Audrey Trovatrova@asu.cas.cz

Astronomical Institute of the Academy of Sciences, Prague, Czech Republic

The aim of the proposal to the COST project MP1304 ”Exploring fundamental physics with compact stars” wasto support a research stay and carry out a collaboration project between my current institute (Astronomical Instituteof the Academy of Sciences of the Czech Republic) and the hosting institution of Laboratoire d’Astrophysique deBordeaux (LAB) in France. During this STSM mission, we worked on equilibrium figures of self-gravitating tori in thecontext of Active Galactic. After a first work to see the impact of the self-gravity on equilibrium of charged tori witha toy-model develloped in [1], we plan to use the self-consistent field method to implement with more accuracy theself-gravity and various equations of state. To begin with, we wanted to study the impact of the central point mass,typically Mc/Mtorus as the parameter. I will summarise the joint research proposal, about loaded self-gravitatingtori by describing the code improved by adding the gravitational field of the central mass and I will show first resultswe obtained.

1 Self-consistent field method

We improved the toy-model by using a SCF code (Self-Consistent Fluid code) [4] for the equilibrium figures of self-gravitating systems in rotation. The numerical code, written in Fortran 90, consist of resolving the equation ofBernoulli for 2D axi-symmetric systems [4, 5, 6], namely

H + Ψ + Φ + Ψc +M = Cste and ∆Ψ = 4πGρ, (1)

where ρ is the mass density of the fluid. The fluid is sensitive to the pressure, represented by H =∫ρ−1dP the

enthalpy, to its own potential Ψ, to Φ the rotational potential, to the potential of the central mass Ψc and to Mthe dipolar ”magnetic potential” of the central mass. The gravitational potential has a central role and has to becalculated with the best accuracy. The system has the following properties :

• symmetry around the rotation axis and the midplane,

• the orbitale velocity is a function only of the distance from the rotation axis,

• The equation of state is barotropic, the pressure P is a function only of the density. By exemple P = kρ1+1n

for polytropes.

After normalization, we obtained the final equation:

aH + Ψ + bΦ + dΨc + eM = c, (2)

where a, b, c, d and e are constants. This approach consists of finding the final density of the equilibrium object by aniterative scheme on the Bernoulli equation (2). We have to set, the rotation law, the polytropic index, the equatorialradius and the constants d and e which represent, respectively, the impact of the gravitational field and the strengthof the magnetic field of the central mass. We have first study the equilibrium without magnetic field. The constantsa, b and c are found by iterative step. We compared our equilibrium figures, with success, to some results publishedin different articles [4, 5].

2 Equilibrium of loaded self-gravitating system

With this code, we have obtained equilibrium structures for various value of d polytropic index n, for differentrotation laws and various equatorial radius ra. We tested various cases. We vary d ∈ [0, 0.01, 0.05, 0.1, 1, 10, 100] andn = 0, 0.5, 1.5, 3. We found that solutions do not exist for any case. For instance, within solid rotation (constantangular velocity) and n = 1.5, amongs the tested values of d, the equilibrium exists only for d < 0.1 and not for thehigher values. In other way, for d = 0.1, solutions exist for n = 0, 0.5, 1.5 and not for n = 3.

2.1 Impact of the central mass on the equilibrium configurations

To analyse the impact of the gravitational field of the central mass on the equilibrium configuration, we choose arotation law, a prolytropic index and an equatorial radius and we varry the value of d. Here, we took a constantangular momemtum law, n = 1.5 and ra = 0.325. In the figure 1, we show, in (a), (b), (c), (d) and (e), the pressureequilibrium maps resulting from the SCF code with the previous paremeters.

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Rigid Rotation law, n = 1.5d range of ra0 [0.325, 0.825]0.05 [0.471, 0.825]0.1 [0.577, 0.825]

Table 1: Range of possible values of ra for three values of d.

0.2 0.4 0.6 0.8 1.0

0.5

0.0

0.5

Pression

radius R

altitu

de

z

1

0.5

1.38778e 16

0.5

1

(a) d = 0.

0.2 0.4 0.6 0.8 1.0

0.5

0.0

0.5

Pression

radius R

altitu

de

z

1

0.5

1.38778e 16

0.5

1

(b) d = 0.1.

0.2 0.4 0.6 0.8 1.0

0.5

0.0

0.5

Pression

radius R

altitu

de

z

1

0.5

1.38778e 16

0.5

1

(c) d = 0.25.

0.2 0.4 0.6 0.8 1.0

0.5

0.0

0.5

Pression

radius R

altitu

de

z

1

0.5

1.38778e 16

0.5

1

(d) d = 0.

