precession.eccentricity: package documentation

eccentricity.affine(vec, low, up)

Perform an affine trasformation to scale a vector given upper and lower limits: rescaled = ( vec - low ) / (up - low). If these are outside the domain of the vector, the result will be in [0,1].

Parameters:

vec: array

Vector in Cartesian coomponents.

low: float

Lower limit.

up: float

Upper limit.

Returns:

rescaled: float, optional

Rescaled vector.

Examples

rescaled = precession.affine(vec,low,up)

eccentricity.angleresonances_endpoint(q, chi1, chi2, chieff)

Small-separation limit of the spin-orbit resonances.

Parameters:

q: float

Mass ratio: 0<=q<=1.

chi1: float

Dimensionless spin of the primary (heavier) black hole: 0<=chi1<=1.

chi2: float

Dimensionless spin of the secondary (lighter) black hole: 0<=chi2<=1.

chieff: float

Effective spin.

Returns:

deltaphiatmax: float

Value of the angle deltaphi at the resonance that maximizes kappa.

deltaphiatmin: float

Value of the angle deltaphi at the resonance that minimizes kappa.

theta1atmax: float

Value of the angle theta1 at the resonance that maximizes kappa.

theta1atmin: float

Value of the angle theta1 at the resonance that minimizes kappa.

theta2atmax: float

Value of the angle theta2 at the resonance that maximizes kappa.

theta2atmin: float

Value of the angle theta2 at the resonance that minimizes kappa.

Examples

theta1atmin,theta2atmin,deltaphiatmin,theta1atmax,theta2atmax,deltaphiatmax = precession.angleresonances_endpoint(q,chi1,chi2,chieff)

eccentricity.angles_to_Jframe(theta1, theta2, deltaphi, a, e, q, chi1, chi2)

Convert the angles (theta1,theta2,deltaphi) to angular momentum vectors (L,S1,S2) in the frame aligned with the total angular momentum. In particular, we set Jx=Jy=Ly=0.

Parameters:

theta1: float

Angle between orbital angular momentum and primary spin.

theta2: float

Angle between orbital angular momentum and secondary spin.

deltaphi: float

Angle between the projections of the two spins onto the orbital plane.

a: float

Semi-major axis.

e: float

Eccentricity: 0<=e<1

q: float

Mass ratio: 0<=q<=1.

chi1: float

Dimensionless spin of the primary (heavier) black hole: 0<=chi1<=1.

chi2: float

Dimensionless spin of the secondary (lighter) black hole: 0<=chi2<=1.

Returns:

Lvec: array

Cartesian vector of the orbital angular momentum.

S1vec: array

Cartesian vector of the primary spin.

S2vec: array

Cartesian vector of the secondary spin.

Examples

Lvec,S1vec,S2vec = precession.angles_to_Jframe(theta1,theta2,deltaphi,a,e,q,chi1,chi2)

eccentricity.angles_to_Lframe(theta1, theta2, deltaphi, a, e, q, chi1, chi2)

Convert the angles (theta1,theta2,deltaphi) to angular momentum vectors (L,S1,S2) in the frame aligned with the orbital angular momentum. In particular, we set Lx=Ly=S1y=0.

Parameters:

theta1: float

Angle between orbital angular momentum and primary spin.

theta2: float

Angle between orbital angular momentum and secondary spin.

deltaphi: float

Angle between the projections of the two spins onto the orbital plane.

a: float

Semi-major axis.

e: float

Eccentricity: 0<=e<1

q: float

Mass ratio: 0<=q<=1.

chi1: float

Dimensionless spin of the primary (heavier) black hole: 0<=chi1<=1.

chi2: float

Dimensionless spin of the secondary (lighter) black hole: 0<=chi2<=1.

Returns:

Lvec: array

Cartesian vector of the orbital angular momentum.

S1vec: array

Cartesian vector of the primary spin.

S2vec: array

Cartesian vector of the secondary spin.

Examples

Lvec,S1vec,S2vec = precession.angles_to_Lframe(theta1,theta2,deltaphi,a,e,q,chi1,chi2)

eccentricity.angles_to_conserved(theta1, theta2, deltaphi, a, e, q, chi1, chi2, full_output=False)

Convert angles (theta1,theta2,deltaphi) into conserved quantities (deltachi,kappa,chieff).

Parameters:

theta1: float

Angle between orbital angular momentum and primary spin.

theta2: float

Angle between orbital angular momentum and secondary spin.

deltaphi: float

Angle between the projections of the two spins onto the orbital plane.

a: float

Semi-major axis.

e: float

Eccentricity: 0<=e<1

q: float

Mass ratio: 0<=q<=1.

chi1: float

Dimensionless spin of the primary (heavier) black hole: 0<=chi1<=1.

chi2: float

Dimensionless spin of the secondary (lighter) black hole: 0<=chi2<=1.

full_output: boolean, optional (default: False)

Return additional outputs.

Returns:

chieff: float

Effective spin.

cyclesign: integer, optional

Sign (either +1 or -1) to cover the two halves of a precesion cycle.

deltachi: float

Weighted spin difference.

kappa: float

Asymptotic angular momentum.

Examples

deltachi,kappa,chieff = precession.angles_to_conserved(theta1,theta2,deltaphi,a,e,q,chi1,chi2) deltachi,kappa,chieff,cyclesign = precession.angles_to_conserved(theta1,theta2,deltaphi,a,e,q,chi1,chi2,full_output=True)

eccentricity.anglesresonances(a, e, chieff, q, chi1, chi2)

Compute the values of the angles corresponding to the two spin-orbit resonances.

Parameters:

a: float

Semi-major axis.

e: float

Eccentricity: 0<=e<1

chieff: float

Effective spin.

q: float

Mass ratio: 0<=q<=1.

chi1: float

Dimensionless spin of the primary (heavier) black hole: 0<=chi1<=1.

chi2: float

Dimensionless spin of the secondary (lighter) black hole: 0<=chi2<=1.

Returns:

deltaphiatmax: float

Value of the angle deltaphi at the resonance that maximizes kappa.

deltaphiatmin: float

Value of the angle deltaphi at the resonance that minimizes kappa.

theta1atmax: float

Value of the angle theta1 at the resonance that maximizes kappa.

theta1atmin: float

Value of the angle theta1 at the resonance that minimizes kappa.

theta2atmax: float

Value of the angle theta2 at the resonance that maximizes kappa.

theta2atmin: float

Value of the angle theta2 at the resonance that minimizes kappa.

Examples

theta1atmin,theta2atmin,deltaphiatmin,theta1atmax,theta2atmax,deltaphiatmax = precession.anglesresonances(r,chieff,q,chi1,chi2)

eccentricity.chiefflimits(q, chi1, chi2)

Limits on the effective spin based only on its definition.

Parameters:

q: float

Mass ratio: 0<=q<=1.

chi1: float

Dimensionless spin of the primary (heavier) black hole: 0<=chi1<=1.

chi2: float

Dimensionless spin of the secondary (lighter) black hole: 0<=chi2<=1.

Returns:

chieffmax: float

Maximum value of the effective spin.

chieffmin: float

Minimum value of the effective spin.

Examples

chieffmin,chieffmax = precession.chiefflimits(q,chi1,chi2)

eccentricity.chip_terms(theta1, theta2, q, chi1, chi2)

Compute the two terms entering the effective precessing spin chip.

Parameters:

theta1: float

Angle between orbital angular momentum and primary spin.

theta2: float

Angle between orbital angular momentum and secondary spin.

q: float

Mass ratio: 0<=q<=1.

chi1: float

Dimensionless spin of the primary (heavier) black hole: 0<=chi1<=1.

chi2: float

Dimensionless spin of the secondary (lighter) black hole: 0<=chi2<=1.

Returns:

chipterm1: float

Term in effective precessing spin.

chipterm2: float

Term in effective precessing spin.

Examples

chipterm1,chipterm2 = precession.chip_terms(theta1,theta2,q,chi1,chi2)

eccentricity.conserved_to_Jframe(deltachi, kappa, a, e, chieff, q, chi1, chi2, cyclesign=1)

Convert the conserved quanties (deltachi,kappa,chieff) to angular momentum vectors (L,S1,S2) in the frame aligned with the total angular momentum. In particular, we set Jx=Jy=Ly=0.

Parameters:

deltachi: float

Weighted spin difference.

kappa: float

Asymptotic angular momentum.

a: float

Semi-major axis.

e: float

Eccentricity: 0<=e<1

chieff: float

Effective spin.

q: float

Mass ratio: 0<=q<=1.

chi1: float

Dimensionless spin of the primary (heavier) black hole: 0<=chi1<=1.

chi2: float

Dimensionless spin of the secondary (lighter) black hole: 0<=chi2<=1.

cyclesign: integer, optional (default: +1)

Sign (either +1 or -1) to cover the two halves of a precesion cycle.

Returns:

Lvec: array

Cartesian vector of the orbital angular momentum.

S1vec: array

Cartesian vector of the primary spin.

S2vec: array

Cartesian vector of the secondary spin.

Examples

Lvec,S1vec,S2vec = precession.conserved_to_Jframe(deltachi,kappa,a,e,chieff,q,chi1,chi2,cyclesign=+1)

eccentricity.conserved_to_Lframe(deltachi, kappa, a, e, chieff, q, chi1, chi2, cyclesign=1)

Convert the conserved quanties (deltachi,kappa,chieff) to angular momentum vectors (L,S1,S2) in the frame aligned with the orbital angular momentum. In particular, we set Lx=Ly=S1y=0.

Parameters:

deltachi: float

Weighted spin difference.

kappa: float

Asymptotic angular momentum.

a: float

Semi-major axis.

e: float

Eccentricity: 0<=e<1

chieff: float

Effective spin.

q: float

Mass ratio: 0<=q<=1.

chi1: float

Dimensionless spin of the primary (heavier) black hole: 0<=chi1<=1.

chi2: float

Dimensionless spin of the secondary (lighter) black hole: 0<=chi2<=1.

cyclesign: integer, optional (default: +1)

Sign (either +1 or -1) to cover the two halves of a precesion cycle.

Returns:

Lvec: array

Cartesian vector of the orbital angular momentum.

S1vec: array

Cartesian vector of the primary spin.

S2vec: array

Cartesian vector of the secondary spin.

Examples

Lvec,S1vec,S2vec = precession.conserved_to_Lframe(deltachi,kappa,a,e,chieff,q,chi1,chi2,cyclesign=+1)

eccentricity.conserved_to_angles(deltachi, kappa, a, e, chieff, q, chi1, chi2, cyclesign=1)

Convert conserved quantities (deltachi,kappa,chieff) into angles (theta1,theta2,deltaphi).

Parameters:

deltachi: float

Weighted spin difference.

kappa: float

Asymptotic angular momentum.

a: float

Semi-major axis.

e: float

Eccentricity: 0<=e<1

chieff: float

Effective spin.

q: float

Mass ratio: 0<=q<=1.

chi1: float

Dimensionless spin of the primary (heavier) black hole: 0<=chi1<=1.

chi2: float

Dimensionless spin of the secondary (lighter) black hole: 0<=chi2<=1.

cyclesign: integer, optional (default: +1)

Sign (either +1 or -1) to cover the two halves of a precesion cycle.

Returns:

deltaphi: float

Angle between the projections of the two spins onto the orbital plane.

theta1: float

Angle between orbital angular momentum and primary spin.

theta2: float

Angle between orbital angular momentum and secondary spin.

Examples

theta1,theta2,deltaphi = precession.conserved_to_angles(deltachi,kappa,a,e,chieff,q,chi1,chi2,cyclesign=+1)

eccentricity.dchidt2_RHS(deltachi, kappa, a, e, chieff, q, chi1, chi2, precomputedroots=None, donotnormalize=False)

Right-hand side of the (ddeltachi/dt)^2 equation. Setting donotnormalize=True is equivalent to mathcalA=1.

Parameters:

deltachi: float

Weighted spin difference.

kappa: float

Asymptotic angular momentum.

a: float

Semi-major axis.

e: float

Eccentricity: 0<=e<1

chieff: float

Effective spin.

q: float

Mass ratio: 0<=q<=1.

chi1: float

Dimensionless spin of the primary (heavier) black hole: 0<=chi1<=1.

chi2: float

Dimensionless spin of the secondary (lighter) black hole: 0<=chi2<=1.

precomputedroots: array, optional (default: None)

Pre-computed output of deltachiroots for computational efficiency.

donotnormalize: boolean, optional (default: False)

If True omit the numerical prefactor.

Returns:

dchidt2: float

Squared time derivative of the weighted spin difference.

