Pre-main sequence evolution
Star formation Lecture Series
28 November 2012
Andrea Stolte
Outline of the lectures
Oct. 10th : Practical details & Introduction
Oct. 17th : Physical processes in the ISM (I): gas + dust radiative processes, solving radiative transfer
Oct. 24th : Physical processes in the ISM (II): thermal balance of the ISM, heating/cooling mechanisms
Oct. 31st : Interstellar chemistry
Nov. 7th : ISM, molecular clouds
Nov. 14th : Equilibrium configuration and collapse
Nov. 21th : Protostars
Nov. 28th : Pre-main sequence evolution
Dec. 5th : Dies Academicus – no class!
Dec. 12th : Discs
Dec. 19st : Planet formation
Jan. 9th : Formation of high-mass stars
Jan. 16th : IMF and star formation on the galactic scale
Jan. 23th : Extragalactic star formation
Jan. 30th : Visit of Effelsberg (?)
Material for today’s lecture
Literature:
Star Formation
Chapter 16
Palla & Stahler 2004, Wiley-VCH (Weinheim)
An Introduction to Star Formation
Ward-Thompson & Whitworth 2011
Chapter 6
Cambridge University Press (Cambridge)
Andrea Stolte
Outline of today’s lecture
Pre-main sequence stars
A brief history & motivation
Discovery of pre-main sequence stars
PMS stars as tracers for star formation
Pre-main sequence evolution
Evolutionary timescales
Derivation of the Kelvin-Helmhotz time
Pre-main sequence evolution of a solar-mass star
The Hayashi track
The Henyey track
Onset of nuclear reactions
Pre-main sequence evolution of low mass stars
Pre-main sequence evolution of high-mass stars
Andrea Stolte
Pre-main sequence stars - why do we care?
Motivation:
• PMS stars are observed in young star clusters,
young associations and the field
we are here right now
Andrea Stolte
Pre-main sequence stars - why do we care?
Motivation:
T Tauri stars are progenitors of the solar system
• T Tauri stars were first observed in 1852 by John Russel Hind
- at the time, T Tau initiated a new class of previously
unknown variable stars, partially associated with nebulae
Image credit: Rector & Schweiker, NOAO
Andrea Stolte
Pre-main sequence stars - why do we care?
Motivation:
T Tauri stars are progenitors of the solar system
• Today, we know T Tauris to be young stars:
- T Tauri spectra show Lithium in absorption
- T Tauris are associated with young groups/clusters/clouds
- classical T Tauri stars have circumstellar discs,
which are believed to be the places of planet formation
- weak-line T Tauri stars (discovered later) host
residual discs, which are the best places to search for
extrasolar planets
Andrea Stolte
Pre-main sequence evolution depends on the stellar mass
• T Tauri stars are solar-type, low-mass stars
0.5-1 < M(T Tauri) < 2-3 Msun
• Herbig Ae/Be stars are their higher-mass counterparts (Herbig 1960)
2-3 < M(HAeBe) < 8 Msun (maybe much more...)
As in T Tau, HAeBe also show disc emission.
Many, but not all, HAeBe are associated with molecular clouds
or young clusters.
Andrea Stolte
What are T Tauri & Herbig Ae/Be stars?
• T Tauri stars are low-mass stars with discs
• Herbig Ae/Be stars are intermediate-mass stars with discs
• in the Hertzsprung-Russel-Diagram (HRD), both are located
above the main sequence, in the region of pre-main sequence evolution
Pre-main sequence:
• main infall & accretion phase has ended
• evolution is driven by the stellar mass
Pre-main sequence stars are in transition
between protostars and the main sequence.
main
sequence
Stars are located above the zero-age main
sequence: compared to their main sequence
counterparts, they have larger radii at lower
temperatures.
They resemble giant stars.
Hernandez et al. 2004
Andrea Stolte
The Birthline - where stars become “visible”
Bochum 6
age: 10 Myr
Herbig Be
star with disc
Pre-main sequence stars
start as stellar objects with
the same radius as a
protostar with the same
mass. The birthline is
defined as the place,
where a star first becomes
visible.
PMS candidate
stars
The pre-main sequence
is the direct continuation
of the protostellar phase.
Mathew et al. 2010
Andrea Stolte
Pre-main sequence stars - why do we care?
Motivation:
• Pre-main sequence stars make up a large fraction of the young stellar
population in the Galaxy, and especially in each young star cluster
Stars with M < 4 Msun are in their PMS phase in all young star clusters.
