Ast 777: Star and Planet Formation Pre-main sequence stars and the IMF

https://universe-review.ca/F08-star05.htm The main phases of star formation

https://www.americanscientist.org/sites/americanscientist.org/files/2005223144527_306.pdf The star has formed* but is not yet on the MS

* it has attained its final mass (or ~99% of it)

https://www.americanscientist.org/sites/americanscientist.org/files/2005223144527_306.pdf The star is optically visible and is a Class II or III star (or TTS or HAeBe)

https://www.americanscientist.org/sites/americanscientist.org/files/2005223144527_306.pdf PMS evolution is described by the equations of stellar evolution

But there are differences from MS evolution: ‣ initial conditions are very different from the MS (and have significant uncertainties) ‣ energy is produced by gravitational collapse and deuterium burning

(This was mostly figured out 30-50 years ago so the references are old, but important for putting current work in context) But see later… How big is a star when the core stops collapsing?

Upper limit can be determined by assuming that all the gravitational energy of the core is used to dissociate and ionized the H2+He gas (i.e., no radiative losses during collapse)

Each H2 requires 4.48eV to dissociate

—> this produces a short-lived intermediate step along the way to a protostar, called the first hydrostatic core (hinted at, though not definitively confirmed, through FIR/mm observations) How big is a star when the core stops collapsing?

Upper limit can be determined by assuming that all the gravitational energy of the core is used to dissociate and ionized the H2+He gas (i.e., no radiative losses during collapse)

Each H2 requires 4.48eV to dissociate and then 2x13.6eV to ionize Each He requires 75eV to (doubly) ionize 46 => ~3x10 erg per M⊙ How big is a star when the core stops collapsing?

Upper limit can be determined by assuming that all the gravitational energy of the core is used to dissociate and ionized the H2+He gas (i.e., no radiative losses during collapse)

Each H2 requires 4.48eV to dissociate and then 2x13.6eV to ionize Each He requires 75eV to (doubly) ionize 46 => ~3x10 erg per M⊙

The total energy of the core, E = T + V = V/2 (since virialized, 2T+V=0) => E = -GM2/2R

=> dissociation+ionization equals the KE+PE of the core at a radius

This is huge!

and wrong… stars would be far too luminous at their observed Teff The assumption of homologous collapse from molecular core to ionized protostar is incorrect End point of protostellar phase = initial conditions for PMS

Recall the Bonnor-Ebert sphere: ‣ increasing density at center ‣ shorter tff ‣ “inside-out” collapse ‣ creation of a hydrostatic core around an infalling envelope

—> a small central object forms early and grows as the core continues to rain down onto it End point of protostellar phase = initial conditions for PMS

The core is very small but luminous => hot Once central T reaches ~106 K, nuclear fusion begins…

Stahler, Shu & Taam 1980 Deuterium burning

nuclear reaction begins at ~106 K

energy per unit mass of ISM

mass fraction

abundance

yield

equivalent temperature

=> self-sustaining reaction

Stahler 1988 Deuterium burning

Stellar structure calculations give the nuclear burning rate

Strong temperature dependence => self-regulating (“thermostat”)

Stahler 1988 Deuterium burning

Star grows in radius approximately linearly with mass, and quickly forgets about initial conditions.

Note that radius is about an order of magnitude smaller than the first calculation that assumes homologous collapse.

Once infall stops, star becomes optically visible and can be plotted on the HR diagram.

Stahler 1988 Deuterium burning

Radius depends on mass accretion rate.

The abrupt increase in radius at 2-4 M⊙ is due to a change from convective to radiative interiors.

Paola & Stahler 1993 Stellar birthline

Stars have a maximum size and therefore luminosity when they become optically visible. This defines a locus in the HR diagram where we should find Birthline newly formed stars.

Birthline meets ZAMS when tKH < tff Main sequence (recall YSO primer lecture)

Behrend & Maeder 2001 PMS evolutionary tracks

Stars contract as they release energy and their D burningBirthline at 10 central temperatures rise until H fusion — they are on the main sequence! 6K H burning at 4-16x10 Main sequence What is the path that stars take from the stellar birthline to the ZAMS? ?