Examples

dchidt2 = precession.dchidt2_RHS(r,chieff,q)

eccentricity.ddchidt_prefactor(a, e, chieff, q)

etamerical prefactor to the ddeltachi/dt derivative.

Parameters:

a: float

Semi-major axis.

e: float

Eccentricity: 0<=e<1.

chieff: float

Effective spin.

q: float

Mass ratio: 0<=q<=1.

Returns:

mathcalA: float

Prefactor in the ddeltachi/dt equation.

Examples

mathcalA = precession.ddchidt_prefactor(a,e,chieff,q)

eccentricity.deltachicubic_coefficients(kappa, u, chieff, q, chi1, chi2)

Coefficients of the cubic equation in deltachi for its time evolution.

Parameters:

kappa: float

Asymptotic angular momentum.

u: float

Compactified separation 1/(2L).

chieff: float

Effective spin.

q: float

Mass ratio: 0<=q<=1.

chi1: float

Dimensionless spin of the primary (heavier) black hole: 0<=chi1<=1.

chi2: float

Dimensionless spin of the secondary (lighter) black hole: 0<=chi2<=1.

Returns:

coeff0: float

Coefficient to the x^0 term in polynomial.

coeff1: float

Coefficient to the x^1 term in polynomial.

coeff2: float

Coefficient to the x^2 term in polynomial.

coeff3: float

Coefficient to the x^3 term in polynomial.

Examples

coeff3,coeff2,coeff1,coeff0 = precession.deltachicubic_coefficients(kappa,u,chieff,q,chi1,chi2)

eccentricity.deltachicubic_rescaled_coefficients(kappa, u, chieff, q, chi1, chi2, precomputedcoefficients=None)

Rescaled coefficients of the cubic equation in deltachi for its time evolution. This is necessary to avoid dividing by (1-q).

Parameters:

kappa: float

Asymptotic angular momentum.

u: float

Compactified separation 1/(2L).

chieff: float

Effective spin.

q: float

Mass ratio: 0<=q<=1.

chi1: float

Dimensionless spin of the primary (heavier) black hole: 0<=chi1<=1.

chi2: float

Dimensionless spin of the secondary (lighter) black hole: 0<=chi2<=1.

precomputedcoefficients: array, optional (default: None)

Pre-computed output of deltachicubic_coefficients for computational efficiency.

Returns:

coeff0: float

Coefficient to the x^0 term in polynomial.

coeff1: float

Coefficient to the x^1 term in polynomial.

coeff2: float

Coefficient to the x^2 term in polynomial.

coeff3: float

Coefficient to the x^3 term in polynomial.

Examples

coeff3,coeff2,coeff1,coeff0 = precession.deltachicubic_rescaled_coefficients(kappa,u,chieff,q,chi1,chi2) coeff3,coeff2,coeff1,coeff0 = precession.deltachicubic_rescaled_coefficients(kappa,u,chieff,q,chi1,chi2,precomputedcoefficients=coeffs)

eccentricity.deltachilimits(kappa=None, a=None, e=None, chieff=None, q=None, chi1=None, chi2=None, precomputedroots=None)

Limits on the weighted spin difference. The contraints considered depend on the inputs provided.

• If q, chi1, chi2 are provided, returns the geometrical limits.

• If chieff, q, chi1, and chi2 are provided, returns the joint ‘rectagle’ limits.

• If kappa, r, chieff, q, chi1, and chi2 are provided, returns the solution of the cubic that sets the precession-timescale evolution.

Parameters:

kappa: float, optional (default: None)

Asymptotic angular momentum.

a: float, optional (default: None)

Semi-major axis.

e: float, optional (default: None)

Eccentricity: 0<=e<1

chieff: float, optional (default: None)

Effective spin.

q: float, optional (default: None)

Mass ratio: 0<=q<=1.

chi1: float, optional (default: None)

Dimensionless spin of the primary (heavier) black hole: 0<=chi1<=1.

chi2: float, optional (default: None)

Dimensionless spin of the secondary (lighter) black hole: 0<=chi2<=1.

precomputedroots: array, optional (default: None)

Pre-computed output of deltachiroots for computational efficiency.

Returns:

deltachimax: float

Maximum value of the weighted spin difference.

deltachimin: float

Minimum value of the weighted spin difference.

Examples

deltachimin,deltachimax = precession.deltachilimits(q=q,chi1=chi1,chi2=chi2) deltachimin,deltachimax = precession.deltachilimits(chieff=chieff,q=q,chi1=chi1,chi2=chi2) deltachimin,deltachimax = precession.deltachilimits(kappa=kappa,a=a,e=e,chieff=chieff,q=q,chi1=chi1,chi2=chi2)

eccentricity.deltachilimits_definition(q, chi1, chi2)

Limits on the weighted spin difference based only on its definition.

Parameters:

q: float

Mass ratio: 0<=q<=1.

chi1: float

Dimensionless spin of the primary (heavier) black hole: 0<=chi1<=1.

chi2: float

Dimensionless spin of the secondary (lighter) black hole: 0<=chi2<=1.

Returns:

deltachimax: float

Maximum value of the effective spin chieff.

deltachimin: float

Minimum value of the effective spin chieff.

Examples

deltachimin,deltachimax = precession.deltachilimits_definition(q,chi1,chi2)

eccentricity.deltachilimits_plusminus(kappa, a, e, chieff, q, chi1, chi2, precomputedroots=None)

Limits on the weighted spin difference for given (kappa, r, chieff, q, chi1, chi2). These are two of the solutions of the underlying cubic polynomial.

Parameters:

kappa: float

Asymptotic angular momentum.

a: float

Semi-major axis.

e: float

Eccentricity: 0<=e<1

chieff: float

Effective spin.

q: float

Mass ratio: 0<=q<=1.

chi1: float

Dimensionless spin of the primary (heavier) black hole: 0<=chi1<=1.

chi2: float

Dimensionless spin of the secondary (lighter) black hole: 0<=chi2<=1.

precomputedroots: array, optional (default: None)

Pre-computed output of deltachiroots for computational efficiency.

Returns:

deltachimax: float

Maximum value of the weighted spin difference.

deltachimin: float

Minimum value of the weighted spin difference.

Examples

deltachimin,deltachimax = precession.deltachilimits_plusminus(kappa,a,e,chieff,q,chi1,chi2)

eccentricity.deltachilimits_rectangle(chieff, q, chi1, chi2)

Limits on the weighted spin difference for given (chieff, q, chi1, chi2). This is a rectangle in the (chieff,deltachi) plane.

Parameters:

chieff: float

Effective spin.

q: float

Mass ratio: 0<=q<=1.

chi1: float

Dimensionless spin of the primary (heavier) black hole: 0<=chi1<=1.

chi2: float

Dimensionless spin of the secondary (lighter) black hole: 0<=chi2<=1.

Returns:

deltachimax: float

Maximum value of the weighted spin difference.

deltachimin: float

Minimum value of the weighted spin difference.

Examples

deltachimin,deltachimax = precession.deltachilimits_rectangle(chieff,q,chi1,chi2)

eccentricity.deltachioft(t, kappa, a, e, chieff, q, chi1, chi2, precomputedroots=None)

Evolution of deltachi on the precessional timescale (without radiation reaction). The broadcasting rules for this function are more general than those of the rest of the code. The variable t is allowed to have shapes (N,M) while all the other variables have shape (N,). This is useful to sample M precession configuration for each of the N binaries specified as inputs.

Parameters:

t: float

Time.

kappa: float

Asymptotic angular momentum.

a: float

Semi-major axis.

e: float

Eccentricity: 0<=e<1

chieff: float

Effective spin.

q: float

Mass ratio: 0<=q<=1.

chi1: float

Dimensionless spin of the primary (heavier) black hole: 0<=chi1<=1.

chi2: float

Dimensionless spin of the secondary (lighter) black hole: 0<=chi2<=1.

precomputedroots: array, optional (default: None)

Pre-computed output of deltachiroots for computational efficiency.

Returns:

deltachi: float

Weighted spin difference.

Examples

deltachi = precession.deltachioft(t,kappa,a,e,chieff,q,chi1,chi2)

eccentricity.deltachirescaling(deltachitilde, kappa, a, e, chieff, q, chi1, chi2, precomputedroots=None)

Compute deltachi from the rescaled parameter 0<=deltachitilde<=1.

Parameters:

deltachitilde: float

Rescaled version of the weighted spin difference.

kappa: float

Asymptotic angular momentum.

a: float

Semi-major axis.

e: float

Eccentricity: 0<=e<1

chieff: float

Effective spin.

q: float

Mass ratio: 0<=q<=1.

chi1: float

Dimensionless spin of the primary (heavier) black hole: 0<=chi1<=1.

chi2: float

Dimensionless spin of the secondary (lighter) black hole: 0<=chi2<=1.

precomputedroots: array, optional (default: None)

Pre-computed output of deltachiroots for computational efficiency.

Returns:

deltachi: float

Weighted spin difference.

Examples

deltachi = precession.deltachirescaling(deltachitilde,kappa,a,e,chieff,q,chi1,chi2)

eccentricity.deltachiresonance(kappa=None, a=None, e=None, u=None, chieff=None, q=None, chi1=None, chi2=None)

Assuming that the inputs correspond to a spin-orbit resonance, find the corresponding value of deltachi. There will be two roots that are coincident if not for numerical errors: for concreteness, return the mean of the real part. This function is dangerous: we do not check that the input is a resonance, that’s up to the user (so use at your own risk). Valid inputs are either (kappa,a,e,chieff,q,chi1,chi2) or (kappa,u,chieff,q,chi1,chi2).

Parameters:

kappa: float, optional (default: None)

Asymptotic angular momentum.

a: float, optional (default: None)

Semi-major axis.

e: float, optional (default: None)

Eccentricity: 0<=e<1

u: float, optional (default: None)

Compactified separation 1/(2L).

chieff: float, optional (default: None)

Effective spin.

q: float, optional (default: None)

Mass ratio: 0<=q<=1.

chi1: float, optional (default: None)

Dimensionless spin of the primary (heavier) black hole: 0<=chi1<=1.

chi2: float, optional (default: None)

Dimensionless spin of the secondary (lighter) black hole: 0<=chi2<=1.

Returns:

deltachi: float

Weighted spin difference.

Examples

deltachi = precession.deltachiresonance(kappa=kappa,a=a,e=e,chieff=chieff,q=q,chi1=chi1,chi2=chi2) deltachi = precession.deltachiresonance(kappa=kappa,u=u,chieff=chieff,q=q,chi1=chi1,chi2=chi2)

eccentricity.deltachiroots(kappa, u, chieff, q, chi1, chi2, full_output=True, precomputedroots=None)

Roots of the cubic equation in deltachi that describes the dynamics on the precession timescale.

Parameters:

kappa: float

Asymptotic angular momentum.

u: float

Compactified separation 1/(2L).

chieff: float

Effective spin.

q: float

Mass ratio: 0<=q<=1.

chi1: float

Dimensionless spin of the primary (heavier) black hole: 0<=chi1<=1.

chi2: float

Dimensionless spin of the secondary (lighter) black hole: 0<=chi2<=1.

full_output: boolean, optional (default: True)

Return additional outputs.

precomputedroots: array, optional (default: None)

Pre-computed output of deltachiroots for computational efficiency.

Returns:

deltachi3: float, optional

Spurious root of the deltachi evolution.

deltachiminus: float

Lowest physical root of the deltachi evolution.

deltachiplus: float

Lowest physical root of the deltachi evolution.

Examples

deltachiminus,deltachiplus,deltachi3 = precession.deltachiroots(kappa,u,chieff,q,chi1,chi2) deltachiminus,deltachiplus,deltachi3 = precession.deltachiroots(kappa,u,chieff,q,chi1,chi2,precomputedroots=roots) deltachiminus,deltachiplus = precession.deltachiroots(kappa,u,chieff,q,chi1,chi2,full_output=False)

eccentricity.deltachisampling(kappa, a, e, chieff, q, chi1, chi2, N=1, precomputedroots=None)

Sample N values of deltachi at fixed separation accoring to its PN-weighted distribution function. Can only be used to sample the same number of configuration for each binary. If the inputs have shape (M,) the output will have shape

• (M,N) if M>1 and N>1;

• (M,) if N=1;

• (N,) if M=1.

Parameters:

kappa: float

Asymptotic angular momentum.

a: float

Semi-major axis.

e: float

Eccentricity: 0<=e<1

chieff: float

Effective spin.

q: float

Mass ratio: 0<=q<=1.

chi1: float

Dimensionless spin of the primary (heavier) black hole: 0<=chi1<=1.

chi2: float

Dimensionless spin of the secondary (lighter) black hole: 0<=chi2<=1.

N: integer, optional (default: 1)

Number of samples.

precomputedroots: array, optional (default: None)

Pre-computed output of deltachiroots for computational efficiency.

Returns:

deltachi: float

Weighted spin difference.

Examples

deltachi = precession.deltachisampling(kappa,a,e,chieff,q,chi1,chi2,N=1)

eccentricity.deltachitildeav(m, tol=1e-07)

Precession average of the rescaled weighted spin difference.

Parameters:

m: float

Parameter of elliptic function(s).

tol: float, optional (default: 1e-7)

Numerical tolerance, see source code for details.

Returns:

coeff: float

Coefficient.

Examples

coeff = precession.deltachitildeav(m,tol=1e-7)

eccentricity.deltachitildeav2(m, tol=1e-07)

Precession average of the rescaled weighted spin difference squared.

Parameters:

m: float

Parameter of elliptic function(s).

tol: float, optional (default: 1e-7)

Numerical tolerance, see source code for details.

Returns:

coeff: float

Coefficient.

Examples

coeff = precession.deltachitildeav2(m,tol=1e-7)

eccentricity.dot_nested(x, y)

Dot product between 2D arrays along last axis.

Parameters:

xarray

Input array.

yarray

Input array.

Returns:

zarray

Dot product array.

Examples

z = dot_nested(x, y)

eccentricity.ellippi(n, phi, m)

Incomplete elliptic integral of the third kind. This is reconstructed using scipy’s implementation of Carlson’s R integrals (arxiv:math/9409227).

Parameters:

n: foat

Characheristic of the elliptic integral.

phi: float

Amplitude of the elliptic integral.

m: float

Parameter of the elliptic integral

Returns:

piintegral: float

Incomplete elliptic integral of the third kind

Examples

piintegral = precession.ellippi(n, phi, m)

eccentricity.elliptic_parameter(kappa, u, chieff, q, chi1, chi2, precomputedroots=None)

Parameter m entering elliptic functions for the evolution of deltachi.

Parameters:

kappa: float

Asymptotic angular momentum.

u: float

Compactified separation 1/(2L).

chieff: float

Effective spin.

q: float

Mass ratio: 0<=q<=1.

chi1: float

Dimensionless spin of the primary (heavier) black hole: 0<=chi1<=1.

chi2: float

Dimensionless spin of the secondary (lighter) black hole: 0<=chi2<=1.

precomputedroots: array, optional (default: None)

Pre-computed output of deltachiroots for computational efficiency.

Returns:

m: float

Parameter of elliptic function(s).

Examples

m = precession.elliptic_parameter(kappa,u,chieff,q,chi1,chi2)

eccentricity.eval_J(theta1=None, theta2=None, deltaphi=None, kappa=None, a=None, e=None, q=None, chi1=None, chi2=None)

Magnitude of the total angular momentum. Provide either (theta1,theta2,deltaphi,a,e,q,chi1,chhi2) or (kappa,a,e,q).

Parameters:

theta1: float, optional (default: None)

Angle between orbital angular momentum and primary spin.

theta2: float, optional (default: None)

Angle between orbital angular momentum and secondary spin.

deltaphi: float, optional (default: None)

Angle between the projections of the two spins onto the orbital plane.

kappa: float, optional (default: None)

Asymptotic angular momentum.

a: float, optional (default: None)

Semi-major axis.

e: float, optional (default: None)

Eccentricity: 0<=e<1

q: float, optional (default: None)

Mass ratio: 0<=q<=1.

chi1: float, optional (default: None)

Dimensionless spin of the primary (heavier) black hole: 0<=chi1<=1.

chi2: float, optional (default: None)

Dimensionless spin of the secondary (lighter) black hole: 0<=chi2<=1.

Returns:

J: float

Magnitude of the total angular momentum.

Examples

J = precession.eval_J(theta1=theta1,theta2=theta2,deltaphi=deltaphi,a=a,e=e,q=q,chi1=chi1,chi2=chi2) J = precession.eval_J(kappa=kappa,a=a,e=e,q=q,chi1=chi1,chi2=chi2)

eccentricity.eval_L(a, e, q)

Newtonian angular momentum of the binary.

Parameters:

a: float

Semi-major axis.

e: float

Eccentricity: 0<=e<1

q: float

Mass ratio: 0<=q<=1.

Returns:

L: float

Magnitude of the Newtonian orbital angular momentum.

Examples

L = precession.eval_L(r,q)

eccentricity.eval_OmegaL(deltachi, kappa, a, e, chieff, q, chi1, chi2)

Evaluate the precession frequency OmegaL along the precession cycle.

Parameters:

deltachi: float

Weighted spin difference.

kappa: float

Asymptotic angular momentum.

a: float

Semi-major axis.

e: float

Eccentricity: 0<=e<1

chieff: float

Effective spin.

q: float

Mass ratio: 0<=q<=1.

chi1: float

Dimensionless spin of the primary (heavier) black hole: 0<=chi1<=1.

chi2: float

Dimensionless spin of the secondary (lighter) black hole: 0<=chi2<=1.

Returns:

OmegaL: float

Precession frequency of L about J.

Examples

OmegaL = precession.eval_OmegaL(deltachi,kappa,a,e,chieff,q,chi1,chi2)

eccentricity.eval_S(theta1=None, theta2=None, deltaphi=None, deltachi=None, kappa=None, a=None, e=None, chieff=None, q=None, chi1=None, chi2=None)

Magnitude of the total spin. Valid inputs are either (theta1, theta2, deltaphi, q, chi1, chi2) or (deltachi, kappa, r, chieff, q).

Parameters:

q: float

Mass ratio: 0<=q<=1.

chi1: float

Dimensionless spin of the primary (heavier) black hole: 0<=chi1<=1.

Returns:

S: float

Magnitude of the total spin.

Examples

S = precession.eval_S(theta1=theta1,theta2=theta2,deltaphi=deltaphi,q=q,chi1=chi1,chi2=chi2) S = precession.eval_S(deltachi=deltachi,kappa=kappa,a=a,e=e,chieff=chieff,q=q)

eccentricity.eval_S1(q, chi1)

Spin angular momentum of the heavier black hole.

Parameters:

q: float

Mass ratio: 0<=q<=1.

chi1: float

Dimensionless spin of the primary (heavier) black hole: 0<=chi1<=1.

Returns:

S1: float

Magnitude of the primary spin.

Examples

S1 = precession.eval_S1(q,chi1)

eccentricity.eval_S2(q, chi2)

Spin angular momentum of the lighter black hole.

Parameters:

q: float

Mass ratio: 0<=q<=1.

chi2: float

Dimensionless spin of the secondary (lighter) black hole: 0<=chi2<=1.

Returns:

S2: float

Magnitude of the secondary spin.

Examples

S2 = precession.eval_S2(q,chi2)

eccentricity.eval_a(L=None, u=None, uc=None, e=None, q=None)

Semi-major axis of the binary. Valid inputs are either (L,e,q) or (u,e,q) or (uc,q).

Parameters:

L: float, optional (default: None)

Magnitude of the Newtonian orbital angular momentum.

u: float, optional (default: None)

Compactified separation 1/(2L).

uc: float, optional (default: None)

Circular compactified separation uc=u(e=0).

e: float (default: 0)

Binary eccentricity: 0<=e<1.

q: float, optional (default: None)

Mass ratio: 0<=q<=1.

Returns:

a: float

Binary semi-major axis.

eccentricity.eval_alpha(kappa, a, e, chieff, q, chi1, chi2, precomputedroots=None)

Evaluate the precession angle spanned by L during an entire cycle.

Parameters:

kappa: float

Asymptotic angular momentum.

a: float

Semi-major axis.

e: float

Eccentricity: 0<=e<1

chieff: float

Effective spin.

q: float

Mass ratio: 0<=q<=1.

chi1: float

Dimensionless spin of the primary (heavier) black hole: 0<=chi1<=1.

chi2: float

Dimensionless spin of the secondary (lighter) black hole: 0<=chi2<=1.

precomputedroots: array, optional (default: None)

Pre-computed output of deltachiroots for computational efficiency.

Returns:

alpha: float

Azimuthal angle spanned by L about J during an entire cycle.

Examples

alpha = precession.eval_alpha(kappa,a,e,chieff,q,chi1,chi2)

eccentricity.eval_bracket_omega(kappa, a, e, chieff, q, chi1, chi2, precomputedroots=None)

Precession average of the precession frequency deltachi as it oscillates from deltachi- to deltachi+ back to deltachi-. This of the five phenomenological parameters from arxiv:2103.03894.

Parameters:

kappa: float

Asymptotic angular momentum.

a: float

Semi-major axis.

e: float

Eccentricity: 0<=e<1

chieff: float

Effective spin.

q: float

Mass ratio: 0<=q<=1.

chi1: float

Dimensionless spin of the primary (heavier) black hole: 0<=chi1<=1.

chi2: float

Dimensionless spin of the secondary (lighter) black hole: 0<=chi2<=1.

precomputedroots: array, optional (default: None)

Pre-computed output of deltachiroots for computational efficiency.

Returns:

bracket_omega: float

Precession-averaged precession frequency.

Examples

bracket_omega = precession.eval_bracket_omega(kappa,a,e,chieff,q,chi1,chi2)

eccentricity.eval_bracket_theta(kappa, a, e, chieff, q, chi1, chi2, **kwargs)

Precession average of precession amplitude of deltachi as it oscillates from deltachi- to deltachi+ back to deltachi-. This of the five phenomenological parameters from arxiv:2103.03894. Additional keywords arguments are passed to precession_average.

Parameters:

kappa: float

Asymptotic angular momentum.

a: float

Semi-major axis.

e: float

Eccentricity: 0<=e<1

chieff: float

Effective spin.

q: float

Mass ratio: 0<=q<=1.

chi1: float

Dimensionless spin of the primary (heavier) black hole: 0<=chi1<=1.

chi2: float

Dimensionless spin of the secondary (lighter) black hole: 0<=chi2<=1.

**kwargs: unpacked dictionary, optional

Additional keyword arguments.

Returns:

bracket_theta: float

Precession-averaged precession amplitude.

Examples

bracket_theta = precession.eval_bracket_theta(kappa,a,e,chieff,q,chi1,chi2)

eccentricity.eval_chi1(q, S1)

Dimensionless spin of the heavier black hole.

Parameters:

q: float

Mass ratio: 0<=q<=1.

S1: float

Magnitude of the primary spin.

Returns:

chi1: float

Dimensionless spin of the primary (heavier) black hole: 0<=chi1<=1.

Examples

chi1 = precession.eval_chi1(q,S1)

eccentricity.eval_chi2(q, S2)

Dimensionless spin of the lighter black hole.

Parameters:

q: float

Mass ratio: 0<=q<=1.

S2: float

Magnitude of the secondary spin.

Returns:

chi2: float

Dimensionless spin of the secondary (lighter) black hole: 0<=chi2<=1.

Examples

chi2 = precession.eval_chi2(q,S2)

eccentricity.eval_chieff(theta1, theta2, q, chi1, chi2)

Eftective spin.

Parameters:

theta1: float

Angle between orbital angular momentum and primary spin.

theta2: float

Angle between orbital angular momentum and secondary spin.

q: float

Mass ratio: 0<=q<=1.

chi1: float

Dimensionless spin of the primary (heavier) black hole: 0<=chi1<=1.

chi2: float

Dimensionless spin of the secondary (lighter) black hole: 0<=chi2<=1.

Returns:

chieff: float

Effective spin.

Examples

chieff = precession.eval_chieff(theta1,theta2,q,chi1,chi2)

eccentricity.eval_chip(theta1=None, theta2=None, deltaphi=None, deltachi=None, kappa=None, a=None, e=None, chieff=None, q=None, chi1=None, chi2=None, which='averaged', **kwargs)

Compute the effective precessing spin. The keyword which one of the following definitions:

• heuristic, as in Schmidt et al 2015. Required inputs are either

• generalized, retail all precession-timescale variations,

• averaged (default), averages over precession-timescale variations,

• rms, compute the root-mean-square average, which is faster to evaluate.

The required inputs are either (theta1,theta2,deltaphi,a,e,q,chi1,chi2) or (deltachi,kappa,chieff,a,e,q,chi1,chi2). In the latter case, deltachi can be omitted and will be resampled from its PN-weighted distribution. Some inputs will be ignored for some of the chip definitions (for instance the heuristic chip does not depend on either r or deltaphi), but dummy values are still requested for code consistency. The keywords arguments are only used for the averaged formulation and are passed to precession_average.

Parameters:

theta1: float, optional (default: None)

Angle between orbital angular momentum and primary spin.

theta2: float, optional (default: None)

Angle between orbital angular momentum and secondary spin.

deltaphi: float, optional (default: None)

Angle between the projections of the two spins onto the orbital plane.

deltachi: float, optional (default: None)

Weighted spin difference.

kappa: float, optional (default: None)

Asymptotic angular momentum.

a: float, optional (default: None)

Semi-major axis.

e: float, optional (default: None)

Eccentricity: 0<=e<1

chieff: float, optional (default: None)

Effective spin.

q: float, optional (default: None)

Mass ratio: 0<=q<=1.

chi1: float, optional (default: None)

Dimensionless spin of the primary (heavier) black hole: 0<=chi1<=1.

chi2: float, optional (default: None)

Dimensionless spin of the secondary (lighter) black hole: 0<=chi2<=1.

which: string, optional (default: “averaged”)

Select function behavior.

**kwargs: unpacked dictionary, optional

Additional keyword arguments.

Returns:

chip: float

Effective precessing spin.

Examples

chip = precession.eval_chip(theta1=theta1,theta2=theta2,deltaphi=deltaphi,a=a,e=e,q=q,chi1=chi1,chi2=chi2,which="heuristic") chip = precession.eval_chip(theta1=theta1,theta2=theta2,deltaphi=deltaphi,a=a,e=e,q=q,chi1=chi1,chi2=chi2,which="generalized") chip = precession.eval_chip(theta1=theta1,theta2=theta2,deltaphi=deltaphi,a=a,e=e,q=q,chi1=chi1,chi2=chi2,which="averaged") chip = precession.eval_chip(theta1=theta1,theta2=theta2,deltaphi=deltaphi,a=a,e=e,q=q,chi1=chi1,chi2=chi2,which="rms") chip = precession.eval_chip(deltachi=deltachi,kappa=kappa,a=a,e=e,chieff=chieff,q=q,chi1=chi1,chi2=chi2,which="heuristic") chip = precession.eval_chip(deltachi=deltachi,kappa=kappa,a=a,e=e,chieff=chieff,q=q,chi1=chi1,chi2=chi2,which="generalized") chip = precession.eval_chip(deltachi=deltachi,kappa=kappa,a=a,e=e,chieff=chieff,q=q,chi1=chi1,chi2=chi2,which="averaged") chip = precession.eval_chip(deltachi=deltachi,kappa=kappa,a=a,e=e,chieff=chieff,q=q,chi1=chi1,chi2=chi2,which="rms")

eccentricity.eval_chip_averaged(kappa, a, e, chieff, q, chi1, chi2, **kwargs)

Averaged definition of the effective precessing spin, see arxiv:2011.11948. Additional keywords arguments are passed to precession_average.

Parameters:

kappa: float

Asymptotic angular momentum.

a: float

Semi-major axis.

e: float

Eccentricity: 0<=e<1

chieff: float

Effective spin.

q: float

Mass ratio: 0<=q<=1.

chi1: float

Dimensionless spin of the primary (heavier) black hole: 0<=chi1<=1.

chi2: float

Dimensionless spin of the secondary (lighter) black hole: 0<=chi2<=1.

**kwargs: unpacked dictionary, optional

Additional keyword arguments.

Returns:

chip: float

Effective precessing spin.

Examples

chip = precession.eval_chip_averaged(kappa,a,e,chieff,q,chi1,chi2)

eccentricity.eval_chip_generalized(theta1, theta2, deltaphi, q, chi1, chi2)

Generalized definition of the effective precessing spin chip, see arxiv:2011.11948. This definition retains all variations on the precession timescale.

Parameters:

theta1: float

Angle between orbital angular momentum and primary spin.

theta2: float

Angle between orbital angular momentum and secondary spin.

deltaphi: float

Angle between the projections of the two spins onto the orbital plane.

q: float

Mass ratio: 0<=q<=1.

chi1: float

Dimensionless spin of the primary (heavier) black hole: 0<=chi1<=1.

chi2: float

Dimensionless spin of the secondary (lighter) black hole: 0<=chi2<=1.

Returns:

chip: float

Effective precessing spin.

Examples

chip = precession.eval_chip_generalized(theta1,theta2,deltaphi,q,chi1,chi2)

eccentricity.eval_chip_heuristic(theta1, theta2, q, chi1, chi2)

Heuristic definition of the effective precessing spin chip (Schmidt et al 2015), see arxiv:2011.11948. This definition inconsistently averages over some, but not all, variations on the precession timescale.

Parameters:

theta1: float

Angle between orbital angular momentum and primary spin.

theta2: float

Angle between orbital angular momentum and secondary spin.

q: float

Mass ratio: 0<=q<=1.

chi1: float

Dimensionless spin of the primary (heavier) black hole: 0<=chi1<=1.

chi2: float

Dimensionless spin of the secondary (lighter) black hole: 0<=chi2<=1.

Returns:

chip: float

Effective precessing spin.

Examples

chip = precession.eval_chip_heuristic(theta1,theta2,q,chi1,chi2)

eccentricity.eval_chip_rms(kappa, a, e, chieff, q, chi1, chi2)

Root-mean-square definition of the effective precessing spin. This is much faster to evaluate than the plain average.

Parameters:

kappa: float

Asymptotic angular momentum.

a: float

Semi-major axis.

e: float

Eccentricity: 0<=e<1

chieff: float

Effective spin.

q: float

Mass ratio: 0<=q<=1.

chi1: float

Dimensionless spin of the primary (heavier) black hole: 0<=chi1<=1.

chi2: float

Dimensionless spin of the secondary (lighter) black hole: 0<=chi2<=1.

Returns:

chip: float

Effective precessing spin.

Examples

chip = precession.eval_chip_rms(kappa,a,e,chieff,q,chi1,chi2)

eccentricity.eval_cosdeltaphi(deltachi, kappa, a, e, chieff, q, chi1, chi2)

Cosine of the angle between the projections of the two spins onto the orbital plane.

Parameters:

deltachi: float

Weighted spin difference.

kappa: float

Asymptotic angular momentum.

a: float

Semi-major axis.

e: float

Eccentricity: 0<=e<1

chieff: float

Effective spin.

q: float

Mass ratio: 0<=q<=1.

chi1: float

Dimensionless spin of the primary (heavier) black hole: 0<=chi1<=1.

chi2: float

Dimensionless spin of the secondary (lighter) black hole: 0<=chi2<=1.

Returns:

cosdeltaphi: float

Cosine of the angle between the projections of the two spins onto the orbital plane.

Examples

cosdeltaphi = precession.eval_cosdeltaphi(deltachi,kappa,a,e,chieff,q,chi1,chi2)

eccentricity.eval_costheta1(deltachi, chieff, q, chi1)

Cosine of the angle between the orbital angular momentum and the spin of the primary black hole.

Parameters:

deltachi: float

Weighted spin difference.

chieff: float

Effective spin.

q: float

Mass ratio: 0<=q<=1.

chi1: float

Dimensionless spin of the primary (heavier) black hole: 0<=chi1<=1.

Returns:

costheta1: float

Cosine of the angle between orbital angular momentum and primary spin.

Examples

costheta1 = precession.eval_costheta1(deltachi,chieff,q,chi1)

eccentricity.eval_costheta12(theta1=None, theta2=None, deltaphi=None, deltachi=None, kappa=None, chieff=None, q=None, chi1=None, chi2=None)

Cosine of the angle between the two spins. Valid inputs are either (theta1,theta2,deltaphi) or (deltachi,kappa,chieff,q,chi1,chi2).

Parameters:

theta1: float, optional (default: None)

Angle between orbital angular momentum and primary spin.

theta2: float, optional (default: None)

Angle between orbital angular momentum and secondary spin.

deltaphi: float, optional (default: None)

Angle between the projections of the two spins onto the orbital plane.

deltachi: float, optional (default: None)

Weighted spin difference.

kappa: float, optional (default: None)

Asymptotic angular momentum.

chieff: float, optional (default: None)

Effective spin.

q: float, optional (default: None)

Mass ratio: 0<=q<=1.

chi1: float, optional (default: None)

Dimensionless spin of the primary (heavier) black hole: 0<=chi1<=1.

chi2: float, optional (default: None)

Dimensionless spin of the secondary (lighter) black hole: 0<=chi2<=1.

Returns:

costheta12: float

Cosine of the angle between the two spins.

Examples

costheta12 = precession.eval_costheta12(theta1=theta1,theta2=theta2,deltaphi=deltaphi) costheta12 = precession.eval_costheta12(deltachi=deltachi,kappa=kappa,chieff=chieff,q=q,chi1=chi1,chi2=chi2)

eccentricity.eval_costheta2(deltachi, chieff, q, chi2)

Cosine of the angle between the orbital angular momentum and the spin of the secondary black hole.

Parameters:

deltachi: float

Weighted spin difference.

chieff: float

Effective spin.

q: float

Mass ratio: 0<=q<=1.

chi2: float

Dimensionless spin of the secondary (lighter) black hole: 0<=chi2<=1.

Returns:

costheta2: float

Cosine of the angle between orbital angular momentum and secondary spin.

Examples

costheta2 = precession.eval_costheta2(deltachi,chieff,q,chi2)

eccentricity.eval_costhetaL(deltachi, kappa, a, e, chieff, q)

Cosine of the angle betwen the orbital angular momentum and the total angular momentum.

Parameters:

deltachi: float

Weighted spin difference.

kappa: float

Asymptotic angular momentum.

a: float

Semi-major axis.

e: float

Eccentricity: 0<=e<1

chieff: float

Effective spin.

q: float

Mass ratio: 0<=q<=1.

Returns:

costhetaL: float

Cosine of the angle betwen orbital angular momentum and total angular momentum.

Examples

costhetaL = precession.eval_costhetaL(deltachi,kappa,a,e,chieff,q)

eccentricity.eval_cyclesign(ddeltachidt=None, deltaphi=None, Lvec=None, S1vec=None, S2vec=None)

Evaluate if the input parameters are in the first of the second half of a precession cycle. We refer to this as the ‘sign’ of a precession cycle, defined as +1 if deltachi is increasing and -1 deltachi is decreasing. Valid inputs are one and not more of the following:

• dSdt

• deltaphi

• Lvec, S1vec, S2vec.

Parameters:

ddeltachidt: float, optional (default: None)

Time derivative of the total spin.

deltaphi: float, optional (default: None)

Angle between the projections of the two spins onto the orbital plane.

Lvec: array, optional (default: None)

Cartesian vector of the orbital angular momentum.

S1vec: array, optional (default: None)

Cartesian vector of the primary spin.

S2vec: array, optional (default: None)

Cartesian vector of the secondary spin.

Returns:

cyclesign: integer

Sign (either +1 or -1) to cover the two halves of a precesion cycle.

Examples

cyclesign = precession.eval_cyclesign(ddeltachidt=ddeltachidt) cyclesign = precession.eval_cyclesign(deltaphi=deltaphi) cyclesign = precession.eval_cyclesign(Lvec=Lvec,S1vec=S1vec,S2vec=S2vec)

eccentricity.eval_delta_omega(kappa, a, e, chieff, q, chi1, chi2, precomputedroots=None)

Variation of the precession frequency of deltachi as it oscillates from deltachi- to deltachi+ back to deltachi-. This of the five phenomenological parameters from arxiv:2103.03894.

Parameters:

kappa: float

Asymptotic angular momentum.

a: float

Semi-major axis.

e: float

Eccentricity: 0<=e<1

chieff: float

Effective spin.

q: float

Mass ratio: 0<=q<=1.

chi1: float

Dimensionless spin of the primary (heavier) black hole: 0<=chi1<=1.

chi2: float

Dimensionless spin of the secondary (lighter) black hole: 0<=chi2<=1.

precomputedroots: array, optional (default: None)

Pre-computed output of deltachiroots for computational efficiency.

Returns:

delta_omega: float

Precession frequency variation due to nutation.

Examples

delta_omega = precession.eval_delta_omega(kappa,a,e,chieff,q,chi1,chi2)

eccentricity.eval_delta_theta(kappa, a, e, chieff, q, chi1, chi2, precomputedroots=None)

Nutation amplitude of deltachi as it oscillates from deltachi- to deltachi+ back to deltachi-. This of the five phenomenological parameters from arxiv:2103.03894.

Parameters:

kappa: float

Asymptotic angular momentum.

a: float

Semi-major axis.

e: float

Eccentricity: 0<=e<1

chieff: float

Effective spin.

q: float

Mass ratio: 0<=q<=1.

chi1: float

Dimensionless spin of the primary (heavier) black hole: 0<=chi1<=1.

chi2: float

Dimensionless spin of the secondary (lighter) black hole: 0<=chi2<=1.

precomputedroots: array, optional (default: None)

Pre-computed output of deltachiroots for computational efficiency.

Returns:

delta_theta: float

Nutation amplitude.

Examples

delta_theta = precession.eval_delta_theta(kappa,a,e,chieff,q,chi1,chi2)

eccentricity.eval_deltachi(theta1, theta2, q, chi1, chi2)

Weighted spin difference.

Parameters:

theta1: float

Angle between orbital angular momentum and primary spin.

theta2: float

Angle between orbital angular momentum and secondary spin.

q: float

Mass ratio: 0<=q<=1.

chi1: float

Dimensionless spin of the primary (heavier) black hole: 0<=chi1<=1.

chi2: float

Dimensionless spin of the secondary (lighter) black hole: 0<=chi2<=1.

Returns:

deltachi: float

Weighted spin difference.

Examples

deltachi = precession.eval_deltachi(theta1,theta2,q,chi1,chi2)

eccentricity.eval_deltachiinf(kappa, chieff, q, chi1, chi2)

Large-separation limit of the weighted spin difference.

Parameters:

kappa: float

Asymptotic angular momentum.

chieff: float

Effective spin.

q: float

Mass ratio: 0<=q<=1.

chi1: float

Dimensionless spin of the primary (heavier) black hole: 0<=chi1<=1.

chi2: float

Dimensionless spin of the secondary (lighter) black hole: 0<=chi2<=1.

Returns:

deltachi: float

Weighted spin difference.

Examples

deltachi = precession.eval_deltachiinf(kappa,chieff,q,chi1,chi2)

eccentricity.eval_deltaphi(deltachi, kappa, a, e, chieff, q, chi1, chi2, cyclesign=1)

Angle between the projections of the two spins onto the orbital plane.

Parameters:

deltachi: float

Weighted spin difference.

kappa: float

Asymptotic angular momentum.

a: float

Semi-major axis.

e: float

Eccentricity: 0<=e<1

chieff: float

Effective spin.

q: float

Mass ratio: 0<=q<=1.

chi1: float

Dimensionless spin of the primary (heavier) black hole: 0<=chi1<=1.

chi2: float

Dimensionless spin of the secondary (lighter) black hole: 0<=chi2<=1.

cyclesign: integer, optional (default: 1)

Sign (either +1 or -1) to cover the two halves of a precesion cycle.

Returns:

deltaphi: float

Angle between the projections of the two spins onto the orbital plane.

Examples

deltaphi = precession.eval_deltaphi(deltachi,kappa,a,e,chieff,q,chi1,chi2,cyclesign=1)

eccentricity.eval_e(L=None, u=None, uc=None, a=None, q=None)

Orbital eccentricity of the binary. Valid inputs are either (L,a,q) or (u,uc,q).

Parameters:

L: float, optional (default: None)

Magnitude of the Newtonian orbital angular momentum.

u: float, optional (default: None)

Compactified separation 1/(2L).

uc: float, optional (default: None)

Circular compactified separation uc=u(e=0).

a: float, optional (default: None)

Binary semi-major axis.

q: float, optional (default: None)

Mass ratio: 0<=q<=1.

Returns:

e: float

Binary eccentricity.

eccentricity.eval_eta(q)

Symmetric mass ratio eta = m1*m2/(m1+m2)^2 = q/(1+q)^2.

Parameters:

q: float

Mass ratio: 0<=q<=1.

Returns:

eta: float

Symmetric mass ratio 0<=eta<=1/4.

Examples

eta = precession.eval_eta(q)

eccentricity.eval_kappa(theta1=None, theta2=None, deltaphi=None, J=None, a=None, e=None, q=None, chi1=None, chi2=None)

Asymptotic angular momentum. Provide either (theta1,theta2,deltaphi,a,e,q,chi1,chhi2) or (J,a,e,q).

Parameters:

theta1: float, optional (default: None)

Angle between orbital angular momentum and primary spin.

theta2: float, optional (default: None)

Angle between orbital angular momentum and secondary spin.

deltaphi: float, optional (default: None)

Angle between the projections of the two spins onto the orbital plane.

J: float, optional (default: None)

Magnitude of the total angular momentum.

a: float, optional (default: None)

Semi-major axis.

e: float, optional (default: None)

Eccentricity: 0<=e<1

q: float, optional (default: None)

Mass ratio: 0<=q<=1.

chi1: float, optional (default: None)

Dimensionless spin of the primary (heavier) black hole: 0<=chi1<=1.

chi2: float, optional (default: None)

Dimensionless spin of the secondary (lighter) black hole: 0<=chi2<=1.

Returns:

kappa: float

Asymptotic angular momentum.

Examples

kappa = precession.eval_kappa(theta1=theta1,theta2=theta2,deltaphi=deltaphi,a=a,e=e,q=q,chi1=chi1,chi2=chi2) kappa = precession.eval_kappa(J=J,a=a,e=e,q=q)

eccentricity.eval_m1(q)

Mass of the heavier black hole in units of the total mass.

Parameters:

q: float

Mass ratio: 0<=q<=1.

Returns:

m1: float

Mass of the primary (heavier) black hole.

Examples

m1 = precession.eval_m1(q)

eccentricity.eval_m2(q)

Mass of the lighter black hole in units of the total mass.

Parameters:

q: float

Mass ratio: 0<=q<=1.

Returns:

m2: float

Mass of the secondary (lighter) black hole.

Examples

m2 = precession.eval_m2(q)

eccentricity.eval_nutation_freq(kappa, a, e, chieff, q, chi1, chi2, precomputedroots=None)

Nutation frequency of deltachi as it oscillates from deltachi- to deltachi+ back to deltachi-. This of the five phenomenological parameters from arxiv:2103.03894.

Parameters:

kappa: float

Asymptotic angular momentum.

a: float

Semi-major axis.

e: float

Eccentricity: 0<=e<1

chieff: float

Effective spin.

q: float

Mass ratio: 0<=q<=1.

chi1: float

Dimensionless spin of the primary (heavier) black hole: 0<=chi1<=1.

chi2: float

Dimensionless spin of the secondary (lighter) black hole: 0<=chi2<=1.

precomputedroots: array, optional (default: None)

Pre-computed output of deltachiroots for computational efficiency.

Returns:

littleomega: float

Squared time derivative of the weighted spin difference.

Examples

littleomega = precession.eval_nutation_freq(kappa,a,e,chieff,q,chi1,chi2)

eccentricity.eval_phiL(deltachi, kappa, a, e, chieff, q, chi1, chi2, cyclesign=1, precomputedroots=None)

Evaluate the azimuthal angle of L in a plane orthogonal to J along the precession cycle.

Parameters:

deltachi: float

Weighted spin difference.

kappa: float

Asymptotic angular momentum.

a: float

Semi-major axis.

e: float

Eccentricity: 0<=e<1

chieff: float

Effective spin.

q: float

Mass ratio: 0<=q<=1.

chi1: float

Dimensionless spin of the primary (heavier) black hole: 0<=chi1<=1.

chi2: float

Dimensionless spin of the secondary (lighter) black hole: 0<=chi2<=1.

cyclesign: integer, optional (default: 1)

Sign (either +1 or -1) to cover the two halves of a precesion cycle.

precomputedroots: array, optional (default: None)

Pre-computed output of deltachiroots for computational efficiency.

Returns:

phiL: float

Azimuthal angle spanned by L about J.

Examples

phiL = precession.eval_phiL(deltachi,kappa,a,e,chieff,q,chi1,chi2,cyclesign=1)

eccentricity.eval_q(m1, m2)

Mass ratio, 0 < q = m2/m1 < 1.

Parameters:

m1: float

Mass of the primary (heavier) black hole.

m2: float

Mass of the secondary (lighter) black hole.

Returns:

q: float

Mass ratio: 0<=q<=1.

Examples

q = precession.eval_q(m1,m2)

eccentricity.eval_tau(kappa, a, e, chieff, q, chi1, chi2, precomputedroots=None, return_psiperiod=False, donotnormalize=False)

Evaluate the nutation period. Setting return_psiperiod=True return the phase-period instead of the time period (i.e. returns tau/2K(m) insteaf of tau). This is useful to avoid the evaluation of an elliptic integral when it’s not needed. Setting donotnormalize=True is equivalent to mathcalA=1.

Parameters:

kappa: float

Asymptotic angular momentum.

a: float

Semi-major axis.

e: float

Eccentricity: 0<=e<1

chieff: float

Effective spin.

q: float

Mass ratio: 0<=q<=1.

chi1: float

Dimensionless spin of the primary (heavier) black hole: 0<=chi1<=1.

chi2: float

Dimensionless spin of the secondary (lighter) black hole: 0<=chi2<=1.

precomputedroots: array, optional (default: None)

Pre-computed output of deltachiroots for computational efficiency.

MISSING: COULD NOT BUILD, optional (default: False)

FILL MANUALLY.

donotnormalize: boolean, optional (default: False)

If True omit the numerical prefactor.

Returns:

tau: float

Nutation period.

Examples

tau = precession.eval_tau(kappa,a,e,chieff,q,chi1,chi2) tau = precession.eval_tau(kappa,a,e,chieff,q,chi1,chi2,return_psiperiod=True)

eccentricity.eval_theta1(deltachi, chieff, q, chi1)

Angle between the orbital angular momentum and the spin of the primary black hole.

Parameters:

deltachi: float

Weighted spin difference.

chieff: float

Effective spin.

q: float

Mass ratio: 0<=q<=1.

chi1: float

Dimensionless spin of the primary (heavier) black hole: 0<=chi1<=1.

Returns:

theta1: float

Angle between orbital angular momentum and primary spin.

Examples

theta1 = precession.eval_theta1(deltachi,chieff,q,chi1)

eccentricity.eval_theta12(theta1=None, theta2=None, deltaphi=None, deltachi=None, kappa=None, chieff=None, q=None, chi1=None, chi2=None)

Angle between the two spins. Valid inputs are either (theta1,theta2,deltaphi) or (deltachi,kappa,chieff,q,chi1,chi2).

Parameters:

theta1: float, optional (default: None)

Angle between orbital angular momentum and primary spin.

theta2: float, optional (default: None)

Angle between orbital angular momentum and secondary spin.

deltaphi: float, optional (default: None)

Angle between the projections of the two spins onto the orbital plane.

deltachi: float, optional (default: None)

Weighted spin difference.

kappa: float, optional (default: None)

Asymptotic angular momentum.

chieff: float, optional (default: None)

Effective spin.

q: float, optional (default: None)

Mass ratio: 0<=q<=1.

chi1: float, optional (default: None)

Dimensionless spin of the primary (heavier) black hole: 0<=chi1<=1.

chi2: float, optional (default: None)

Dimensionless spin of the secondary (lighter) black hole: 0<=chi2<=1.

Returns:

theta12: float

Angle between the two spins.

Examples

theta12 = precession.eval_theta12(theta1=theta1,theta2=theta2,deltaphi=deltaphi) theta12 = precession.eval_theta12(deltachi=deltachi,kappa=kappa,chieff=chieff,q=q,chi1=chi1,chi2=chi2)

eccentricity.eval_theta2(deltachi, chieff, q, chi2)

Angle between the orbital angular momentum and the spin of the secondary black hole.

Parameters:

deltachi: float

Weighted spin difference.

chieff: float

Effective spin.

q: float

Mass ratio: 0<=q<=1.

chi2: float

Dimensionless spin of the secondary (lighter) black hole: 0<=chi2<=1.

Returns:

theta2: float

Angle between orbital angular momentum and secondary spin.

Examples

theta2 = precession.eval_theta2(deltachi,chieff,q,chi2)

eccentricity.eval_thetaL(deltachi, kappa, a, e, chieff, q)

Angle betwen the orbital angular momentum and the total angular momentum.

Parameters:

deltachi: float

Weighted spin difference.

kappa: float

Asymptotic angular momentum.

a: float

Semi-major axis.

e: float

Eccentricity: 0<=e<1

chieff: float

Effective spin.

q: float

Mass ratio: 0<=q<=1.

Returns:

thetaL: float

Angle betwen orbital angular momentum and total angular momentum.

Examples

thetaL = precession.eval_thetaL(deltachi,kappa,a,e,chieff,q)

eccentricity.eval_u(a, e, q)

Compactified orbital separation.

Parameters:

a: float

Semi-major axis.

e: float

Eccentricity: 0<=e<1

q: float

Mass ratio: 0<=q<=1.

Returns:

u: float

Compactified separation 1/(2L).

Examples

u = precession.eval_u(r,q)

eccentricity.eval_v(a, e)

Newtonian orbital velocity of the binary.

Parameters:

a: float

Semi-major axis.

e: float

Eccentricity: 0<=e<1

Returns:

v: float

Newtonian orbital velocity.

