 0.3 Myr
• Mass of the protostar:

0.19±0.04 M⊙

• Protostar/envelope

mass ratio ∼ 0.2

• Luminosity: accretion

(6.6 ×10−7 M⊙/yr)

9 Star formation 9.1. Introduction

13 Protostar

9 Star formation 9.1. Introduction

14 Protostar

9 Star formation 9.1. Introduction

15 Protostar

9 Star formation 9.1. Introduction

16 Outflows and Herbig-Haros objects

HH212: molecular hydrogen

9 Star formation 9.1. Introduction

17 Outflows and Herbig-Haros objects

HH212

9 Star formation 9.1. Introduction

18 TTauri stars and circumstellar disks

HL Tau protoplanetary disk with ALMA

9 Star formation 9.1. Introduction

19 Primitive solar system

Comet McNaught

9 Star formation 9.1. Introduction

20 Primitive solar system

Comet 67P/C-P with ESA/Rosetta

9 Star formation 9.1. Introduction

21 Exoplanetary systems

The Trappist-1 system with NASA/Spitzer

9 Star formation 9.1. Introduction

22 Exoplanetary systems

9 Star formation 9.1. Introduction

23 From clouds to stars to planets

• All stars form in molecular clouds
• Molecular clouds: n ∼ 103 cm−3, T ∼ 30 K, ∼1-10 Myr ?
• Prestellar cores: n ∼ 104 cm−3, T ∼ 10 K, ∼1 Myr ?
• Protostars: n > 105 cm−3, short phase, ∼ few 0.1 Myr ?
• Protoplanetary disks: ∼10 Myr before gas dissipation
• Planet formation: rapid (less than few 100 Myr)
• Life: less than 1 Gyr ? (oldest microfossils: at least 3.7 Gyr
• ld)

9 Star formation 9.2. Collapse of spherical gaseous configurations

9– Star formation

Introduction Collapse of spherical gaseous configurations Stability of isothermal self-gravitating spheres Numerical simulations Formation of the protostar Current view of star formation Initial mass function Open questions

9 Star formation 9.2. Collapse of spherical gaseous configurations

25 Jeans analysis

From molecular clouds to protostars

• Uniform density
• Jeans criterium: a cloud of mass M > MJ is unstable

MJ =

• 3kT

αGµma 3/2 3 4πρ0 1/2

• Jeans Length

RJ =

• 9kT

4παGµmaρ0 1/2

• Jeans density

ρJ = 3 4πM2

• 3kT

αµmaG 3

• Note: α = 3/5 for a spherical distribution

9 Star formation 9.2. Collapse of spherical gaseous configurations

26 Jeans analysis

nH T M R M†

J

R†

J

cm−3 K M⊙ pc msol pc Diffuse WNM 1 8000 – – 1.3 107 400 GMC (H2) 50 30 > 104 30 400 3 Molecular clouds 500 25 3 ×103 10 100 0.9 Prestellar cores 104 10 few 0.1 5.5 0.1

• Values for spherical geometry and molecular gas
• Note: observationally, prestellar cores have a typical size

∼ 0.1pc

9 Star formation 9.2. Collapse of spherical gaseous configurations

27 Fragmentation

• Note: Jeans analysis does not include gas dynamics, no mag.

pressure, no rotation, etc.

• How do we form stars with mass much lower than that of the
• riginal cloud ? No definitive answer: fragmentation,

competitive accretion. Fragmentation

• Collapse if M > MJ; during collapse, ρ increases; regions of

the cloud become Jeans-unstable; they collapse, separate (motions); etc...