6 K

Behrend & Maeder 2001 Energy transport

Radiative or convective depending on the temperature gradient (Schwarzchild criterion)

Low mass stars have relatively weak energy production => relatively dense cores & high opacity to radiation => energy is transported by convection

-5 Above ~2.4 M⊙ (for 10 accretion rate), the core is radiative

See Jen’s course for details :) From Rolf Kudritzki “Stellar atmospheres” course http://www.ifa.hawaii.edu/users/kud/teaching/4.Convection.pdf Convection on the solar surface

https://nso.edu/telescopes/dkist/first-light-cropped-field-movie/ Hussain & Aleutian 2014 Hayashi track

Low mass PMS stars with convective cores have almost vertical tracks from birthline to ZAMS

For derivation see Jen’s class else wikipedia https://en.wikipedia.org/wiki/Hayashi_track

Hayashi 1961 Henyey track

Intermediate mass PMS stars with radiative cores have predominantly horizontal tracks from birthline to ZAMS

For derivation see Jen’s class and edit the wikipedia page! https://en.wikipedia.org/wiki/Henyey_track

Henyey et al. 1955 Henyey tracks Hayashi tracks

Hussain & Aleutian 2014 PMS evolution is described by the equations of stellar evolution

(This was mostly figured out 30-50 years ago so the references are old, but important for putting current work in context)

I may have overstated a bit…

“hot or cold accretion” Using PMS tracks to determine masses and ages

0.1 Myr

1 Myr

ZAMS Models show systematic differences: masses are reasonably well determined (to ~20%) but ages are uncertain by ~0.5 Myr and unreliable for 10 Myr < 1 Myr.

Differences come from the 0.1 Myr poor knowledge of the 1 Myr accretion and treatment of convection.

BUT… all agree that there is an age spread within the cluster and that there are many more low mass than high mass stars. 10 Myr

ZAMS

Hillenbrand 1997 Using PMS tracks to determine masses and ages

PMS tracks show Class II YSO lifetime of ~ 2 Myr

Andrews et al. 2013 Brown dwarfs (a brief aside)

Below 0.08 M⊙, the central temperature does not get high enough to start H-fusion; such objects never reach the main sequence.

Corresponds to spectral types M6.5 and later (Teff < 2200 K)

Lower mass / older objects extend into new spectral classes, L, T, Y

BDs are fully convective (and therefore mixed); will burn through all their deuterium, and then slowly cool and fade over Gyr. (Thus their spectral classification is a function of mass and age.)

Their atmospheres are molecular rich & cloudy: a useful proxy for giant exoplanet atmospheres (—> Mike Liu) Luminosity evolution

Stars Hydrogen burning limit

Deuterium burning limitBrown Dwarfs

Planets

Burrows et al. 1997 4000 Teff evolution

211 MJ

3000

(K) 2000 eff T 13 MJ 73 MJ

1000 1 MJ

0.3 MJ

106 107 108 109 1010 Age (yr)

http://irtfweb.ifa.hawaii.edu/~cushing/resources.html Condensation of molecules in cool atmospheres

4 Na K HD 76151 (G2 V) H2O

3 H2O vB10 (M8 V)

CH4 CH4 m) + Constant µ 2

CH CH

(1.07 4 4 λ

/ f Gl 229B λ f 1

Jupiter

1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4 Wavelength (µm) http://irtfweb.ifa.hawaii.edu/~cushing/resources.html The initial mass function

Cluster observations of L-Teff => ΔN in mass bins ΔM

=> initial mass function, dN/dM often plotted in dimensionless form, dN/dlnM

Stars are born over a range of masses, M★ ~ 0.1-100 M⊙ Median M★ ~ 0.5 M⊙ High mass stars are rare; ~ 100 M⊙ in low mass stars for every 10 M⊙ star Low mass stars are rare; the mass function turns over below ~ 0.2 M⊙ (SpT M4)

For M★ > 1 M⊙ IMF has a power law form

“Salpeter IMF”

At low masses, dN/dlnM is approximately Gaussian, i.e., IMF is “log-normal" The initial mass function

Most stars (by number and total mass) lie within a factor of 3 of 0.5 M⊙

Williams The initial mass function

log-Normal is expected if there are many essentially independent random variables that each multiplicatively affect M lower limit to stellar mass set by ability of cloud fragments to radiate

upper limit to stellar mass may be determined by feedback or stellar structure

Williams Lower mass limit

Gravitational PE loss is bounded by BB emission rate

You should verify this yourselves!

Depending on temperature and efficiency of radiative loss, ε, this gives a range for the lowest stellar mass, M★ ~ 0.02-0.2 M⊙ Universality

The IMF is remarkably invariant, at least for M★ > 0.2 M⊙

=> environment is not important

Many theories, but no agreement yet, on why Γ=1.35 and what sets the stellar mass scale: this is probably the biggest question in star formation…

…and the subject of the asynchronous lecture

Bastian, Covey, & Meyer 2010 Further reading / viewing

Krumholz Chapter 12/13

and/or

Offner et al. 2014, Protostars and Planets VI https://ui.adsabs.harvard.edu/abs/2014prpl.conf...53O/abstract

with associated talk https://www.youtube.com/watch?v=Wd6Pq6Ymc_c Asynchronous lecture

Excerpt from Krumholz Chapter 12

See class website for pdf with questions in blue boxes