Examples

v = precession.eval_v(r)

eccentricity.implicit(u, uc)

LHS of eq. (28) in Fumagalli & Gerosa, arXiv:2310.16893. This is used to compute the implicit function u(uc) in the inspiral_precav function. :param u: Compactified separation 1/(2L). :type u: float :param uc: Circular compactified separation 1/(2L(e=0)). :type uc: float

eccentricity.inspiral(*args, which=None, **kwargs)

Generic wrapper for binary inspirals. The keyword which selects between the different approximation:

• If which=precession use precession-averaged inspiral (i.e. run inspiral_precav)

• If which=orbit use orbit-averaged inspiral (i.e. run inspiral_orbav)

• If which=hybrid use hybrid inspiral (i.e. run inspiral_hybrid)

All other aguments and keyword arguments are passed on to the relevant inspiral function; see their docs for details.

Parameters:

**args: unpacked dictionary, optional

Additional arguments.

which: string, optional (default: None)

Select function behavior.

**kwargs: unpacked dictionary, optional

Additional keyword arguments.

Returns:

outputs: dictionary, optional

Set of outputs.

Examples

outputs = precession.inspiral(*args,which='precession',**kwargs) outputs = precession.inspiral(*args,which='orbit',**kwargs) outputs = precession.inspiral(*args,which='hybrid',**kwargs)

eccentricity.inspiral_hybrid(theta1=None, theta2=None, deltaphi=None, deltachi=None, kappa=None, a=None, aswitch=None, e=None, uc=None, ucswitch=None, u=None, chieff=None, q=None, chi1=None, chi2=None, requested_outputs=None, **odeint_kwargs)

Perform hybrid inspirals, i.e. evolve the binary at large separation with a pression-averaged evolution and at small separation with an orbit-averaged evolution, matching the two. The variables q, chi1, and chi2 must always be provided. The integration range must be specified using either a or uc (and not both); provide also ucswitch and aswitch consistently. Either a of uc needs to be arrays with lenght >=1, where e.g. a[0] corresponds to the initial condition and a[1:] corresponds to the location where outputs are returned. It does not work at past time infinity if e !=0. The function is vectorized: evolving N multiple binaries with M outputs requires kappainitial, chieff, q, chi1, chi2 to be of shape (N,) and u of shape (M,N). The initial conditions must be specified in terms of one an only one of the following:

• theta1,theta2, and deltaphi (but note that deltaphi is not necessary if integrating from infinite separation).

• kappa, chieff.

The desired outputs can be specified with a list e.g. requested_outputs=[‘theta1’,’theta2’,’deltaphi’]. All the available variables are returned by default. These are: [‘theta1’, ‘theta2’, ‘deltaphi’, ‘deltachi’, ‘kappa’, ‘a’,’e’, ‘uc’,’u’, ‘deltachiminus’, ‘deltachiplus’, ‘deltachi3’, ‘chieff’, ‘q’, ‘chi1’, ‘chi2’]. The flag enforce allows checking the consistency of the input variables. Additional keywords arguments are passed to scipy.integrate.odeint after some custom-made default settings.

Parameters:

theta1: float, optional (default: None)

Angle between orbital angular momentum and primary spin.

theta2: float, optional (default: None)

Angle between orbital angular momentum and secondary spin.

deltaphi: float, optional (default: None)

Angle between the projections of the two spins onto the orbital plane.

deltachi: float, optional (default: None)

Weighted spin difference.

kappa: float, optional (default: None)

Asymptotic angular momentum.

a: float, optional (default: None)

Binary semi-major axis.

e: float, optional (default: None)

Binary eccentricity: 0<=e<1 .

aswitch: float, optional (default: None)

Matching separation between the precession- and orbit-averaged chunks.

eswitch: float, optional (default: None)

Matching separation between the precession- and orbit-averaged chunks.

u: float, optional (default: None)

Compactified separation 1/(2L).

uc: float, optional (default: None)

Compactified circular separation 1/(2L).

uswitch: float, optional (default: None)

Matching compactified separation between the precession- and orbit-averaged chunks.

chieff: float, optional (default: None)

Effective spin.

q: float, optional (default: None)

Mass ratio: 0<=q<=1.

chi1: float, optional (default: None)

Dimensionless spin of the primary (heavier) black hole: 0<=chi1<=1.

chi2: float, optional (default: None)

Dimensionless spin of the secondary (lighter) black hole: 0<=chi2<=1.

requested_outputs: list, optional (default: None)

Set of outputs.

**odeint_kwargs: unpacked dictionary, optional

Additional keyword arguments.

Returns:

outputs: dictionary

Set of outputs.

Examples

outputs = precession.inspiral_hybrid(theta1=theta1,theta2=theta2,deltaphi=deltaphi,a=a,e=e,q=q,chi1=chi1,chi2=chi2) outputs = precession.inspiral_hybrid(theta1=theta1,theta2=theta2,deltaphi=deltaphi,uc=uc,u=u,q=q,chi1=chi1,chi2=chi2) outputs = precession.inspiral_hybrid(kappa,a=a,e=e,chieff=chieff,q=q,chi1=chi1,chi2=chi2) outputs = precession.inspiral_hybrid(kappa,uc=uc,u=u,chieff=chieff,q=q,chi1=chi1,chi2=chi2)

eccentricity.inspiral_orbav(theta1=None, theta2=None, deltaphi=None, Lh=None, S1h=None, S2h=None, delta_lambda=0, deltachi=None, kappa=None, a=None, e=None, uc=None, u=None, chieff=None, q=None, chi1=None, chi2=None, cyclesign=1, PNorderpre=[0, 0.5], PNorderrad=[0, 1, 1.5, 2, 2.5, 3, 3.5], requested_outputs=None, **odeint_kwargs)

Perform precession-averaged inspirals. The variables q, chi1, and chi2 must always be provided. The integration range must be specified using either (a,e) or (uc,u) (and not both). These need to be arrays with lenght >=1, where e.g. a[0] corresponds to the initial condition and a[1:] corresponds to the location where outputs are returned. The function is vectorized: evolving N multiple binaries with M outputs requires kappainitial, chieff, q, chi1, chi2 to be of shape (N,) and uc of shape (M,N). The initial conditions must be specified in terms of one an only one of the following:

• Lh, S1h, and S2h

• theta1,theta2, and deltaphi.

• deltachi, kappa, chieff, cyclesign.

The desired outputs can be specified with a list e.g. requested_outputs=[‘theta1’,’theta2’,’deltaphi’]. All the available variables are returned by default. These are: [‘theta1’, ‘theta2’, ‘deltaphi’, ‘deltachi’, ‘kappa’, ‘r’, ‘u’, ‘deltachiminus’, ‘deltachiplus’, ‘deltachi3’, ‘chieff’, ‘q’, ‘chi1’, ‘chi2’]. The flag enforce allows checking the consistency of the input variables. Additional keywords arguments are passed to scipy.integrate.odeint after some custom-made default settings.

Parameters:

theta1: float, optional (default: None)

Angle between orbital angular momentum and primary spin.

theta2: float, optional (default: None)

Angle between orbital angular momentum and secondary spin.

deltaphi: float, optional (default: None)

Angle between the projections of the two spins onto the orbital plane.

Lh: array, optional (default: None)

Direction of the orbital angular momentum, unit vector.

S1h: array, optional (default: None)

Direction of the primary spin, unit vector.

S2h: array, optional (default: None)

Direction of the secondary spin, unit vector.

deltachi: float, optional (default: None)

Weighted spin difference.

kappa: float, optional (default: None)

Asymptotic angular momentum.

a: float, optional (default: None)

Binary semi-major axis.

e: float, optional (default: None)

Eccentricity 0<=e<1.

uc: float, optional (default: None)

Circualr compactified separation 1/(2L(e=0)).

u: float, optional (default: None)

Compactified separation 1/(2L).

chieff: float, optional (default: None)

Effective spin.

q: float, optional (default: None)

Mass ratio: 0<=q<=1.

chi1: float, optional (default: None)

Dimensionless spin of the primary (heavier) black hole: 0<=chi1<=1.

chi2: float, optional (default: None)

Dimensionless spin of the secondary (lighter) black hole: 0<=chi2<=1.

cyclesign: integer, optional (default: +1)

Sign (either +1 or -1) to cover the two halves of a precesion cycle.

PNorderpre: array (default: [0,0.5])

PN orders considered in the spin-precession equations.

PNorderrad: array (default: [0,0.5])

PN orders considered in the radiation-reaction equation.

requested_outputs: list, optional (default: None)

Set of outputs.

**odeint_kwargs: unpacked dictionary, optional

Additional keyword arguments.

Returns:

outputs: dictionary

Set of outputs.

Examples

outputs = precession.inspiral_orbav(Lh=Lh,S1h=S1h,S2h=S2h,a=a,e=e,q=q,chi1=chi1,chi2=chi2) outputs = precession.inspiral_orbav(Lh=Lh,S1h=S1h,S2h=S2h,uc=uc,u=u,q=q,chi1=chi1,chi2=chi2) outputs = precession.inspiral_orbav(theta1=theta1,theta2=theta2,deltaphi=deltaphi,a=a,e=e,q=q,chi1=chi1,chi2=chi2) outputs = precession.inspiral_orbav(theta1=theta1,theta2=theta2,deltaphi=deltaphi,uc=uc,u=u,q=q,chi1=chi1,chi2=chi2) outputs = precession.inspiral_orbav(deltachi=deltachi,kappa=kappa,a=a,e=e,chieff=chieff,q=q,chi1=chi1,chi2=chi2)

eccentricity.inspiral_precav(theta1=None, theta2=None, deltaphi=None, deltachi=None, kappa=None, a=None, e=None, u=None, uc=None, chieff=None, q=None, chi1=None, chi2=None, requested_outputs=None, enforce=False, **odeint_kwargs)

The integration range must be specified using either (a,e) or (uc, u), (and not both). These need to be arrays with lenght >=1, where e.g. a[0] corresponds to the initial condition and a[1:] corresponds to the location where outputs are returned. Do not go to past time infinity with e!=0. The function is vectorized: evolving N multiple binaries with M outputs requires kappainitial, chieff, q, chi1, chi2 to be of shape (N,) and u of shape (M,N). The initial conditions must be specified in terms of one an only one of the following:

• theta1,theta2, and deltaphi.

• kappa, chieff.

The desired outputs can be specified with a list e.g. requested_outputs=[‘theta1’,’theta2’,’deltaphi’]. All the available variables are returned by default. These are: [‘theta1’, ‘theta2’, ‘deltaphi’, ‘deltachi’, ‘kappa’, ‘a’, ‘e’, ‘u’, ‘deltachiminus’, ‘deltachiplus’, ‘deltachi3’, ‘chieff’, ‘q’, ‘chi1’, ‘chi2’]. The flag enforce allows checking the consistency of the input variables. Additional keywords arguments are passed to scipy.integrate.odeint after some custom-made default settings.

Parameters:

theta1: float, optional (default: None)

Angle between orbital angular momentum and primary spin.

theta2: float, optional (default: None)

Angle between orbital angular momentum and secondary spin.

deltaphi: float, optional (default: None)

Angle between the projections of the two spins onto the orbital plane.

kappa: float, optional (default: None)

Asymptotic angular momentum.

r: float, optional (default: None)

Binary separation.

u: float, optional (default: None)

Compactified separation 1/(2L).

chieff: float, optional (default: None)

Effective spin.

q: float, optional (default: None)

Mass ratio: 0<=q<=1.

chi1: float, optional (default: None)

Dimensionless spin of the primary (heavier) black hole: 0<=chi1<=1.

chi2: float, optional (default: None)

Dimensionless spin of the secondary (lighter) black hole: 0<=chi2<=1.

requested_outputs: list, optional (default: None)

Set of outputs.

enforce: boolean, optional (default: False)

If True raise errors, if False raise warnings.

**odeint_kwargs: unpacked dictionary, optional

Additional keyword arguments.

Returns:

outputs: dictionary

Set of outputs.

Examples

outputs = precession.inspiral_precav(theta1=theta1,theta2=theta2,deltaphi=deltaphi,a=a,e=e,q=q,chi1=chi1,chi2=chi2) outputs = precession.inspiral_precav(theta1=theta1,theta2=theta2,deltaphi=deltaphi,uc=uc,u=u,q=q,chi1=chi1,chi2=chi2) outputs = precession.inspiral_precav(kappa,a=a,e=e,chieff=chieff,q=q,chi1=chi1,chi2=chi2) outputs = precession.inspiral_precav(kappa,uc=uc,u=u,chieff=chieff,q=q,chi1=chi1,chi2=chi2)

eccentricity.integrator_orbav(Lhinitial, S1hinitial, S2hinitial, delta_lambda, a, e, q, chi1, chi2, PNorderpre=[0, 0.5], PNorderrad=[0, 1, 1.5, 2, 2.5, 3], **odeint_kwargs)

Integration of the systems of ODEs describing orbit-averaged inspirals. Additional keywords arguments are passed to scipy.integrate.odeint after some custom-made default settings.

Parameters:

Lhinitial: array

Initial direction of the orbital angular momentum, unit vector.

S1hinitial: array

Initial direction of the primary spin, unit vector.

S2hinitial: array

Initial direction of the secondary spin, unit vector.

a: float

Bianary semi-major axis.

efloat

Eccentricty: 0<=e<1. .

q: float

Mass ratio: 0<=q<=1.

chi1: float

Dimensionless spin of the primary (heavier) black hole: 0<=chi1<=1.

chi2: float

Dimensionless spin of the secondary (lighter) black hole: 0<=chi2<=1.

PNorderpre: array (default: [0,0.5])

PN orders considered in the spin-precession equations.

PNorderrad: array (default: [0,0.5])

PN orders considered in the radiation-reaction equation.

**odeint_kwargs: unpacked dictionary, optional

Additional keyword arguments.

Returns:

ODEsolution: array

Solution of the ODE.

Examples

ODEsolution = precession.integrator_orbavintegrator_orbav(Lhinitial,S1hinitial,S2hinitial,v,q,chi1,chi2)

eccentricity.integrator_precav(kappainitial, u, chieff, q, chi1, chi2, **odeint_kwargs)

Integration of ODE dkappa/du describing precession-averaged inspirals. Here u needs to be an array with lenght >=1, where u[0] corresponds to the initial condition and u[1:] corresponds to the location where outputs are returned. For past time infinity use u=0. The function is vectorized: evolving N multiple binaries with M outputs requires kappainitial, chieff, q, chi1, chi2 to be of shape (N,) and u of shape (M,N). Additional keywords arguments are passed to scipy.integrate.odeint after some custom-made default settings.

Parameters:

kappainitial: float

Initial value of the regularized momentum kappa.

u: float

Compactified separation 1/(2L).

chieff: float

Effective spin.

q: float

Mass ratio: 0<=q<=1.

chi1: float

Dimensionless spin of the primary (heavier) black hole: 0<=chi1<=1.

chi2: float

Dimensionless spin of the secondary (lighter) black hole: 0<=chi2<=1.

**odeint_kwargs: unpacked dictionary, optional

Additional keyword arguments.

Returns:

kappa: float

Asymptotic angular momentum.

Examples

kappa = precession.integrator_precav(kappainitial,u,chieff,q,chi1,chi2)

eccentricity.intertial_ingredients(kappa, a, e, chieff, q, chi1, chi2)

Numerical prefactors entering the precession frequency.

Parameters:

kappa: float

Asymptotic angular momentum.

a: float

Semi-major axis.

e: float

Eccentricity: 0<=e<1

chieff: float

Effective spin.

q: float

Mass ratio: 0<=q<=1.

chi1: float

Dimensionless spin of the primary (heavier) black hole: 0<=chi1<=1.

chi2: float

Dimensionless spin of the secondary (lighter) black hole: 0<=chi2<=1.

Returns:

bigC0: float

Prefactor in the OmegaL equation.

bigCminus: float

Prefactor in the OmegaL equation.

bigCplus: float

Prefactor in the OmegaL equation.

bigRminus: float

Prefactor in the PhiL equation.

bigRplus: float

Prefactor in the PhiL equation.

Examples

bigC0,bigCplus,bigCminus,bigRplus,bigRminus = precession.intertial_ingredients(kappa,a,e,chieff,q,chi1,chi2)

eccentricity.inverseaffine(rescaled, low, up)

Perform an inferse affine trasformation to scale a vector from some upper and lower limits: vec = low + rescaled*(up-low). If rescaled is defined in [0,1], the result will be defined in [low,up].

Parameters:

rescaled: float

Rescaled vector.

low: float

Lower limit.

up: float

Upper limit.

Returns:

vec: array

Vector in Cartesian coomponents.

Examples

vec = precession.inverseaffine(rescaled,low,up)

eccentricity.ismonotonic(vec, which)

Check if an array is monotonic. The parameter which can takes the following values:

• < check array is strictly increasing.

• <= check array is increasing.

• > check array is strictly decreasing.

• >= check array is decreasing.

Parameters:

vec: array

Input array.

which: string

Select function behavior.

Returns:

check: boolean

Result

Examples

check = ismonotonic(vec, which)

eccentricity.isotropic_angles(N=1)

Sample the angles theta1, theta2, and deltaphi from an isotropic distribution.

Parameters:

N: integer, optional (default: 1)

Number of samples.

Returns:

vec: array

Vector in Cartesian coomponents.

Examples

vec = precession.isotropic_angles(N=1)

eccentricity.kappadiscriminant_coefficients(u, chieff, q, chi1, chi2)

Coefficients of the quintic equation in kappa that defines the spin-orbit resonances.

Parameters:

u: float

Compactified separation 1/(2L).

chieff: float

Effective spin.

q: float

Mass ratio: 0<=q<=1.

chi1: float

Dimensionless spin of the primary (heavier) black hole: 0<=chi1<=1.

chi2: float

Dimensionless spin of the secondary (lighter) black hole: 0<=chi2<=1.

Returns:

coeff0: float

Coefficient to the x^0 term in polynomial.

coeff1: float

Coefficient to the x^1 term in polynomial.

coeff2: float

Coefficient to the x^2 term in polynomial.

coeff3: float

Coefficient to the x^3 term in polynomial.

coeff4: float

Coefficient to the x^4 term in polynomial.

coeff5: float

Coefficient to the x^5 term in polynomial.

Examples

coeff5,coeff4,coeff3,coeff2,coeff1,coeff0 = precession.kappadiscriminant_coefficients(u,chieff,q,chi1,chi2)

eccentricity.kappalimits(a=None, e=None, chieff=None, q=None, chi1=None, chi2=None, enforce=False, **kwargs)

Limits on the asymptotic angular momentum. The contraints considered depend on the inputs provided.

• If r, q, chi1, chi2 are provided, returns the geometrical limits.

• If r, chieff, q, chi1, and chi2 are provided, returns the spin-orbit resonances.

The boolean flag enforce raises an error in case the inputs are not compatible. Additional kwargs are passed to kapparesonances.

Parameters:

a: float, optional (default: None)

Semi-major axis.

e: float, optional (default: None)

Eccentricity: 0<=e<1

chieff: float, optional (default: None)

Effective spin.

q: float, optional (default: None)

Mass ratio: 0<=q<=1.

chi1: float, optional (default: None)

Dimensionless spin of the primary (heavier) black hole: 0<=chi1<=1.

chi2: float, optional (default: None)

Dimensionless spin of the secondary (lighter) black hole: 0<=chi2<=1.

enforce: boolean, optional (default: False)

Perform additional checks.

**kwargs: unpacked dictionary, optional

Additional keyword arguments.

Returns:

kappamax: float

Maximum value of the asymptotic angular momentum kappa.

kappamin: float

Minimum value of the asymptotic angular momentum kappa.

Examples

kappamin,kappamax = precession.kappalimits(a=a,e=e,q=q,chi1=chi1,chi2=chi2) kappamin,kappamax = precession.kappalimits(a=a,e=e,chieff=chieff,q=q,chi1=chi1,chi2=chi2) kappamin,kappamax = precession.kappalimits(a=a,e=e,chieff=chieff,q=q,chi1=chi1,chi2=chi2,enforce=True)

eccentricity.kappalimits_geometrical(a, e, q, chi1, chi2)

Limits in kappa conditioned on r, q, chi1, chi2.

Parameters:

a: float

Semi-major axis.

e: float

Eccentricity: 0<=e<1

q: float

Mass ratio: 0<=q<=1.

chi1: float

Dimensionless spin of the primary (heavier) black hole: 0<=chi1<=1.

chi2: float

Dimensionless spin of the secondary (lighter) black hole: 0<=chi2<=1.

Returns:

kappamax: float

Maximum value of the asymptotic angular momentum kappa.

kappamin: float

Minimum value of the asymptotic angular momentum kappa.

Examples

kappamin,kappamax = precession.kappalimits_geometrical(r,q,chi1,chi2)

eccentricity.kapparescaling(kappatilde, a, e, chieff, q, chi1, chi2)

Compute kappa from the rescaled parameter 0<=kappatilde<=1.

Parameters:

kappatilde: float

Rescaled version of the asymptotic angular momentum.

a: float

Semi-major axis.

e: float

Eccentricity: 0<=e<1

chieff: float

Effective spin.

q: float

Mass ratio: 0<=q<=1.

chi1: float

Dimensionless spin of the primary (heavier) black hole: 0<=chi1<=1.

chi2: float

Dimensionless spin of the secondary (lighter) black hole: 0<=chi2<=1.

Returns:

kappa: float

Asymptotic angular momentum.

Examples

kappa = precession.kapparescaling(kappatilde,a,e,chieff,q,chi1,chi2)

eccentricity.kapparesonances(a, e, chieff, q, chi1, chi2, tol=0.0001)

asymptotic angular momentum of the two spin-orbit resonances. The resonances minimizes and maximizes kappa for given values of r, chieff, q, chi1, chi2. The minimum corresponds to deltaphi=pi and the maximum corresponds to deltaphi=0.

Parameters:

a: float

Semi-major axis.

e: float

Eccentricity: 0<=e<1

chieff: float

Effective spin.

q: float

Mass ratio: 0<=q<=1.

chi1: float

Dimensionless spin of the primary (heavier) black hole: 0<=chi1<=1.

chi2: float

Dimensionless spin of the secondary (lighter) black hole: 0<=chi2<=1.

tol: float, optional (default: 1e-4)

Numerical tolerance, see source code for details.

Returns:

kappamax: float

Maximum value of the asymptotic angular momentum kappa.

kappamin: float

Minimum value of the asymptotic angular momentum kappa.

Examples

kappamin,kappamax = precession.kapparesonances(u,chieff,q,chi1,chi2,tol=1e-4)

eccentricity.morphology(kappa, a, e, chieff, q, chi1, chi2, simpler=False, precomputedroots=None)

Evaluate the spin morphology. Returns:

• L0 for librating about deltaphi=0,

• Lpi for librating about deltaphi=pi

• C- for circulating from deltaphi=pi to deltaphi=0,

• C+ for circulating from deltaphi=0 to deltaphi=pi.

If simpler=True, do not distinguish between the two circulating morphologies and return C for both.

Parameters:

kappa: float

Asymptotic angular momentum.

a: float

Semi-major axis.

e: float

Eccentricity: 0<=e<1

chieff: float

Effective spin.

q: float

Mass ratio: 0<=q<=1.

chi1: float

Dimensionless spin of the primary (heavier) black hole: 0<=chi1<=1.

chi2: float

Dimensionless spin of the secondary (lighter) black hole: 0<=chi2<=1.

simpler: boolean, optional (default: False)

If True simplifies output.

precomputedroots: array, optional (default: None)

Pre-computed output of deltachiroots for computational efficiency.

Returns:

morph: string

Spin morphology.

Examples

morph = precession.morphology(kappa,a,e,chieff,q,chi1,chi2,simpler=False)

eccentricity.norm_nested(x)

Norm of 2D array of shape (N,3) along last axis.

Parameters:

xarray

Input array.

Returns:

narray

Norm of the input arrays.

Examples

n = norm_nested(x)

eccentricity.normalize_nested(x)

Normalize 2D array of shape (N,3) along last axis.

Parameters:

xarray

Input array.

Returns:

yarray

Normalized array.

Examples

y = normalize_nested(x)

eccentricity.omegasq_aligned(a, e, q, chi1, chi2, which)

Squared oscillation frequency of a given perturbed aligned-spin binary. The flag which needs to be set to uu for up-up, ud for up-down, du for down-up or dd for down-down where the term before (after) the hyphen refers to the spin of the heavier (lighter) black hole.

Parameters:

a: float

Semi-major axis.

e: float

Eccentricity: 0<=e<1

q: float

Mass ratio: 0<=q<=1.

chi1: float

Dimensionless spin of the primary (heavier) black hole: 0<=chi1<=1.

chi2: float

Dimensionless spin of the secondary (lighter) black hole: 0<=chi2<=1.

which: string

Select function behavior.

Returns:

omegasq: float

Squared frequency.

Examples

omegasq = precession.omegasq_aligned(r,q,chi1,chi2,which)

eccentricity.precession_average(kappa, a, e, chieff, q, chi1, chi2, func, *args, method='quadrature', Nsamples=10000.0)

Average a generic function over a precession cycle. The function needs to have call: func(deltachi, *args). Keywords arguments are not supported. There are integration methods implemented:

• method=’quadrature’ uses scipy.integrate.quad. This is set by default and should be preferred.