9 Star formation 9.2. Collapse of spherical gaseous configurations General equations

28 Special case: Isothermal sphere

• Prestellar cores: isothermal evolution towards protostar

formation (efficient cooling by molecular line emission)

• Pressure: P = nkT + aT 4/3; cold prestellar cores, radiative

pressure is negligible

• EOS: P = c2

s ρ

9 Star formation 9.2. Collapse of spherical gaseous configurations General equations

29 Special case: Isothermal sphere

• Self-gravitating sphere of an isothermal, perfect, gas
• Start with dP/dr = −Gm(r)ρ(r)/r2 and dm/dr = 4πr2ρ(r)

plus the EOS

• Usual change of variable:
• ρ = ρce−ψ
• r = aξ, a = (kT/4πGµmHρc)1/2 = cs/(4πGρc)1/2
• Particular instance of the Lane-Emden equation for isothermal

case: 1 ξ2 d dξ

• ξ2 dψ

dξ

• = e−ψ
• Mass: (show that) M(ξ) = M0m(ξ) with M0 = c3

s /(G 3/2ρ1/2 c

) and m(ξ) = ξ2ψ′(ξ)/ √ 4π

9 Star formation 9.2. Collapse of spherical gaseous configurations General equations

30 Singular isothermal sphere

• Particular solution of the Lane-Emden equation: infinite

central density ρ(r) = c2

s

2πGr2

• Equilibrium solutions: infinite extent, infinite mass !

9 Star formation 9.2. Collapse of spherical gaseous configurations General equations

31 Non-singular solutions

• Boundary conditions:

ψ = ψ′ = 0 at ξ = 0

• Numerical integration

(using RK4 scheme)

• Two 1st order ODE:

u = ψ, v = ξ2ψ′

9 Star formation 9.2. Collapse of spherical gaseous configurations General equations

32 Non-singular solutions

• Family of solutions (central density, nc, in cm−3)
• µ = 2.33 (fully molecular); T = 10 K

9 Star formation 9.3. Stability of isothermal self-gravitating spheres

9– Star formation

Introduction Collapse of spherical gaseous configurations Stability of isothermal self-gravitating spheres Numerical simulations Formation of the protostar Current view of star formation Initial mass function Open questions

9 Star formation 9.3. Stability of isothermal self-gravitating spheres Bonnor-Ebert spheres

34 Role of the external pressure: Bonnor-Ebert sphere

• Pre-stellar core embedded in a

molecular cloud

• Realistic clouds are pressure

bounded: pext = c2

s ρ0

• External radius: ξ0
• Boundary conditions: ψ = 0,

ψ′ = 0 at ξ = 0

• Additional boundary condition:

total mass, M = M(ξ0)

• Typical values: L1498,

nc = 105 cm−3, next ≈ 500 cm−3

• New picture: physical parameters

are M and pext

9 Star formation 9.3. Stability of isothermal self-gravitating spheres Bonnor-Ebert spheres

35 Bonnor-Ebert sphere

• From M and pext
• find ρc
• find external radius ξ0 hence cloud size R
• Total mass: show that M ≡ M(ξ0) = M0m(ξ0) with
• M0 = c3

s /(G 3/2ρ1/2

) = c4

s /(G 3/2p1/2 ext )

• dimensionless mass: m(ξ0) = (4πρc/ρ0)−1/2ξ2

0ψ′(ξ0) or

m(ξ0) = e−ψ0/2 ξ2

0ψ′(ξ0)/

√ 4π = MG 3/2p1/2

ext

c4

s

• External pressure: pext = ρ0c2

s so pext = c8 s /(G 3M2 0), such

that pext = c8

s

G 3M2 m2(ξ0)

9 Star formation 9.3. Stability of isothermal self-gravitating spheres Bonnor-Ebert spheres

36 Bonnor-Ebert sphere

• Maximum allowed mass:

MBE = 1.18

c4

s

G 3/2p1/2

ext

• M > MBE: no hydrostatic

equilibrium possible

• Cloud must evolve

dynamically

9 Star formation 9.3. Stability of isothermal self-gravitating spheres Bonnor-Ebert spheres

37 Bonnor-Ebert sphere

9 Star formation 9.3. Stability of isothermal self-gravitating spheres Bonnor-Ebert spheres

38 Stability of BE spheres

• Start from a stable BE sphere; external pressure pext, mass M
• Let us increase pext: increase internal pressure, hence

re-expand the BE sphere; increase pext,

• ⇒ m(ξ0) = MG 3/2p1/2

ext

c4

s

increases

• hence density contrast must increase
• Internal pressure (P = ρc2

s ) also increases; central and

average pressure increase more than pext;

• Increase of ρc/ρ0 increases ξ0: however, how does

R = aξ0 = ξ0cs/√4πGρc vary ?