Image credit: Gendler & Hannahoe / ESO
Andrea Stolte
Outline of today’s lecture
Pre-main sequence stars
A brief history & motivation
Discovery of pre-main sequence stars
PMS stars as tracers for star formation
Pre-main sequence evolution
Evolutionary timescales
Derivation of the Kelvin-Helmhotz time
Pre-main sequence evolution of a solar-mass star
The Hayashi track
The Henyey track
Onset of nuclear reactions
Pre-main sequence evolution of low mass stars
Pre-main sequence evolution of high-mass stars
Andrea Stolte
From protostar to pre-main sequence star
Protostar:
- opaque, hydrostatic object with gaseous envelope
Surface luminosity is given by Stefan's Law:
L *=4 π R2* σ T 4eff
- L* is determined by accretion luminosity:
Lacc = GM* Ṁ
R*
Ṁ : accretion rate
- the protostellar phase ends as infall stops
Pre-main sequence star:
- stellar mass is fixed (for all practical purposes)
Energy sources on the PMS:
- L* is provided by gravitational contraction in the early phase
- nuclear reactions contribute an increasing amount of energy
Andrea Stolte
Pre-main sequence evolution
What is the difference to protostars?
Evolutionary timescales:
Protostar:
1 ⎤1/2 ~ 104 yrs (ρ0 ~ 2 10-18 g cm-3 )
⎣32 Gρ0 ⎦
tff = ⎡3π
Timescale for a star to thermally adjust to contraction:
Pre-main seq star:
Note:
2
tKH ~ G M* ~ 107 yrs (solar-type star)
R* L*
tff ~ 1/ρ ~ V/M ~ R3/M
tKH ~ 1/R
decreases during collapse!
increases during contraction
Andrea Stolte
Derivation of the Kelvin-Helmholtz Timescale
What is the energy source of PMS objects?
Gravitational energy during contraction:
dF = G M(r) dm
r2
is the gravitational force on each mass
element dm (outside of r)
dU = — G M(r) dm
r
is the gravitational potential of the point
mass particle dm
dm = 4π r2 ρ dr
for a shell build up uniformly of point masses
dU = — G M(r) 4π r2ρ dr
r
can be integrated to obtain Ugrav
R
U = — 4π G0 ∫ M(r) r ρ dr
in principle, calculating U requires knowledge of the density profile...
Andrea Stolte
Derivation of the Kelvin-Helmholtz Timescale
What is the energy source of PMS objects?
Gravitational energy during contraction:
here, we use the average density to estimate the gravitational energy
generated by the contraction of a PMS star
ρ ~ <ρ> = 3M*
4π R3
M(r) ~ 4π r3 ρ
3
and
R
U = — 4π G0 ∫ M(r) r ρ dr ~ — 16π2 G <ρ>2R5 = — 3G M2
15
5 R
virial theorem: 1/2 Egrav heats the star, 1/2 can be radiated away
Energy output:
E ~ — 3 G M2
10 R
Andrea Stolte
Derivation of the Kelvin-Helmholtz Timescale
What is the energy source of PMS objects?
Gravitational energy during contraction:
E ~ — 3 G M2
10 R
The Kelvin-Helmhotz timescale is the time during which the luminosity
can be sustained with this energy source.
tKH = ΔEg
L*
where
ΔEg = Einitial - Efinal
tKH = 3 G M*2
10 R* L*
Andrea Stolte
Outline of today’s lecture
Pre-main sequence stars
A brief history & motivation
Discovery of pre-main sequence stars
PMS stars as tracers for star formation
Pre-main sequence evolution
Evolutionary timescales
Derivation of the Kelvin-Helmhotz time
Pre-main sequence evolution (of a solar-mass star)
The Hayashi track
The Henyey track
Onset of nuclear reactions
Pre-main sequence evolution of low mass stars
Pre-main sequence evolution of high-mass stars
Andrea Stolte
Stellar structure equations
Pre-main sequence stars are objects in hydrostatic equilibrium.
Their evolution with time, and their internal structure, can be calculated using
the stellar structure equations. Here, the independent variable that determines
stellar evolution is the mass.