0.2 0.4 0.6 0.8 1.0

0.5

0.0

0.5

Pression

radius R

altitu

de

z

1

0.5

1.38778e 16

0.5

1

(e) d = 0.1.

0 0.2 0.4 0.6 0.8R

0

0.2

0.4

0.6

0.8

1

ρ(R,0)

d=0

d=10

(f) Density profile in the equatorialplane of the torus

Figure 1: (a), (b), (c), (d) and (e) represent the pressure equilibrium maps resulting from the SCF code with aconstant angular momemtum, n = 1.5 and ra = 0.325 for various values of d.

We can see that when we increase the value of d, meaning increasing the value of the central mass, the torusbecomes thinner and the maximum of pressure moves close to the center mass. It makes sense because higher is thecentral mass higher is the attraction.

2.2 Impact of the central mass on the ”one-ring” sequences

By varying the equatorial radius, ra, and the mass ratio, d, and imposing all the other parameters, we can see theimpact of the value of d on the ”one-ring” sequence presented in [4] for self-gravitating disk only (d = 0). We testthe rigid rotation law (d = 0, 0.01, 0.05, 0.1) for ra = [0.325, 0.825] and the constant angular momemtum rotation law(d = 0, 0.05, 0.1) for ra = [0.05, 0.941]. The results are, respectively, presented in the figures 2 and 3. They are donefor n = 1.5. We can directly see the change on the sequences. Another interesting point is that for each value of d itdoes not exist solutions for each equatorial radius for the rigid rotation law (see Table 1), which is not the case forthe constant angular momemtum rotation law, where solutions exist for each equatorial radius in the given range.

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0 0.1 0.2 0.3 0.4 0.5

J2/4πGM

10/3ρ-1/3

0

0.02

0.04

0.06

0.08

0.1

Ω2/4

πG

ρ

(a) Ω2/4πGρ as function of J2/4πGM10/3ρ−1/3.

0 0.1 0.2 0.3 0.4 0.5

J2/4πGM

10/3ρ-1/3

0.3

0.35

0.4

0.45

0.5

T/|W|

(b) T/|W | as function of J2/4πGM10/3ρ−1/3.

Figure 2: Graphics are done for d = 0 (full line), 0.01 (dashed line), 0.05 (dot-dashed line), 0.1 (dot line), for n = 1.5and a rigid rotation law.

3 Conclusion

To conclude, we found that the mass of the central object influences the configuration of the torus, the ”one-ring”sequence is modified as the existence of equilibrium configurations. There is a lot of parameters to explore, as othervalues for the polytropic index, or using other rotation law (as the constant orbital velocity law). This work can beextended by adding a magnetic field of the central mass by giving a value to the constant e and studied the impact.A final step is the addition of the magnetic field of the torus itself which could have an impact on the structure ofthe tori.

0 0.1 0.2 0.3 0.4 0.5

J2/4πGM

10/3ρ-1/3

0

0.01

0.02

0.03

0.04

0.05

Ω2/4

πG

ρ

(a) Ω2/4πGρ as function of J2/4πGM10/3ρ−1/3.

0 0.1 0.2 0.3 0.4 0.5

J2/4πGM

10/3ρ-1/3

0

0.1

0.2

0.3

0.4

0.5

T/|W|

(b) T/|W | as function of J2/4πGM10/3ρ−1/3.

Figure 3: Graphics are done for d = 0 (full line), 0.05 (dashed line), 0.1 (dot-dashed line), for n = 1.5 and a constantangular momentum rotation law.

References

[1] A. Trova, V. Karas, P. Slany, and J. Kovar. Toy-model for self-gravitating charged tori in dipolar magnetic field.Manuscript submitted for publication.

[2] P. Slany, J. Kovar, Z. Stuchlık, and V. Karas. Charged Tori in Spherical Gravitational and Dipolar MagneticFields. ApJS, 205:3, March 2013.

[3] J. Kovar, P. Slany, C. Cremaschini, Z. Stuchlık, V. Karas, and A. Trova. Electrically charged matter in rigidrotation around magnetized black hole. Phys. Rev. D, 90(4):044029, August 2014.

[4] I. Hachisu. A versatile method for obtaining structures of rapidly rotating stars. ApJS, 61:479–507, July 1986.

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[5] M. Ansorg, A. Kleinwachter, and R. Meinel. Uniformly rotating axisymmetric fluid configurations bifurcatingfrom highly flattened Maclaurin spheroids. MNRAS, 339:515–523, February 2003.

[6] Y. Tomimura and Y. Eriguchi. A new numerical scheme for structures of rotating magnetic stars. MNRAS,359:1117–1130, May 2005.

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