# Non-Keplerian Motion of the star S2

Stellar proper motions and accelerations during the past ten years allowed Schoedel et al. (2002, 2003 ), Ghez et al. (2005 ) and Eisenhauer et al. (2005 ) to fit Keplerian orbits to the trajectories of 6 of the S-stars. These are currently the innermost observed stars, their observed proper motions and accelerations are in agreement with orbits around a central, dark, and massive object (Mass ~ 3.7 million solar masses) coincident with the position of the radio source SgrA*. Based on these results, and considering that these stars resolve the central stellar cusp mass inside their apoastron separation from the center, it is possible then to account for the fraction of the extended mass present at these distances. In order to get e better insight into the nature of possible stellar orbits in realistic potentials in the vicinity of a central compact mass at the centre of a dense stellar cluster we perform orbital integrations using a Fourth-Order Hermite Integrator. Rubilar and Eckart (2001) have shown that the detection of relativistic or Newtonian peri-astron-shifts from three different stars will allow us to determine the amount, size, and shape of the enclosed central mass.

However the star S2 which has the shortest orbital period (15 years) and the most complete orbit of the S-stars (see Schoedel et al. (2002, 2003 ), Ghez et al. (2005 ) and Eisenhauer et al. (2005 ) is only candidate which can give us informations about the amount of mass distribution inside its orbit. Thus, with only one candidate, it is not possible to determine the three required paramters. Nevertheless, if the shape and size of the mass disribution is fixed to the stellar light curve, it is possible to constraint the amount on the extended mass. The motion of the star in an extended mass potentail will result in a rosetta-shaped orbit. The figure below shows that both Keplerian and non-Keplerian orbits fit very well the data, and it still not possible with the present observations to descriminate between both potentials.

* Three exemplary fitting orbits. Upper left panel: Keplerian orbit with 3.65 10 ^{6} M_{sol} point mass. Upper middle panel, non-Keplerian orbits with 4.1 10^{6} M_{sol} total mass of which 0.4 10^{6} M_{sol} are extended. Upper right panel, non-Keplerian orbits with 4.8 10^{6} M_{sol} total mass of which 1.2 10^{6} M_{sol} are extended. Here the central mass is at the offset position, 0.082 mpc east and 0.112 mpc south from the nominal radio position of SgrA*. The lower panels show the velocity in the relative R.A, the relative Dec., and along the line-of-sight as a function of time for the case of 4.1 10^{6} M_{sol} central mass + 10% extended component. This case is representative for the other cases. The direction of the angle of the peri-centre-shift is shown by an arrow.*

This figure illustrates results obtained by Mouawad et al. (2005). They give an upper limit of the order of 5% of the fraction of the extended mass inside the sphere of radius the apocenter of S2 over the total central mass of 3.8 million solar masses. In other words, they find that the data allow in principle for an unobserved extended mass component of several 10^{5} solar masses in the cusp centered on the black hole position. Considering only the fraction of the cusp mass M_{S2apo} within the apo-center of the S2 orbit Mouawad et al. (2005) derive as an upper limit that M_{S2apo}/(MBH +M_{S2apo} ) ≤ 0.05. A large extended cusp mass, if present, is unlikely to be composed of sub-solar mass constituents, but could be explained rather well by a cluster of high M/L stellar remnants.

## Hermite Integrator

The fourth-order Hermite integrator uses higher order derivatives wich are explicitly calulated from the Hermite interpolation formula , in order to construct interpolation polynomials of the force.

Particle motion is followed using predictor-corrector scheme (Makino and Aarseth 1992).During a time step, particle positions and velocities are first predicted to fourth order using the known acceleration and the time derivative of the acceleration, wich are then computed, and the motion is then corrected using the additional derivative information thereby obtained.

To construct a Hermite interpolation formula, both the values of the function and its derivatives are used. For example, if we know the values of the acceleration and its first time derivative at two points in time, we can construct a third-order interpolation polynomial. The timestep can be calculated using the higher order terms of the acceleration, and an accurancy parameter that we can vary (Aarseth 1985).The calculation cost and accurancy are then controlled by this parameter.

The fourth-order scheme is close to optimal for a wide range of required accuracy, efficiency, and simplicity of the algorithm.

## Rosetta Orbit

There exists only two types of central gravitational fields where bound particles move on closed orbits. These are the spherical Keplerian and harmonic potentials. In traveling from pericentre to apo-centre and back, the azimuthal angle Φ increases by an amount where (r, Φ) are the planar polar coordinates in which the centre of the attraction is at r=0 and Φ is the azimuthal angle in the orbital plane, r_{peri} and r_{apo} represent the closest and the furthest seperations from the centre of force respectively. E and U are the total and potential energy respectively, M the total mass of the spherical system and m the mass of the particle (Binney & Tremaine 1987). In general, (Δ Φ)/(2π) will not be an integer number. Hence the orbit will not be closed. In this case, the trajectory will be indefinitely spanning an area between two circular boundaries determined by the inner radius r_{peri} and the outer radius r_{apo} (see figure below).

An a simplified way, the Newtonian orbital shift can be modelled by considering the case of a spherically symmetric mass distribution. One can assume that a given star can penetrate the extended mass distribution, and neglect any collisional interaction. The (Newtonian) gravitational force on a given star depends only on the enclosed mass within the radius corresponding to the position of the star. Therefore, as the star moves towards the centre of forces, the gravitational force and hence the curvature of the orbit is smaller than for the case in which the whole mass is concentrated within a radius smaller than the periastron radius of the stellar orbit. This leads to orbits with retrograde orbital shifts, that is, shifts in the opposite direction as compared to the relativistic orbital shift.

Jiang & Lin (1985) presented an analytical treatment of the orbits of a test particle which is allowed to enter into the inner region of a sphere with uniform matter distribution without collision. Only the Newtonian gravitational force was considered and the relativistic effect has been neglected. They found that the major axis precesses about an axis passing through the centre of the system and perpendicular to the plane of motion of the object.

## Orbital Elements

The trajectory of a test mass *m* bound in the gravitational field of a central total mass *M* (*m* << *M*), composed of a point mass and an extended component of a fixed radius, will result in unclosed orbits known as Rosetta-shaped orbits. These can be defined by 8 orbital elements. These can be defined this way:

Five orbital elements, T_{peri}, Ω, ω, *i*, and the *a* conserve the same definition as for the case of a Keplerian orbit, because the precession of the orbit occurs in the orbital plane, whereas for the other 3 elements we give the definion below:

- The period, P, of the orbit, defined as the time between two consecutive apocentre passage.
- The eccentricity or ellipticity 0 < e < 1 of the orbit, eccentricity of the equivalent closed Keplerian orbit between two consecutive apo-centre passages.
- The periastron-shift angle, Δ α which is the precession angle of the major-axis per revolution. This angle depends on the amount of extended mass present within the radius determined by the position of the star.