∂r
1
=
∂ M r 4 π r 2ρ
∂ P −G M r
=
4
∂Mr
4πr
in radiatively stable regions of the star:
∂ L intern
∂s
=ϵ−T
∂Mr
∂t
in convectively unstable regions:
−3 κ Lintern
∂T
T
=
∂ M r 256 π 2 σ r 4
∂s
=0
∂Mr
3
Additionally, the ideal gas law and Stefan's Law enter the numerical calculations:
Ry
P= μ ρ T
L *=4 π R2* σ T 4eff
As for protostars, these equations have to be solved with appropritate boundary
conditions. Note that one of the surface conditions requires:
L *= Lintern (M * )
Andrea Stolte
Overview of PMS evolution
log L/Lsun
Henyey track
(radiative)
Hayashi track
(convective)
Zero-age
main sequence
Iben 1965
log Teff
Andrea Stolte
First stage of the PMS: the Hayashi track
Transition from a protostar to a pre-main sequence object
Protostar:
opacity source = dust
κ d ~ a ρd
a: grain size
ρd : dust density
Pre-main sequence star:
T ≥ 2000 K => dust evaporates
opacity source = H- ion
κH- ~ ρ3/2 T9/2
after dust opacity is eliminated, H- and metal ions
provide the major sources of opacity.
The opacity varies strongly with temperature.
Andrea Stolte
First stage of the PMS: the Hayashi track
Reminder: flux, opacity, and optical depth
Gravitational energy is generated in the interior of the pre-main sequence star
=> flux that reaches the surface is constant across sphere:
F = ‒ 4 σSB T3
κH-(ρ,T)
with dτ = - κH- dR
κ-1
dT
dR
=>
τ(r) = ‒ r=R∫
r=R*
κH-(r) dr
optical depth
is the mean free path of a photon.
The optical depth τ is the number of mean free pathes of a photon along the line of sight.
At the surface, this flux is related to the effective temperature T* :
F(R*) ≃ ‒ σSB dT4 ≃ σSB T*4
dτ
(with solution:
T4 (τ) = T*4 (τ +1)
as expected, T increases inwards )
The opacity determines whether radiation can escape from the interior of the
star (radiative interior) or whether energy has to be transported outwards
by rising stellar material (convective interior).
Andrea Stolte
First stage of the PMS: the Hayashi track
Transition from a protostar to a pre-main sequence object
Pre-main sequence star:
T ≥ 2000 K => dust evaporates
opacity source = H- ion
κH- ~ ρ3/2 T9/2
The opacity increases with ρ & T
=> κ becomes very large
=> radiation cannot escape freely from the core
The pre-main sequence stellar core on the Hayashi
track becomes a radiation trap
=> the core on the Hayashi track is convective
Andrea Stolte
∂s
>0
∂Mr
Entropy s
Condition for radiative stability
What is the maximum luminosity that can be carried by radition?
−3 κ Lintern
∂T
T
=
∂ M r 256 π 2 σ r 4
∂ P −G M r
=
4
∂Mr
4πr
3
3 κ Lintern
∂T
T
=
∂ P 64 π σ G M r
As long as the entropy s is increasing outwards,
ds/dMr > 0, the region is stable against convection.
3
At the stability limit:
∂s
=0
∂Mr
64 π σ T 3 G M r ∂ T
L crit =
3κ
∂P
=> s = constant
( )
s =const
At any radius r inside the star, where Lintern < Lcrit, the generated energy
is radiated freely outwards.
At any radius where Lintern > Lcrit, the energy cannot be radiated away,
and the region becomes convectively unstable.
Andrea Stolte
Critical luminosity for convection
For the entire star, we can make the following approximations:
T=
μ GM *
3 Ry R*
∂T
∂P
( )
M*
2
→
s =⟨s ⟩
T
P
3M*
ρ=
3
4 π R*
∂P
GM*
P=∫
dM r =−
8 π R 4*
0 ∂ Mr
As H- in the core is ionised, the opacity is dominated
by bound-free/free-free collisions:
κ∼ρT −7 /2
Inserting into the equation of Lcrit, the dependence on mass and radius is found:
11 / 2
L crit ∼C M *
−1/ 2
R*
where C is a lengthy constant
We know that the Sun is stable against
convection throughout most of its volume,
so approximately Lcrit can be simplified to:
Andrea Stolte
L crit ≈1 L⊙
M*
M⊙
11/ 2
R*
R⊙
−1/ 2
( ) ( )
Critical luminosity for convection: Interpretation
In any star, as soon as the internal luminosity exceeds the critical value for this
mass and radius, convection develops. Condition for convection:
L intern > L crit =1 L⊙
M*
M⊙
11/ 2
R*
R⊙
−1/ 2
( ) ( )
higher mass stars:
Lcrit is higher => it is easier for the star to remain radiatively stable
larger stars (pre-main sequence vs main sequence):
Lcrit is lower => convective instability evolves
=> All low-mass PMS stars start fully convective on the Hayashi track.
