Star Formation and the Pre-Main Sequence

Additional Resource: Theory of Star Formation and Ch. 9 of Notes by Onno Pols.

Day 13 - October, 9, 2025

Agenda:

• Updates/Reminders - HW2 - Due (extended): Oct. 14 (2m)
• Lecture (20m)
• ICA 12 - 4 Groups - Due: end of day. (30m)
• ICA 12 Group Discussion (10m)

Star formation

• Some general caveats: Star-formation is not very well understood compared to stellarevolution. Key uncertainties still lie in the resulting:

  • star formation efficiency: the fraction of gas of the interstellar cloud that is turned into stars,
  • initial mass function: the spectrum and relative probability of stellar masses that are formed.

• Whenever possible, we use observational efforts to constrain this and many other uncertainties in star formation.

Star formation

Observations demonstrate that stars are formed out of molecular clouds, typcially giant molecular clouds:

• masses
• about 10 parsecs in size
• 10 - 100 K
• densitities of 10-300 molecules .
• few percent of dust, causing them to be very opaque to visible
• roughly in HSE with surrounding interstellar medium (ISM)

The Jeans Criterion

• a criterion will be found that can be used to predict under what physical conditions an interstellar cloud can collapse. This criterion is a product of the British scientist Sir James Hopwood Jeans (1877 – 1946).

• Start with a homogeneous cloud of mass and radius in hydrostatic equilibrium and obeys virial theorem ().

The Jeans Criterion

• Stable - Start with a homogeneous cloud of mass and radius in hydrostatic equilibrium and obeys virial theorem ().

• Unstable If at some point ,

  • gravitational energy dominates the thermal energy
  • cloud will be unstable and collapse due to gravitational pull

Recall we can define the gravitational energy for the cloud:

The Jeans Criterion

Assume isothermal and have constant density, the thermal energy is:

where is the total number of particles in the cloud and can be written in terms of the mean molecular weight: .

Collapse can occur when: , or when

The Jeans Criterion

We can further simplify this by expressing in terms of only :

However, recall that the radius is , plugging in,

The Jeans Criterion

• In order for a cloud to collapse, its mass must be larger than Jeans’ mass: .
• Clouds with larger have smaller Jeans' mass, easier to collapse
• Clouds with larger have larger Jeans' mass, harder to collapse

Interstellar cloud collapse

We can describe approximately six stages of Star Formation.

Star formation stars with a perturbation: a nearby shock/collision moving the cloud out of HSE and inducing collapse is the first stage.

The Jeans mass concerns the mass involved in such a perturbation. If not, the filiments will undergo free-fall collapse described by the dynamical timescale, . May also be written as:

Cloud fragmentation

As the density of the collapsing material increases, the Jeans mass decreases.

• The cloud begins to fragment into small pieces, which individually continue to collapse until fragments are .

• This process makes it possible for a large number of stars to form from a single molecular cloud

M16: Pillars of Star Creation - Image Credit: NASA, ESA

Formation of a protostellar core

Looking only at collapsing cloud fragment, as the density continues to increase, the gas becomes opaque to infrared photons.

• Radiation is trapped causing heating and increase in gas pressure that slows collapse and the and the fragment comes into HSE.

• The fragment continues in HSE and quasi-static contraction. A protostar is born!

Herbig-Haro 24 Protostar - Image Credit: NASA, ESA

Accretion

Next phase is dominated by accretion as gas falls onto the protostar. Due to conservation of angular momentum, the accretion leads to the formation of an accretion disk.

• Accretion disks are observed around most young stars, primarily in inrared and submillimeter wavelengths.

The protostar within the dark cloud L1527 is embedded within a cloud of material feeding its growth. - Image Credit: NASA, NIRCam

Accretion

The luminosity of this object during this phase is powered by the accretion:

during which the core continues to heat up adiabitcally ().

The protostar within the dark cloud L1527 is embedded within a cloud of material feeding its growth. - Image Credit: NASA, NIRCam

Dissociation and ionization

As the protostellar core continues to contract the gas behaves like an ideal gas.

• However, at K molecular hydrogen starts to disassociate, and HSE is again lost and dynamical collapse begins again.

• Once is fully disassociated into atomic H, HSE is restored.

• The final result is ionization of the H and He at K, the protostar is back in HSE at a much reduced radius.

Pre-main sequence phase

Eventually, the accretion will slow down and stop. The protostar is now a pre-main sequence star.

It's luminosity is now given by the gravitational contraction and according to the virial theorem follows

• At such low temperatures, the opacity of the of pre-MS star is very high throughout, radiative heat transfer is inefficient and the star is fully convective throughout.

Fully convective stars: the Hayashi line (track)

Fully convective stars of a given mass occupy a nearly vertical line in the HR diagram ( constant). This is the Hayashi Line/Track.

• the region to the right is now as the forbidden region and fully convective stars in HSE cannot ocupy.

• to the left, these stars are not fully convective and some regions must be radiative.

• We can esimate the lifetime on the PMS as

Fully convective stars: the Hayashi line (track)

The position of the Hayashi lines in the H-R diagram.

• The slope is not exactly constant due in part to neglect of ionization zones and superadiabicity regions in the outer layers. Credit: Onno Pols.

• massive protostars reach the ZAMS much earlier than lower-mass stars

Fully convective stars: the Hayashi line (track)

A protostar leaves the Hayashi track

• when it either deviates to the Henyey track

  • characterized by a slow collapse in near hydrostatic equilibrium, approaching th main sequence almost horizontally in the HR diagram (i.e. luminosity remains almost constant)

• nuclear fusion occurs and the star reaches the main-sequence.

• This differences are all reliant of the initial mass of the star!

Fully convective stars: the Hayashi line (track)

HRD from Stahler 1988 showing low mass stellar evolution tracks along the Hayashi and Henyey tracks until reaching the Main Sequence.

• Also shown are the observed locations of a number of T Tauri stars.
  • T Tauri Stars are those that are still contracting towards the main-sequence.

In-Class Assignment 12

In class: Work on ICA here with groups per usual. Discuss conceptual questions together and prepare answers to share at the end of class.

• Choose someone that will report out the groups responses ahead of time!

After Class: Due: End of Day to D2L

Note: ICAs will be shorter with the goal of: reducing focus on coding, increasing time for discussion and interpretation of results / plots in groups and as a class.