MESA Common Envelope ejection

Stability

Current implemented stability criteria are:

  • Mdot: lg_mstar_dot_1 > value

  • delta: mass_transfer_delta > value

  • J_div_Jdot_div_P: 10**log10_J_div_Jdot_div_P < value

  • M_div_Mdot_div_P: 10**log10_M_div_Mdot_div_P < value

  • R_div_SMA: star_1_radius / binary_separation > value

An up to date list of all stability criteria can be obtained with:

from nnaps.mesa.common_envelope import STABILITY_CRITERIA
print(STABILITY_CRITERIA)

If you want to check a model against one of those stability criteria you can use the is_stable() function. For example the code below will ready a compressed mesa model and check if it is stable using the criterion that a model is unstable if J_div_Jdot_div_P < 10:

from nnaps.mesa import fileio, common_envelope

history, _ = fileio.read_compressed_track(<path_to_model.h5>)
stable = common_envelope.is_stable(history, criterion='J_div_Jdot_div_P', value=10)

if stable:
    print('The model is stable')
else:
    print('The model is unstable')

Common Envelope

The following formalisms are defined in NNaPS:

Name

Function

Parameters

Profile Integration

Iben & Tutukov 1984

iben_tutukov1984()

\(\alpha_{\rm CE}\)

No

Webbink 1984

webbink1984()

\(\alpha_{\rm CE}\), \(\lambda{\rm CE}\)

No

Demarco et al 2011

demarco2011()

\(\alpha_{\rm CE}\), \(\lambda_{\rm CE}\)

No

Dewi & Tauris 2000

dewi_tauris2000()

\(\alpha_{\rm CE}\), \(\alpha_{\rm TH}\)

Yes

An up to date list of all recognized CE formalisms including possible experimental formalisms can also be obtained with:

from nnaps.mesa.common_envelope import CE_FORMALISMS
print(CE_FORMALISMS)

Details of the different formalisms, which parameters are required from MESA for them to work, and which parameters can be varied are given below.

Iben & Tutukov 1984

The CE ejection formalism of Iben & Tutukov 1984, ApJ, 284, 719, one of the earlier CE formalisms using only the alpha parameter. The separation after the CE phase is calculated as follows:

\[a_{\rm final} = \alpha \frac{M_{\rm core} \cdot M_2}{M_1^2} a_{\rm init}\]

Where \(M_1\) and \(M_2\) are the donor and companion mass just before the CE starts, \(M_{\rm core}\) is the core mass of the donor that remains after the CE is ejected. This is assumed the same as the core mass before the CE starts. \(a_{\rm init}\) is the separation just before the CE, \(a_{\rm final}\) is the final separation and \(\alpha\) is the alpha parameter of the formalism.

Required history parameters:

  • star_1_mass

  • star_2_mass

  • he_core_mass

  • binary_separation

Webbink 1984

The CE formalism from Webbink 1984, ApJ, 277, 355, using both the alpha and lambda parameter. The separation after the CE phase is calculated as follows:

\[a_{\rm final} = \frac{a_{\rm init}\ \alpha\ \lambda\ RL_1\ M_{\rm core}\ M_2} {2\ a_{\rm init}\ M_1\ M_{\rm env} + \alpha\ \lambda\ RL_1\ M_1\ M_2}\]

Where \(M_1\) and \(M_2\) are the donor and companion mass just before the CE starts, \(M_{\rm core}\) and \(M_{\rm env}\) are the core mass and envelope mass of the donor just before the CE starts. \(RL_1\) is the Roche lobe radius of the donor at the start of CE. \(a_{\rm init}\) is the separation at the start of CE, \(a_{\rm final}\) is the final separation and \(\alpha\) and \(\gamma\) are the alpha and gamma parameters of this formalism.

Parameters:

  • \(\alpha\): the efficiency parameter for the CE formalism

  • \(\lambda\): the mass distribution factor of the primary envelope: lambda * Rl = the effective mass-weighted mean radius of the envelope at the start of CE.

Required history parameters:

  • star_1_mass

  • star_2_mass

  • he_core_mass

  • binary_separation

  • rl_1

Demarco et al 2011

The CE formalism from De Marco et al. 2011, MNRAS, 411, 2277, using an alpha and lambda parameter. This formalism is a slight variation on the formalisms of Webbink 1984. The separation after the CE phase is calculated as follows:

\[a_{\rm final} = \frac{a_{\rm init}\ \alpha\ \lambda\ RL_1\ M_{\rm core}\ M_2} {M_{\rm env}\ (M_{\rm env} / 2.0 + M_{\rm core})\ a_{\rm init} + \alpha\ \lambda\ RL_1\ M_1\ M_2}\]

Where \(M_1\) and \(M_2\) are the donor and companion mass just before the CE starts, \(M_{\rm core}\) and \(M_{\rm env}\) are the core mass and envelope mass of the donor just before the CE starts. \(RL_1\) is the Roche lobe radius of the donor at the start of CE. \(a_{\rm init}\) is the separation at the start of CE, \(a_{\rm final}\) is the final separation and \(\alpha\) and \(\gamma\) are the alpha and gamma parameters of this formalism.

Parameters:

  • \(\alpha\): the efficiency parameter for the CE formalism

  • \(\lambda\): the mass distribution factor of the primary envelope: lambda * Rl = the effective mass-weighted mean radius of the envelope at the start of CE.

Required history parameters:

  • star_1_mass

  • star_2_mass

  • he_core_mass

  • binary_separation

  • rl_1

Dewi & Tauris 2000

This CE formalism is presented in Dewi and Tauris 2000, A&A, 360, 1043, based on the idea of obtaining the binding energy by integrating the stellar profile from Han et al 1995, MNRAS, 272, 800

The CE formalism will integrate over the profile provided. For each cell in the profile it will remove the mass in that cell and then calculate the change in separation due to the removal of that mass. The integration will continue until the donor star will stop overfilling it’s Roche lobe or until both stars merge. The change in separation in each cell step is calculated as follows:

\[da = dm \cdot (\frac{G M_1}{R_c} - \alpha_{\rm TH} U + \frac{\alpha_{\rm CE} G M_2}{2 a}) \frac{2 a^2} {\alpha_{\rm CE} G M_1 M_2}\]
Parameters:

  • \(\alpha_{\rm CE}\): the efficiency of ce

  • \(\alpha_{\rm TH}\): the efficiency of binding energy

Required history parameters:

  • star_2_mass

  • binary_separation

Required profile parameters:

  • mass

  • logR

  • logP

  • logRho

If you want to apply the CE formalism yourself on a mesa model you can use the apply_ce() function: