geoffryan / fdlo Goto Github PK

Low tech Fermi-Dirac integrals.

Just a handful of astrophysically relevant Fermi-Dirac integrals computed with direct integration, no fancy asymptotic approximations or fitting formulae.

With default settings, evaluates an integral in about ~ 7 μs. This is slow compared to a fitting formula, but much faster than a pure Python approach with SciPy which takes ~ 450 μs per evaluation with the same desired tolerance. About a 64x speedup!

Note this package only computes the dimensionless Fermi-Dirac integrals. It does not include the various constants (c, m_e, ħ, etc) or the numerical prefactors (1/3, 2π, etc).

Install locally with pip:

$ git clone [email protected]:geoffryan/fdlo
$ cd fdlo
$ pip3 install -e .

fdlo functions can be called with just a temperature θ (in electron masses) and degeneracy η (dimensionless). Additional parameters are the desired relative tolerance rtol (defaults to 10^-12) and integration routine method (defaults to fdlo.gk1021: adaptive quadrature with the Gauss-Kronrod 10-21 rule stencil).

The functions return a tuple (result, error, numEvals):

• result is the desired value, a Fermi-dirac type integral
• error is the (approximate) uncertainty in result
• numEvals is the number of function evaluations used.

Here's an example

>>> import fdlo
>>> theta = 3.0
>>> eta = 0.5
>>> fdlo.P_eminus(theta, eta)
(720.8874231042558, 2.3028377235560364e-10, 273)
>>>

The integral has a value of 720.8874231042558 with an approximate error 2.30e-10 after 273 function evaluations.

Increasing the tolerance will use less function evals at the cost of more error:

>>> import fdlo
>>> fdlo.P_eminus(3, 0.5, rtol=1.0e-5)
(720.8874230739101, 0.001521589698322771, 189)

This only took 189 evaluations, with potentially much larger error (although comparing this to the previous value the apprpximation appears to be an over-estimate in this case)

EOS integrals currently included:

• n_eminus: Number density of electrons.
• n_eplus: Number density of positrons.
• P_eminus: Pressure of electrons.
• P_eplus: Pressure of positrons.
• U_eminus: Energy density of electrons.
• U_eplus: Energy density of positrons.

Neutrino integrals in transparent limit currently included (all use eta_e):

• Ndot_nu_transp: Number flux of neutrinos in transparent limit.
• Ndot_nubar_transp: Number flux of anti-neutrinos in transparent limit.
• F_nu_transp: Energy flux of neutrinos in transparent limit.
• F_nubar_transp: Energy flux of anti-neutrinos in transparent limit.

Neutrino integrals in opaque limit currently included (all use eta_nu):

• n_nu_opaque: Number density of neutrinos in opaque limit.
• n_nubar_opaque: Number density of anti-neutrinos in opaque limit.
• U_nu_opaque: Energy density of neutrinos in opaque limit.
• U_nubar_opaque: Energy density of anti-neutrinos in opaque limit.

Check out examples/