Andrea Stolte
When does convection end?
As the temperature in the core of the pre-main sequence star increases,
the opacity drops steeply:
κbf ∼T
−7 /2
but Lcrit increases with decreasing opacity:
3
64 π σ T G M r ∂ T
L crit =
3κ
∂P
( )
∼T
13/ 2
s =const
=> an increasing fraction of Lintern can be carried by radiation
=> PMS stars with sufficient mass develop radiatively stable cores,
as soon as nuclear reactions start in the core.
=> Radiative stability resumes earlier in stars with higher mass.
=> If the mass is too low, T and hence Lcrit do not grow sufficiently
to overcome the convective instability.
Stars with M < 0.5 Msun remain convective throughout their lives.
Andrea Stolte
First stage of the PMS: the Hayashi track
Quasi-static contraction on the Hayashi track:
Using the stellar structure equations, the condition for convection,
and the opacity near the surface of the stellar atmosphere, κH- ~ ρ3/2 T9/2 ,
the effective temperature can be derived:
T* ~ (M*7 R* )1/31
and with Stefan's law:
L* ~ (M*28 R*66 )1/31
re-arranging, these dependencies can be written as:
L* ~ M*-14 T*66
d ln(L*)
d ln(T*)
|~
|
T* ~ M*0.212 L*0.015
66
M*
Temperature stable at
d ln(T*)
d ln(M*)
|~
|
0.212
L*
T ~ 4000 K
Luminosity changes dramatically as the star contracts.
Andrea Stolte
Second stage of the PMS: the Henyey track
Pre-main sequence evolution changes as nuclear reactions ignite.
Qualitatively, the evolution progresses as:
T increases → ionisation increases
→ opacity decreases
⇒ radiative core develops
⇒ luminosity is transported into
the convective envelope
For a star in radiative equilibrium:
L* = 4π R*2 σSB T*4
R continues to shrink:
L up, R down → T* has to increase!
Andrea Stolte
Onset of first nuclear
reactions in the core
Second stage of the PMS: the Henyey track
The star develops close to radiative equilibrium.
For a star in radiative equilibrium:
L* = 4π R*2 σSB T*4 = 4π σSB R* <T>4
<κ>
with mean temperature <T>
mean opacity
<κ>
using the opacity in the interior of the star, κbf ~ ρ2 T ‒7/2 :
T* ~ M*11/8 R*-5/8
d ln(L*)
d ln(T*)
|~
|
M*
4/5
L* ~ M*22/5 T*4/5
d ln(L*)
d ln(M*)
|~
|
22/5
T*
At fixed mass, L increases slightly with increasing T.
At a constant temperature, L increases substantially with M.
=> Stars with higher mass have to evolve at higher L.
Andrea Stolte
Onset of first nuclear
reactions in the core
Outline of today’s lecture
Pre-main sequence stars
A brief history & motivation
Discovery of pre-main sequence stars
PMS stars as tracers for star formation
Pre-main sequence evolution
Evolutionary timescales
Derivation of the Kelvin-Helmhotz time
Pre-main sequence evolution (of a solar-mass star)
The Hayashi track
The Henyey track
Onset of nuclear reactions
Pre-main sequence evolution of low mass stars
Pre-main sequence evolution of high-mass stars
Andrea Stolte
Nuclear reactions in the interior of the PMS star
The first nuclear reactions are ignited while the star is still on
the Hayashi track.
Deuterium burning requires the lowest temperature to ignite:
2
H
+ H →
He + γ
3
( Tcore ~ 106 K)
However: the small fraction of 2H implies: this reaction does not decrease
the collapse rate significantly.
On the Henyey track, the PPI cycle is completed (but not in equilibrium!):
H
+ H →
H + e- + νe
2
( Tcore ~ 107 K)
Massive stars are hotter ⇒ the first steps of the CNO cycle ignite: ( Tcore ~ 3 107 K)
C+p →
12
13
N (+ γ) →
13
C (+ e+ + νe) |+p →
14
N+γ
Nuclear reactions provide additional heating & thermal pressure
=> gravitational collapse slows
=> nuclear reactions increasingly contribute to energy release
Andrea Stolte
Henyey
Hayashi
Henyey
High- and low-mass stars
Andrea Stolte
High-mass stars & very little in solar-type stars
Main seq.
Henyey
Andrea Stolte
NASA/ESA, L. Ricci / ESO
Outline of today’s lecture
Pre-main sequence stars
A brief history & motivation
Discovery of pre-main sequence stars
PMS stars as tracers for star formation
Pre-main sequence evolution
Evolutionary timescales
Derivation of the Kelvin-Helmhotz time
Pre-main sequence evolution (of a solar-mass star)
The Hayashi track
The Henyey track
Onset of nuclear reactions
Pre-main sequence evolution of low mass stars
Pre-main sequence evolution of high-mass stars
Andrea Stolte
PMS evolution: The case of a very low-mass star
Equilibration to the main sequence for M ≲ 0.5 Msun
• central Temperature does not reach 3 x 107 K
⇒ nuclear reactions develop late & slowly
⇒ in particular: the CNO cycle never ignites
• in stars of higher masses, a steep temperature gradient
decreases the opacity in the stellar interior and allows a
radiative core to develop
radiative
core
• qualitatively, a steep temperature gradient evolves when
nuclear reactions start in solar-type stars
⇒ in low-mass stars, a steep dT/dR does not develop,
and the radiative core defining the Henyey track
cannot form
⇒ the core remains convective at all times!
hydrogen
burning
Low-mass stars stay on the Hayashi track until the pp hydrogen burning
temperature is reached, and the cycle is fully equilibrated.
Andrea Stolte
PMS evolution: The case of a very high-mass star
Equilibration to the main sequence for M > 10 Msun
• central Temperature reaches 3 x 107 K within ~1000 years!
⇒ CNO is ignited early (but far from equilibrium!)
⇒ a steep dT/dR gradient develops, the opacity sinks (core!)
⇒ a radiative core forms very early
• a radiative core means the star already left its Hayashi track
⇒ massive stars are almost instantaneously on the Henyey track!
• opacticy switches to electron scattering
which simply implies:
=> L* ~ M*3
κ~ρ
and
τ = κR* ~ M*/R*2
independent of T
(with
L* = 4πR*2σSBT*4)
PMS stars shrink, R decreases, but M & L stay constant: T has to increase!
High-mass stars evolve horizontally on their Henyey track.
Andrea Stolte
Late stages of the PMS: the PMS/MS transition
Equilibration to the main sequence for M > 10 Msun
• in the PMS to MS transition zone, the CNO cycle becomes
the dominant source of energy for high-mass stars
• on the main sequence, the CNO cycle stays the major energy
contributor
=> strong temperature dependence on energy output
=> convective core re-develops
When approaching the MS, high-mass stars develop - and maintain! a convective core, surrounded by a radiative envelope.
This is exactly the opposite interior structure as in solar-type stars.
Andrea Stolte
Convective vs radiative phases on the PMS
Fully convective:
Radiative core:
Hayashi track:
Henyey track:
all stars
PMS evolution & MS:
M < 0.5 Msun
all M > 0.5 Msun
PMS transition & MS:
0.5 < M < 1.5 Msun
Convetive core w/
radiative shell:
PMS transition to MS:
M > 1.5 Msun
remains like this on MS
(needs strong CNO cycle!)
Andrea Stolte
Summary
1. Pre-main sequence evolution is driven by changes in temperature and
density, which lead to changes in the dominant opacity contribution.
=> Stars alternate between convective and radiative phases.
2. The exact path of a PMS star in the Hertzsprung-Russel-Diagram,
i.e. its luminosity-temperature evolution, depends on the stellar mass.
Solar-type stars develop radiative cores after the convective Hayashi phase.
High-mass stars re-develop convective cores again during the radiatively-driven
Henyey phase.
Low-mass stars never develop radiative cores -- they remain convective all their lifes.
3. The timescale that drives early PMS evolution is the Kelvin-Helmholtz timescale.
This timescale is very different for stars with different masses.
Solar-type stars spend about 3 x 107 yrs on the pre-main sequence.
High-mass stars with more than 10 Msun less than 105 yrs!
Low-mass stars can require more than 108 yrs to reach hydrogen burning equilibrium.
Andrea Stolte
Next lecture... 12 December!
T Tauri & Herbig Ae/Be stars are young stars with discs.
Next time, we will study the disc properties & disc evolution.
Andrea Stolte