9 Star formation 9.3. Stability of isothermal self-gravitating spheres Bonnor-Ebert spheres

39 Stability of BE spheres

• external radius: R = GM

c2

s [ξ0ψ′(ξ0)]−1

• R decreases with ξ0; initially R ∼ p−1/3

ext

• Stable branch for ∂pext/∂R < 0

9 Star formation 9.3. Stability of isothermal self-gravitating spheres Bonnor-Ebert spheres

40 Stability of BE spheres

• One necessary condition for stability: increase of pext leads to

decrease of R0 ∂pext ∂R < 0

• Condition for stability may be written:

∂pext ∂ρc/ρ0 > 0

9 Star formation 9.3. Stability of isothermal self-gravitating spheres Bonnor-Ebert spheres

41 Bonnor-Ebert sphere

• Critical dimensionless radius: stability iif ξ < ξc = 6.5
• Mass M < MBE must have a radius larger than critical value
• Critical radius: R > RBE = 0.414GM/c2

s

• This translates into critical density contrast: ρc/ρ0 < 14.1
• External pressure: pext < 1.39c8

s /G 3M2

Numerical values

• RBE = 0.12 (M/1 M⊙)(T/10)−1 (µ/2.33) pc
• MBE = 5.3 (T/10)3/2 (n0/500 cm−3)−1/2 (µ/2.33)−3/2 M⊙
• PBE = 1.4 ×104 (T/10)4 (M/1 M⊙)−2 (µ/2.33)−4 K cm−3

9 Star formation 9.3. Stability of isothermal self-gravitating spheres Bonnor-Ebert spheres

42 B68: a Bonnor-Ebert sphere

Alves et al. (2001)

• ξ = 6.9 > 6.5; ρc/ρ0 = 16.5 > 14.1; pext = 2.5 ×10−12 Pa
• Additional support: magnetic pressure ?
• critically stable

9 Star formation 9.3. Stability of isothermal self-gravitating spheres Bonnor-Ebert spheres

43 Collapsing Bonnor-Ebert spheres

• initial: critical B-E sphere
• expansion (rarefaction) wave propagating outwards at cs
• inside-out collapse
• central region: free-fall collapse

9 Star formation 9.3. Stability of isothermal self-gravitating spheres Bonnor-Ebert spheres

44 Collapsing isothermal sphere: numerical model

Larson (1969)

9 Star formation 9.3. Stability of isothermal self-gravitating spheres Bonnor-Ebert spheres

45 The r −2 density profile

• How does a r−2 emerge from a wide variety of initial

conditions ?

• convergence to r−2: approach to mechanical balance for an

isothermal sphere

• requires subsonic velocities in the envelope during early phases
• self-similar collapse then follows

9 Star formation 9.4. Numerical simulations

9– Star formation

Introduction Collapse of spherical gaseous configurations Stability of isothermal self-gravitating spheres Numerical simulations Formation of the protostar Current view of star formation Initial mass function Open questions

9 Star formation 9.4. Numerical simulations

47 Numerical simulations

• M = 50 M⊙; diameter: 0.375 pc; T = 10 K, µ = 2.46,

MJ = 1 M⊙

• n=3 ×104 cm−3;
• Free-fall time: τff = 0.19 Myr; simulation duration: 0.27 Myr
• Initial supersonic perturbation: random Gaussian,

P(k) ∝ k−4: highly dynamical (∼ Burgers turb.)

