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Comparisons of $E$ , $L$ and $\Omega$ between Bardeen et al, Toshmatov et al and Lamy et al

This Jupyter/SageMath notebook is related to the article Lamy et al, arXiv:1802.01635.

In [1]:

version()

Out[1]:

'SageMath version 8.1, Release Date: 2017-12-07'

In [2]:

%display latex

In [3]:

from sage.manifolds.utilities import simplify_chain_real as simpl

$E$ , $L$ and $\Omega$ in Bardeen '72 (Kerr case)

In [4]:

var('r a b m0', domain='real')
var('th', latex_name=r'\theta', domain='real')
assume(r>0)
assume(m0>0)
assume(a>=0)

In [5]:

omega_pos_B=sqrt(m0)/(r^(3/2)+a*sqrt(m0)) # Omega computed in Bardeen et al '72 (Kerr) for co-rotating orbits
omega_neg_B=-sqrt(m0)/(r^(3/2)-a*sqrt(m0))# Omega computed in Bardeen et al '72 (Kerr) for counter-rotating orbits

In [6]:

E_pos_B=(r^(3/2)-2*m0*sqrt(r)+a*sqrt(m0))/(r^(3/4)*sqrt(r^(3/2)-3*m0*sqrt(r)+2*a*sqrt(m0))) # E computed in Bardeen et al '72 (Kerr) for co-rotating orbits
E_neg_B=(r^(3/2)-2*m0*sqrt(r)-a*sqrt(m0))/(r^(3/4)*sqrt(r^(3/2)-3*m0*sqrt(r)-2*a*sqrt(m0))) # E computed in Bardeen et al '72 (Kerr) for counter-rotating orbits

In [7]:

L_pos_B=(sqrt(m0)*(r^2-2*a*sqrt(m0)*sqrt(r)+a^2))/(r^(3/4)*sqrt(r^(3/2)-3*m0*sqrt(r)+2*a*sqrt(m0))) # L computed in Bardeen et al '72 (Kerr) for co-rotating orbits
L_neg_B=(-sqrt(m0)*(r^2+2*a*sqrt(m0)*sqrt(r)+a^2))/(r^(3/4)*sqrt(r^(3/2)-3*m0*sqrt(r)-2*a*sqrt(m0))) # L computed in Bardeen et al '72 (Kerr) for counter-rotating orbits

In [8]:

ratio_pos_B=L_pos_B/E_pos_B

In [9]:

ratio_pos_B.simplify_full()

Out[9]:

$E$ , $L$ and $\Omega$ in Toshmatov et al '17 (Kerr with $M(r)$ )

Definitions

In [10]:

M=m0*r^3/(r^3+2*m0*b^2) #Let's assume the Hayward form for M(r), which boils down to Kerr for b=0
Sigma = r^2 + a^2*cos(th)^2
Delta = r^2 - 2*M*r + a^2
g00=-(1 - 2*r*M/Sigma)
g03=-2*a*r*sin(th)^2*M/Sigma
g33=(r^2 + a^2 + 2*r*(a*sin(th))^2*M/Sigma)*sin(th)^2
dg00=diff(g00,r)
dg03=diff(g03,r)
dg33=diff(g33,r)
Mdot=diff(M,r)

In [11]:

omega_pos_T=(sqrt(M/r-Mdot)/(a*sqrt(M/r-Mdot)+r)) # Omega computed in Toshmatov et al '17
omega_neg_T=sqrt(M/r-Mdot)/(a*sqrt(M/r-Mdot)-r)

In [12]:

E_pos_T=(r-2*M+a*sqrt(M/r-Mdot))/(r*(r-3*M+2*a*sqrt(M/r-Mdot)+r*Mdot)) # E computed in Toshmatov et al '17
E_neg_T=-(r-2*M+a*sqrt(M/r-Mdot))/(r*(r-3*M+2*a*sqrt(M/r-Mdot)+r*Mdot))

In [13]:

L_pos_T=(2*a*M-(r^2+a^2)*sqrt(M/r-Mdot))/(r*(r-3*M+2*a*sqrt(M/r-Mdot)+r*Mdot)) # L computed in Toshmatov et al '17
L_neg_T=-(2*a*M-(r^2+a^2)*sqrt(M/r-Mdot))/(r*(r-3*M+2*a*sqrt(M/r-Mdot)+r*Mdot))

Comparison with Bardeen '72

$\Omega$ : agreement

In [14]:

omega_pos_B

Out[14]:

In [15]:

simpl(omega_pos_T.subs({b:0, th:pi/2}))

Out[15]:

In [16]:

bool(omega_pos_B == simpl(omega_pos_T.subs({b:0, th:pi/2})))

Out[16]:

$E$ : disagreement

In [17]:

simpl(E_pos_B)

Out[17]:

In [18]:

simpl(E_pos_T.subs({b:0, th:pi/2}))

Out[18]:

In [19]:

E_pos_B.subs({a:1},{m0:1},{r:3}).n()

Out[19]:

In [20]:

