mcfacts.physics.feedback

Module for calculating corrections to migration due to feedback models.

mcfacts.physics.feedback.feedback_bh_hankla(disk_bh_pro_orbs_a, disk_surf_density_func, disk_opacity_func, disk_bh_eddington_ratio, disk_alpha_viscosity, disk_radius_outer)

Calculate the ratio of radiative feedback torque to migration torque.

This feedback model uses Eqn. 28 in Hankla, Jiang & Armitage (2020) which yields the ratio of heating torque to migration torque. Heating torque is directed outwards. So, Ratio < 1, slows the inward migration of an object. Ratio > 1 sends the object migrating outwards. The direction & magnitude of migration (effected by feedback) will be executed in type1.py.

Parameters:

  • disk_bh_pro_orbs_a (numpy.ndarray) – Orbital semi-major axes [r_{g,SMBH}] of prograde singleton BH at start of a timestep (math:r_g=GM_{SMBH}/c^2) with float type

  • disk_surf_density_func (function) – Returns AGN gas disk surface density [kg/m^2] given a distance [r_{g,SMBH}] from the SMBH can accept a simple float (constant), but this is deprecated

  • disk_opacity_model (lambda) – Opacity as a function of radius

  • disk_bh_eddington_ratio (float) – Accretion rate of fully embedded stellar mass black hole [Eddington accretion rate]. 1.0=embedded BH accreting at Eddington. Super-Eddington accretion rates are permitted. User chosen input set by input file

  • disk_alpha_viscosity (float) – Disk gas viscocity [units??] alpha parameter

  • disk_radius_outer (float) – Outer radius [r_{g,SMBH}] of the disk

Returns:

ratio_feedback_migration_torque – Ratio of feedback torque to migration torque with float type

Return type:

numpy.ndarray

Notes

The ratio of torque due to heating to Type 1 migration torque is calculated as R = Gamma_heat/Gamma_mig

~ 0.07 (speed of light/ Keplerian vel.)(Eddington ratio)(1/optical depth)(1/alpha)^3/2

where Eddington ratio can be >=1 or <1 as needed, optical depth (tau) = Sigma* kappa alpha = disk viscosity parameter (e.g. alpha = 0.01 in Sirko & Goodman 2003) kappa = 10^0.76 cm^2 g^-1=5.75 cm^2/g = 0.575 m^2/kg for most of Sirko & Goodman disk model (see Fig. 1 & sec 2) but e.g. electron scattering opacity is 0.4 cm^2/g So tau = Sigma*0.575 where Sigma is in kg/m^2. Since v_kep = c/sqrt(a(r_g)) then R ~ 0.07 (a(r_g))^{1/2}(Edd_ratio) (1/tau) (1/alpha)^3/2 So if assume a=10^3r_g, Sigma=7.e6kg/m^2, alpha=0.01, tau=0.575*Sigma (SG03 disk model), Edd_ratio=1, R ~5.5e-4 (a/10^3r_g)^(1/2) (Sigma/7.e6) v.small modification to in-migration at a=10^3r_g

~0.243 (R/10^4r_g)^(1/2) (Sigma/5.e5) comparable. >1 (a/2x10^4r_g)^(1/2)(Sigma/) migration is outward at >=20,000r_g in SG03 >10 (a/7x10^4r_g)^(1/2)(Sigma/) migration outwards starts to runaway in SG03

mcfacts.physics.feedback.feedback_stars_hankla(disk_stars_pro_orbs_a, disk_surf_density_func, disk_opacity_func, disk_stars_eddington_ratio, disk_alpha_viscosity, disk_radius_outer)

Calculate the ratio of radiative feedback torque to migration torque.

This feedback model uses Eqn. 28 in Hankla, Jiang & Armitage (2020) which yields the ratio of heating torque to migration torque. Heating torque is directed outwards. So, Ratio < 1, slows the inward migration of an object. Ratio > 1 sends the object migrating outwards. The direction & magnitude of migration (effected by feedback) will be executed in type1.py.

Parameters:

  • disk_bh_pro_orbs_a (numpy.ndarray) – Orbital semi-major axes [r_{g,SMBH}] of prograde singleton BH at start of a timestep (math:r_g=GM_{SMBH}/c^2) with float type

  • disk_surf_density_func (function) – Returns AGN gas disk surface density [kg/m^2] given a distance [r_{g,SMBH}] from the SMBH can accept a simple float (constant), but this is deprecated

  • disk_opacity_model (lambda) – Opacity as a function of radius

  • disk_bh_eddington_ratio (float) – Accretion rate of fully embedded stellar mass black hole [Eddington accretion rate]. 1.0=embedded BH accreting at Eddington. Super-Eddington accretion rates are permitted. User chosen input set by input file

  • disk_alpha_viscosity (float) – Disk gas viscocity [units??] alpha parameter

  • disk_radius_outer (float) – Outer radius [r_{g,SMBH}] of the disk

Returns:

ratio_feedback_to_mig – Ratio of feedback torque to migration torque with float type

Return type:

numpy.ndarray

Notes

The ratio of torque due to heating to Type 1 migration torque is calculated as R = Gamma_heat/Gamma_mig

~ 0.07 (speed of light/ Keplerian vel.)(Eddington ratio)(1/optical depth)(1/alpha)^3/2

where Eddington ratio can be >=1 or <1 as needed, optical depth (tau) = Sigma* kappa alpha = disk viscosity parameter (e.g. alpha = 0.01 in Sirko & Goodman 2003) kappa = 10^0.76 cm^2 g^-1=5.75 cm^2/g = 0.575 m^2/kg for most of Sirko & Goodman

disk model (see Fig. 1 & sec 2)

but e.g. electron scattering opacity is 0.4 cm^2/g So tau = Sigma*0.575 where Sigma is in kg/m^2. Since v_kep = c/sqrt(a(r_g)) then R ~ 0.07 (a(r_g))^{1/2}(Edd_ratio) (1/tau) (1/alpha)^3/2 So if assume a=10^3r_g, Sigma=7.e6kg/m^2, alpha=0.01, tau=0.575*Sigma (SG03 disk model),

Edd_ratio=1,

R ~5.5e-4 (a/10^3r_g)^(1/2) (Sigma/7.e6) v.small modification to in-migration at a=10^3r_g

~0.243 (R/10^4r_g)^(1/2) (Sigma/5.e5) comparable. >1 (a/2x10^4r_g)^(1/2)(Sigma/) migration is outward at >=20,000r_g in SG03 >10 (a/7x10^4r_g)^(1/2)(Sigma/) migration outwards starts to runaway in SG03