1 Shocks develop dissipating initial supersonic motions 2 Pre-stellar cores: supersonic support dissipated, gravity takes

• ver

3 competitive accretion: ejection vs accretion

9 Star formation 9.4. Numerical simulations Simulation Matthew Bate, University of Exeter Visualisation Richard West, UKAFF, 2002

9 Star formation 9.4. Numerical simulations

49 Star formation sequence

1 2e5 yr: formation of a binary 2 +12 kyr: filaments, discs, brown

dwarfs; pp-disks, protostars

3 +20 kyr: ⋆ and BD fall into a

cluster

4 +26 kyr: unstable system breaks

apart, ejecting ⋆ from the cloud

5 +41 kyr: ⋆-formation again, even

in a massive disk (3 ⋆)

6 +50 kyr: ejection; competitive

accretion;

9 Star formation 9.5. Formation of the protostar

9– Star formation

Introduction Collapse of spherical gaseous configurations Stability of isothermal self-gravitating spheres Numerical simulations Formation of the protostar Current view of star formation Initial mass function Open questions

9 Star formation 9.5. Formation of the protostar

51 Formation of the protostar: the first Larson core

Larson (1969)

• ρ < 10−13g cm−3

(≈ 2.6 ×1010 cm−3): nearly isothermal contraction

• 10−13 < ρ < 10−8g cm−3: opaque

core, adiabatic phase; T, P rise: stop collapse; core in hydrostatic equilibrium (first Larson core); γ = 5/4 − 5/3;

• m1 ∼ 0.05 M⊙, r1 ∼ 6 ×1013cm

(4-5 au), ρ1 ∼ 2(−10)g cm−3 (∼ 5 ×1013 cm−3), T1 ≈ 200 K

9 Star formation 9.5. Formation of the protostar

52 Formation of the protostar: the first Larson core

Larson (1969)

• core embedded in the envelope
• envelope: T ∼cst, ∼ free-fall
• accretion shock
• rebound when collapse first stops

+ oscillations around eq.

• m1 increases, r1 decreases

(radiative losses)

9 Star formation 9.5. Formation of the protostar

53 Evolution of the 1st core: dissociation of H2

• Grav. energy goes into internal and kinetic energy
• Ionization: increases # of free particles, hence P
• Ionization potential: χ; internal energy

u = 3/2P/ρ + (x/mH)χ

• H2: IP=4.5 eV≈ 5 ×104 K; χ/kT ≫ 1 in the 1st core γ close

to one; T rises slowly

9 Star formation 9.5. Formation of the protostar

54 Formation of the protostar: the second Larson core

Larson (1969)

• Mass of 1st core increases at ∼ cst

T: becomes unstable

• 1st collapses: ρ and T increase
• ρ > 10−8g cm−3: dissociation of

H2; T ∼cst

• T ∼ 2000 K, cst
• γ = 1.1: thermal support is weak;

unstable γ < 4/3

• collapse starts again, until all H2 is

dissociated, γ > 4/3

• second Larson core: birth of the

protostar

• m2 ∼ 3(30)g, r2 ∼ 9(10)cm,

−3

9 Star formation 9.6. Current view of star formation

9– Star formation

Introduction Collapse of spherical gaseous configurations Stability of isothermal self-gravitating spheres Numerical simulations Formation of the protostar Current view of star formation Initial mass function Open questions

9 Star formation 9.6. Current view of star formation

56 The star formation rate: ˙ M⋆

Krumholz (2014)

Depletion timescale (tdep = M/ ˙ M⋆) larger than 1 Gyr

9 Star formation 9.6. Current view of star formation

57 The star formation rate: ˙ M⋆

• A question of timescales...
• Depletion timescale: tdep = M/ ˙

M⋆ ∼ few Gyr

• Free-fall timescale: τff = (3π/32Gρ)1/2
• for atomic gas: τff = 46n−1/2 Myr; ≪ age of the Galaxy
• Free-fall timescale for molecular clouds: τff = 1 − 10 Myr;
• Depletion timescale are much larger than the free-fall time
• Average star formation: few M⊙/yr

What prevented the Galactic gas to convert entirely, and very quickly, into stars ?

9 Star formation 9.6. Current view of star formation

58 Scaling laws

Cloud of initial radius R that contracts: R decreases Thermal support

• Etherm/Egrav ∼ PV /(GM2/R) ∼ ργR4 ∝ R4−3γ
• If γ < 4/3, thermal support decreases wrt gravity

Centrifugal support

• j = R2ω
• Erot/Egrav ∼ MR2ω2/(GM2/R) ∝ 1/R
• Erot/Egrav increases

Magnetic support

2

9 Star formation 9.7. Initial mass function

9– Star formation

Introduction Collapse of spherical gaseous configurations Stability of isothermal self-gravitating spheres Numerical simulations Formation of the protostar Current view of star formation Initial mass function Open questions

9 Star formation 9.7. Initial mass function

60 The initial mass function

• Distribution of stellar masses at birth
• ξ(M)dM = dN ∝ M−αdM: number of stars with mass within

[M : M + dM]

• Convert colors and magnitudes into luminosity: distance
• Convert luminosity into masses: stellar evolution models
• Convert the present-day MF into IMF
• Samples are limited: low-luminosity (not visible), high-mass

(short lifetime); evolutionary model dependent; galactic evolution

• Clusters: some advantage, and some biases
• Historical IMF: E. Salpeter, 1955

ξ(M) = dN/dM ∝ M−2.35

9 Star formation 9.7. Initial mass function

61 The initial mass function

da Rio 2012

• The IMF in the Orion

Nebula Cluster

• ONC: few 103 ⋆, 1-3 Myr
• ld, M 50 M⊙
• log-normal fit
• Conversion from colors to

masses

• Same dataset, different PMS

evolutionary stellar models

9 Star formation 9.7. Initial mass function

62 The initial mass function

Offner 2015

• Peak mass at few 0.1 M⊙:

uncertain

• Disagreement below BD

limit (0.08 M⊙)

• invariant over a broad range
• f environments
• Is it universal ? open

question

• What is the origin of the

IMF ? pre-stellar core mass function

• Link with SF mechanisms:

ejection, fragmentation, accretion

9 Star formation 9.8. Open questions

9– Star formation

Introduction Collapse of spherical gaseous configurations Stability of isothermal self-gravitating spheres Numerical simulations Formation of the protostar Current view of star formation Initial mass function Open questions

9 Star formation 9.8. Open questions

64 Magnetic fields

• B in the ISM: few µG (1G = 10−4 T)
• Magnetic flux conservation: BR2 is constant
• Radius: from 0.1 pc to 7 ×108 m
• Increase of B by 1013
• Sunspot magnetic field ∼ 0.1 T
• Dissipation of B ?

9 Star formation 9.8. Open questions

65 The angular momentum problem

• j = R2ω
• prestellar cores: rotation ω ∼ 2 km s−1/pc
• take 2R = 0.1 pc: v = Rω = 0.1 km s−1
• Equatorial velocity for a TTauri star (R⋆ ≈ 3R⊙) assuming

j = R2ω = Rv is conserved: vf /vi = Ri/Rf = 7 ×104 km s−1

• Escape velocity (0.5mve = GmM⋆/R⋆):

ve =

• 2GM⋆/R⋆ ≈ 300 km s−1
• Observed rotation for TTauri: ≈ 10 km s−1
• Angular momentum must be dissipated from core to TTauri

9 Star formation 9.9. Open questions

66 Bibliography

Alves, J. F., Lada, C. J., & Lada, E. A. 2001, Nature, 409, 159 Dame, T. M., Hartmann, D., & Thaddeus, P. 2001, ApJ, 547, 792 Krumholz, M. R. 2014, MNRAS, 437, 1662 Larson, R. B. 1969, MNRAS, 145, 271