E_pos_T.subs({a:1},{b:0},{m0:1},{r:3},{th:pi/2}).n() #b=0 boils down to Kerr

Out[20]:

In [21]:

E_neg_B.subs({a:1},{m0:1},{r:3}).n()

Out[21]:

In [22]:

E_neg_T.subs({a:1},{b:0},{m0:1},{r:3},{th:pi/2}).n() #b=0 boils down to Kerr

Out[22]:

$L$ : disagreement

In [23]:

simpl(L_pos_B)

Out[23]:

In [24]:

simpl(L_pos_T.subs({b:0, th:pi/2}))

Out[24]:

In [25]:

L_pos_B.subs({a:1},{m0:1},{r:3}).n()

Out[25]:

In [26]:

L_pos_T.subs({a:1},{b:0},{m0:1},{r:3},{th:pi/2}).n() #b=0 boils down to Kerr

Out[26]:

$E$ , $L$ and $\Omega$ in Lamy et al '18 (Kerr with $M(r)$ )

Definitions

In [27]:

omega_pos=(-dg03+sqrt(dg03^2-dg00*dg33))/dg33 # Omega computed in Lamy et al '18
omega_neg=(-dg03-sqrt(dg03^2-dg00*dg33))/dg33

In [28]:

E_pos=-(g00+g03*omega_pos)/sqrt(-(g00+2*g03*omega_pos+g33*omega_pos^2)) #E computed in Lamy et al '18 
E_neg=-(g00+g03*omega_neg)/sqrt(-(g00+2*g03*omega_neg+g33*omega_neg^2))

In [29]:

L_pos=(g03+g33*omega_pos)/sqrt(-(g00+2*g03*omega_pos+g33*omega_pos^2)) # L computed in Lamy et al '18
L_neg=(g03+g33*omega_neg)/sqrt(-(g00+2*g03*omega_neg+g33*omega_neg^2))

Comparison with Bardeen '72

$\Omega$ : agreement

In [30]:

simpl(omega_pos_B)

Out[30]:

In [31]:

simpl(omega_pos.subs({b:0, th:pi/2}))

Out[31]:

In [32]:

bool(simpl(omega_pos_B) == simpl(omega_pos.subs({b:0},{th:pi/2})))

Out[32]:

$E$ : agreement

In [33]:

E_pos_B1 = simpl(E_pos_B)
E_pos_B1

Out[33]:

In [34]:

E_pos1 = simpl(E_pos.subs({b:0, th:pi/2})) #b=0 boils down to Kerr 
E_pos1

Out[34]:

In [35]:

bool(E_pos1^2 == E_pos_B1^2)

Out[35]:

$L$ : agreement

In [36]:

L_pos_B1 = simpl(L_pos_B)
L_pos_B1

Out[36]:

In [37]:

L_pos1 = simpl(L_pos.subs({b:0, th:pi/2})) #b=0 boils down to Kerr 
L_pos1

Out[37]:

In [38]:

bool(L_pos1^2 == L_pos_B1^2)

Out[38]:

Comparisons of $E$ , $L$ and $\Omega$ between Bardeen et al, Toshmatov et al and Lamy et al

$E$ , $L$ and $\Omega$ in Bardeen '72 (Kerr case)

$E$ , $L$ and $\Omega$ in Toshmatov et al '17 (Kerr with $M(r)$ )

Definitions

Comparison with Bardeen '72

$\Omega$ : agreement

$E$ : disagreement

$L$ : disagreement

$E$ , $L$ and $\Omega$ in Lamy et al '18 (Kerr with $M(r)$ )

Definitions

Comparison with Bardeen '72

$\Omega$ : agreement

$E$ : agreement

$L$ : agreement

Product

Resources

Company

Comparisons of EEE, LLL and Ω\OmegaΩ between Bardeen et al, Toshmatov et al and Lamy et al

EEE, LLL and Ω\OmegaΩ in Bardeen '72 (Kerr case)

EEE, LLL and Ω\OmegaΩ in Toshmatov et al '17 (Kerr with M(r)M(r)M(r))

Definitions

Comparison with Bardeen '72

Ω\OmegaΩ: agreement

EEE: disagreement

LLL: disagreement

EEE, LLL and Ω\OmegaΩ in Lamy et al '18 (Kerr with M(r)M(r)M(r))

Definitions

Comparison with Bardeen '72

Ω\OmegaΩ: agreement

EEE: agreement

LLL: agreement

Comparisons of $E$ , $L$ and $\Omega$ between Bardeen et al, Toshmatov et al and Lamy et al

$E$ , $L$ and $\Omega$ in Bardeen '72 (Kerr case)

$E$ , $L$ and $\Omega$ in Toshmatov et al '17 (Kerr with $M(r)$ )

$\Omega$ : agreement

$E$ : disagreement

$L$ : disagreement

$E$ , $L$ and $\Omega$ in Lamy et al '18 (Kerr with $M(r)$ )

$\Omega$ : agreement

$E$ : agreement

$L$ : agreement