In this section we will study the tidal effects that are the cause of the interesting dynamics of an NSBH binary. First, we will study these tidal effects in a classical Newtonian mechanics formalism. In the next section we will study the tidal effects in a classical Lagrangian mechan-ics formalism which can be easily carried over to a general relativistic description of the system.
In section 2.1we have seen observational evidence that at a certain point during the inspiral of an NSBH binary, the masses of the stellar objects can not be estimated as point masses anymore and that one of the main processes that becomes of great significance during the final phase of the inspiral is tidal disruption. All bodies that are not point masses are subject to tidal forces. The difference between the gravitational force of the black hole on a point mass in the centre of the neutron star and a point mass at the surface of the neutron star is what is known as the differential tidal force ∆ ⃗F or just tidal force. This is the force relative to the centre of the neutron star, which we can derive using Newtonian theory.
We will derive the tidal force field in two spatial dimensions, which can be straightforwardly extended to three spatial dimensions. Consider a neutron star in the form of a spherical body with radius RN S, subject to the gravitational field of a black hole with mass MBH. The orbital
Figure 2.5: Geometry for calculating the differential tidal force. The left object is considered to be the neutron star that will be subject to tidal effects from the black hole.
separation of the two objects is given by r, which will be assumed to be much greater than RN S
(r ≫ RN S). A test-mass m in the centre of the neutron star is subject to the gravitational force of the black hole by:
F⃗C= MBHm
r2 ˆex, (2.1)
where G is Newtons gravitational constant and ˆexis the unit vector in the x-direction. Simi-larly, a test-mass m on the surface of the neutron star is subject to the gravitational force of the black hole. We will use θ as the angle between the neutron star-black hole line and the radial line to m from the neutron star, ϕ as the same angle but with the radial line to m from the black hole and finally s as the radial line to m from the black hole, see Figure 2.5. We then have:
F⃗S =MBHm
s2 (cos(ϕ)ˆex− sin(ϕ)ˆey). (2.2) The difference between the gravitational force from the black hole on the test-mass on the surface of the neutron star and in the centre of the neutron star is then given by:
∆ ⃗F = MBHm cos(ϕ) s2 − 1
r2
ˆ
ex−sin(ϕ) s2 ˆey
. (2.3)
Since we have r ≫ RN S, we can approximate cos(ϕ) ≈ 1. From trigonometry we also have:
sin(ϕ) = RBHsin(θ)
s ≈ RNSsin(θ)
r , (2.4)
as well as:
s2= (r − RNScos(θ))2+ (RNSsin(θ))2= r2
1 −2RNScos(θ) r +R2NS
r2
1 s2 = 1
r2
1 −2RNScos(θ) r
−1
≈ 1 r2
1 +2RNScos(θ) r
,
(2.5)
where we neglected the RN S2 /r2 term and Taylor expanded the third term because r ≫ RN S. We can write equation (2.3) as:
∆ ⃗F = MBHmRNS
r3
2 cos(θ)ˆex− sin(θ)ˆey
. (2.6)
Here we can see that for θ = 0 the tidal force is directed at the black hole, while for θ = π, counter-intuitively, the tidal force is directed away from the black hole. The net force is still
Figure 2.6: Tidal tidal displacement field of a neutron star from the presence of a companion black hole in the body frame of the neutron star. Note that ϕ is now the angle denoting the phase of the orbit of the black hole around the neutron star in anticipation of next sections.
directed in the direction of the black hole, but the tidal force denotes the force on the surface of the neutron star relative to the force on the centre of the neutron star because of the gravita-tional pull of the black hole. We can see that the tidal force wants to pull the neutron star into an ellipsoidal form. See Figure2.6 for a visualisation of the tidal displacement field. As long as the self-gravitational force of the neutron star can counteract the tidal force, the neutron star will remain in compact form, but when the tidal force starts to become much greater than the self-gravitational force from the neutron star, the star starts to become tidally disrupted and will no longer be close to a spherical object in space.
We can note that the tidal force is maximal for θ = 0. Before any tidal disruption, the neutron star will still be spherical of form, and therefore the self-gravitational force will be constant along the surface of the neutron star. We will therefore compare the self-gravitational force of the neutron star with the tidal force along the neutron star-black hole line. When the magnitude of the tidal force at θ = 0 exceeds the magnitude of the self-gravitational force, the tidal disruption process will start taking place, beginning along the θ = 0 line before slowly expanding to the rest of the neutron star. This also confirms the time evolution of the tidal disruption prematurely depicted in the previous section in Figure2.4.
The force balance of the self-gravitational force with the tidal force at the the surface of the neutron star along the common axis of the neutron star and the black hole is given by:
2MBHmRNS
r3 = MNSm
R2NS . (2.7)
We can therefore estimate that tidal disruption starts taking place for orbital radii rtidal that are approximately larger than:
rtidal= RNS
2MBH
MNS
1/3
. (2.8)
We should keep in mind that we made the assumption that r ≫ RN S, which might not hold during the final phase of the inspiral. Also, we know that Newtonian theory does not apply
to compact objects and to relativistic motion. Ref. [33] proposes a correction based on a comparison of the estimated tidal deformation radius with NR results. This correction ensures that compact objects are more strongly bound:
˜
rtidal= rtidal(1 − 2MNS/RNS). (2.9) A much better way to incorporate effects from the compactness MNS/RNSof the objects as well relativistic effects is to derive the mechanisms that are responsible for these effects ourselves.
This can be done more easily using another formalism, namely that of an action principle which will be discussed in the next section. This formalism has the advantage that it easily carries over to a general relativistic formulation of the problem. Another advantage of this approach is that we will be able to derive the energy of the system, which we can use to estimate a more accurate tidal disruption radius. The radius we have estimated now is calculated from a force balance evaluated at one certain point at the surface of the neutron star, while an energy balance can be used to take the entire system into account.