• method=’montecarlo’ samples t(deltachi) and approximate the integral with a Monte Carlo sum. The number of samples can be specifed by Nsamples.

Additional keyword arguments are passed to scipy.integrate.quad.

Parameters:

kappa: float

Asymptotic angular momentum.

a: float

Semi-major axis.

e: float

Eccentricity: 0<=e<1

chieff: float

Effective spin.

q: float

Mass ratio: 0<=q<=1.

chi1: float

Dimensionless spin of the primary (heavier) black hole: 0<=chi1<=1.

chi2: float

Dimensionless spin of the secondary (lighter) black hole: 0<=chi2<=1.

func: function

Function to precession-average.

*args: tuple

Extra arguments to pass to func.

method: string (default: ‘quadrature’)

Either ‘quadrature’ or ‘montecarlo’

Nsamples: integer, optional (default: 1e4)

Number of Monte Carlo samples.

Returns:

func_av: float

Precession average of func.

Examples

func_av = precession.precession_average(kappa,a,e,chieff,q,chi1,chi2,func,*args,method='quadrature',Nsamples=1e4)

eccentricity.rhs_orbav(allvars, a, q, m1, m2, eta, chi1, chi2, S1, S2, PNorderpre=[0, 0.5], PNorderrad=[0, 1, 1.5, 2, 2.5, 3])

Right-hand side of the systems of ODEs describing orbit-averaged inspiral. The equations are reported in Sec 4A of Gerosa and Kesden, arXiv:1605.01067. The format is d[allvars]/dv=RHS where allvars=[Lhx,Lhy,Lhz,S1hx,S1hy,S1hz,S2hx,S2hy,S2hz,t], h indicates unit vectors, v is the orbital velocity, and t is time. This is an internal function used by the ODE integrator and is not array-compatible.

Parameters:

allvars: array

Packed ODE input variables.

a: float

Newtonian orbital velocity.

q: float

Mass ratio: 0<=q<=1.

m1: float

Mass of the primary (heavier) black hole.

m2: float

Mass of the secondary (lighter) black hole.

eta: float

Symmetric mass ratio 0<=eta<=1/4.

chi1: float

Dimensionless spin of the primary (heavier) black hole: 0<=chi1<=1.

chi2: float

Dimensionless spin of the secondary (lighter) black hole: 0<=chi2<=1.

S1: float

Magnitude of the primary spin.

S2: float

Magnitude of the secondary spin.

PNorderpre: array (default: [0,0.5])

PN orders considered in the spin-precession equations.

PNorderrad: array (default: [0,0.5])

PN orders considered in the radiation-reaction equation.

Returns:

RHS: float

Right-hand side.

Examples

RHS = precession.rhs_orbav(allvars,a,q,m1,m2,eta,chi1,chi2,S1,S2,PNorderpre=[0,0.5],PNorderrad=[0,1,1.5,2,2.5,3])

eccentricity.rhs_precav(kappa, u, chieff, q, chi1, chi2)

Right-hand side of the dkappa/du ODE describing precession-averaged inspiral. This is an internal function used by the ODE integrator and is not array-compatible.

Parameters:

kappa: float

Asymptotic angular momentum.

u: float

Compactified separation 1/(2L).

chieff: float

Effective spin.

q: float

Mass ratio: 0<=q<=1.

chi1: float

Dimensionless spin of the primary (heavier) black hole: 0<=chi1<=1.

chi2: float

Dimensionless spin of the secondary (lighter) black hole: 0<=chi2<=1.

Returns:

RHS: float

Right-hand side.

Examples

RHS = precession.rhs_precav(kappa,u,chieff,q,chi1,chi2)

eccentricity.roots_vec(p)

Locate roots of polynomial using a vectorized version of numpy.roots. Credit to stackoverflow user ‘pv’ (https://stackoverflow.com/a/35853977) for some of this code, which we generalize to accomodate equations of different degrees.

Parameters:

p: float

Polynomial coefficients.

Returns:

roots: float

Polynomial roots.

Examples

roots = precession.roots_vec(p)

eccentricity.rotate_nested(vec, align_zaxis, align_xzplane)

Rotate a given vector vec to a frame such that the vector align_zaxis lies along z and the vector align_xzplane lies in the xz plane.

Parameters:

vecarray

Input array.

align_zaxisarray

Reference array.

align_xzplanearray

Reference array.

Returns:

vecrotarray

Rotated array.

Examples

vecrot = rotate_nested(vec, align_zaxis, align_xzplane)

eccentricity.rupdown(q, chi1, chi2)

The critical separations r_ud+/- marking the region of the up-down precessional instability.

Parameters:

q: float

Mass ratio: 0<=q<=1.

chi1: float

Dimensionless spin of the primary (heavier) black hole: 0<=chi1<=1.

chi2: float

Dimensionless spin of the secondary (lighter) black hole: 0<=chi2<=1.

Returns:

rudm: float

Inner orbital separation in the up-down instability.

rudp: float

Outer orbital separation in the up-down instability.

Examples

rudp,rudm = precession.rupdown(q,chi1,chi2)

eccentricity.sample_unitsphere(N=1)

Sample points uniformly on a sphere of unit radius. Returns array of shape (N,3).

Parameters:

N: integer, optional (default: 1)

Number of samples.

Returns:

vec: array

Vector in Cartesian coomponents.

Examples

vec = sample_unitsphere(N=1)

eccentricity.scalar_nested(k, x)

Nested scalar product between a 1D and a 2D array.

Parameters:

kfloat

Input scalar.

xarray

Input array.

Returns:

yarray

Scalar product array.

Examples

y = scalar_nested(k, x)

eccentricity.tiler(thing, shaper)

Repeat the quantity thing a numer of times give by the shape of shaper.

Parameters:

thing: float

Quantity to be repeated

shaper: array

Quantity providing the shape.

Returns:

thing: float

Quantity to be repeated

Examples

thing = precession.tiler(thing,shaper)

eccentricity.tofdeltachi(deltachi, kappa, a, e, chieff, q, chi1, chi2, cyclesign=1, precomputedroots=None)

Time as a function of deltachi on the precessional timescale (without radiation reaction).

Parameters:

deltachi: float

Weighted spin difference.

kappa: float

Asymptotic angular momentum.

a: float

Semi-major axis.

e: float

Eccentricity: 0<=e<1

chieff: float

Effective spin.

q: float

Mass ratio: 0<=q<=1.

chi1: float

Dimensionless spin of the primary (heavier) black hole: 0<=chi1<=1.

chi2: float

Dimensionless spin of the secondary (lighter) black hole: 0<=chi2<=1.

cyclesign: integer, optional (default: 1)

Sign (either +1 or -1) to cover the two halves of a precesion cycle.

precomputedroots: array, optional (default: None)

Pre-computed output of deltachiroots for computational efficiency.

Returns:

t: float

Time.

Examples

t = precession.tofdeltachi(deltachi,kappa,a,e,chieff,q,chi1,chi2,cyclesign=1)

eccentricity.updown_endpoint(q, chi1, chi2)

Endpoint of the up-down precessional instability.

Parameters:

q: float

Mass ratio: 0<=q<=1.

chi1: float

Dimensionless spin of the primary (heavier) black hole: 0<=chi1<=1.

chi2: float

Dimensionless spin of the secondary (lighter) black hole: 0<=chi2<=1.

Returns:

deltaphi: float

Angle between the projections of the two spins onto the orbital plane.

theta1: float

Angle between orbital angular momentum and primary spin.

theta2: float

Angle between orbital angular momentum and secondary spin.

Examples

theta1,theta2,deltaphi = precession.updown_endpoint(q,chi1,chi2)

eccentricity.vectors_to_Jframe(Lvec, S1vec, S2vec)

Rotate vectors of the three momenta onto a frame where J is along z and L lies in the x-z plane.

Parameters:

Lvec: array

Cartesian vector of the orbital angular momentum.

S1vec: array

Cartesian vector of the primary spin.

S2vec: array

Cartesian vector of the secondary spin.

Returns:

Lvec: array

Cartesian vector of the orbital angular momentum.

S1vec: array

Cartesian vector of the primary spin.

S2vec: array

Cartesian vector of the secondary spin.

Examples

Lvec,S1vec,S2vec = precession.vectors_to_Jframe(Lvec,S1vec,S2vec)

eccentricity.vectors_to_Lframe(Lvec, S1vec, S2vec)

Rotate vectors of the three momenta onto a frame where L is along z and S1 lies in the x-z plane.

Parameters:

Lvec: array

Cartesian vector of the orbital angular momentum.

S1vec: array

Cartesian vector of the primary spin.

S2vec: array

Cartesian vector of the secondary spin.

Returns:

Lvec: array

Cartesian vector of the orbital angular momentum.

S1vec: array

Cartesian vector of the primary spin.

S2vec: array

Cartesian vector of the secondary spin.

Examples

Lvec,S1vec,S2vec = precession.vectors_to_Lframe(Lvec,S1vec,S2vec)

eccentricity.vectors_to_angles(Lvec, S1vec, S2vec)

Convert cartesian vectors (L,S1,S2) into angles (theta1,theta2,deltaphi).

Parameters:

Lvec: array

Cartesian vector of the orbital angular momentum.

S1vec: array

Cartesian vector of the primary spin.

S2vec: array

Cartesian vector of the secondary spin.

Returns:

deltaphi: float

Angle between the projections of the two spins onto the orbital plane.

theta1: float

Angle between orbital angular momentum and primary spin.

theta2: float

Angle between orbital angular momentum and secondary spin.

Examples

theta1,theta2,deltaphi = precession.vectors_to_angles(Lvec,S1vec,S2vec)

eccentricity.vectors_to_conserved(Lvec, S1vec, S2vec, a, e, q, full_output=False)

Convert vectors (L,S1,S2) to conserved quanties (deltachi,kappa,chieff).

Parameters:

Lvec: array

Cartesian vector of the orbital angular momentum.

S1vec: array

Cartesian vector of the primary spin.

S2vec: array

Cartesian vector of the secondary spin.

q: float

Mass ratio: 0<=q<=1.

full_output: boolean, optional (default: False)

Return additional outputs.

Returns:

chieff: float

Effective spin.

cyclesign: integer, optional

Sign (either +1 or -1) to cover the two halves of a precesion cycle.

deltachi: float

Weighted spin difference.

kappa: float

Asymptotic angular momentum.

Examples

deltachi,kappa,chieff = precession.vectors_to_conserved(Lvec,S1vec,S2vec,q) deltachi,kappa,chieff,cyclesign = precession.vectors_to_conserved(Lvec,S1vec,S2vec,q,full_output=True)

eccentricity.widenutation_condition(a, e, q, chi1, chi2)

Conditions for wide nutation to take place. The returned flag which is

• wide1 if wide nutation is allowed for the primary BH.

• wide2 if wide nutation is allowed for the secondary BH.

• nowide if wide nutation is not allowed.

Parameters:

a: float

Semi-major axis.

e: float

Eccentricity: 0<=e<1

q: float

Mass ratio: 0<=q<=1.

chi1: float

Dimensionless spin of the primary (heavier) black hole: 0<=chi1<=1.

chi2: float

Dimensionless spin of the secondary (lighter) black hole: 0<=chi2<=1.

Returns:

chieff: float

Effective spin.

kappa: float

Asymptotic angular momentum.

which: string

Select function behavior.

Examples

which,kappa,chieff = precession.widenutation_condition(r,q,chi1,chi2)

eccentricity.widenutation_separation(q, chi1, chi2)

The critical separation r_wide below which the binary component with smaller dimensionless spin may undergo wide nutations.

Parameters:

q: float

Mass ratio: 0<=q<=1.

chi1: float

Dimensionless spin of the primary (heavier) black hole: 0<=chi1<=1.

chi2: float

Dimensionless spin of the secondary (lighter) black hole: 0<=chi2<=1.

Returns:

r_wide: float

Orbital separation where wide nutations becomes possible.

Examples

r_wide = precession.widenutation_separation(q,chi1,chi2)

eccentricity.wraproots(coefficientfunction, *args, **kwargs)

Find roots of a polynomial given coefficients, ordered according to their real part. Complex roots are masked with nans. This is essentially a wrapper of numpy.roots.

Parameters:

coefficientfunction: callable

Function returning the polynomial coefficients ordered from highest to lowest degree.

*args, **kwargs:

Parameters of coefficientfunction.

Returns:

sols: array

Roots of the polynomial.

Examples

sols = precession.wraproots(coefficientfunction, *args, **kwargs)