# The Effect of General Relativity on Hyperbolic Orbits and  Its Application to the Flyby Anomaly

Authors: Lorenzo Iorio
Journal: Scholarly Research Exchange
 (2009)

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## Abstract

We investigate the impact of the general relativistic gravitoelectromagnetic forces on hyperbolic orbits around a massive spinning body. The gravitomagnetic field, causing the
well-known Lense-Thirring precessions of elliptic orbits, is generated by the spinS of the central body. It deflects and displaces the trajectories differently according to the mutual orientation of S and the orbital angular momentum L of the test particle. The gravitoelectric force, which induces the Einstein precession of the perihelion of the orbit of Mercury, always deflects the trajectories inward irrespective of the L−S orientation. We numerically compute their effect on the range r, radial and transverse components vr and vτ of the velocity, and speed v of the NEAR spacecraft at its closest approach with the Earth in January 1998 when it experienced an anomalous increase of its asymptotic outgoing velocity v∞o of 13.46±0.01 mm sec−1; while the gravitoelectric force was modeled in the software used to process the NEAR data, this was not done for the gravitomagnetic one. The range rate and the speed are affected by general relativistic gravitoelectromagnetism at 10−2 (gravitoelectric) to 10−5 (gravitomagnetic) mm sec−1 levels. The changes in the range are of the order of 10−2 (gravitomagnetic) to 101 (gravitoelectric) mm.


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## Body

## 1. Introduction

We  will investigate the effects of general  relativity, in its weak-field and
slow-motion approximation, on unbound, hyperbolic orbits of test particles
approaching a body of massM and angular momentum S.
We will consider both the gravitomagnetic and the gravitoelectric relativistic
forces induced by the stationary and static components, respectively, of the
field of M [1].The results obtained will be applied to some realistic
planet-spacecraft scenarios in the Solar System to see if the predicted effects
fall within the current or future sensitivity level of the ranging techniques.
In particular, we will examine the so-called flyby anomaly consisting of
unexplained changes of the asymptotic outgoing velocities of some spacecraft
(Galileo, NEAR, Cassini, and MESSENGER) that occurred at their closest
approaches with the Earth [2, 3].Let us recall some basics of the Newtonian hyperbolic
orbit [4] which represents, in this case, the reference unperturbed path; see
Figure 1. Its equation is(1)r=p1+ecosf,where p=a(e2−1) is the semilatus rectum, and f is the true anomaly considered positive in the
anticlockwise direction from the point of closest approach A; a is the semimajor axis and e>1.
The speed is(2)v=GM(2r+1a),so that(3)v∞=GMa.The asymptotic ingoing and
outgoing speeds, which are equal to v∞ in the unperturbed case, are denoted with v∞i and v∞o,
respectively. The angle ϕ between the r and v is defined by(4)sinϕ=1+ecosf1+e2+2ecosf,cosϕ=−esinf1+e2+2ecosf. For x=0, y=±p,
that is, f=f¯=±π/2,
it is(5)vx0=v¯sinϕ¯=GMp,vy0=v¯cosϕ¯=∓eGMp.Figure 1
Essential features of the unperturbed Newtonian hyperbola.P is the
position of a test particle at time t with respect to the body located at the origin
of the chosen reference frame. The smallest distance is rmin=a(e−1).
The angle between r and v is ϕ.
M here is not to be confused with the central body of mass M.


## 2. The Gravitomagnetic Force

In this
section, we will deal with the effect of the general relativistic
gravitomagnetic force on the hyperbolic motion of a test particle approaching a
spinning body of massM and angular momentum S.Let us briefly recall that, in the weak-field and
slow-motion linear approximation of general relativity, the off-diagonal
componentsg0i,i=1,2,3 of the space-time metric tensor, related to
the mass-energy currents of the source, induce a gravitomagnetic field Bg [1] by analogy with the magnetic field caused
by moving electric charges in the linear Maxwellian electromagnetism. Far from
an isolated rotating body, the gravitomagnetic field becomes [5](6)Bg=−Gcr3[S−3(S⋅r^)r^],where G is the Newtonian gravitational constant and c is the speed of light in vacuum. It exerts the
noncentral Lorentz-like acceleration [1](7)AGM=−2cv×Bgupon a test particle moving with
velocity v.
For ordinary astronomical bodies like, for example, the Earth and the Sun, AGM is many orders of magnitude smaller than the
Newtonian monopole AN=GM/r2,
so that it can be considered as a small perturbation.The action of (7) in the case of unperturbed close
orbits, giving rise to, for example, the Lense-Thirring precession of the ellipse
of a test particle [6–8], has been the subject of intense activity, both
theoretically and observationally, in recent times [9].Here, we will consider as reference path a Newtonian hyperbolic trajectory. In order to work out the effects of the
gravitomagnetic field on it, we will numerically integrate the equations of
motion in Cartesian rectangular coordinates [10]
for some particular orbital geometries over a time span including the epoch of
closest approach to M which is assumed located at the origin of the
coordinate system.


### 2.1. Qualitative Features for Equatorial and Polar Osculating Orbits

First, we will
consider a trajectory lying in the equatorial plane of the rotating body for
the cases of corotation (Figure2) and counterrotation (Figure 3) of the
particle radius vector r with respect to the diurnal rotation of M whose spin S is assumed to be directed along the positive z axis (anticlockwise diurnal rotation).
Just for illustrative purposes, we
assumed the Earth as source of the gravitational field and rescaled its
gravitomagnetic force by a factor 1010 in such a way to still keeping the condition AGM/AN<1 valid over the entire orbit. It turns out that
the perturbed trajectory remains confined in the equatorial plane of the central
body; for corotation, the path is deflected outward with respect to the
unperturbed hyperbola, while for counterrotation,
it is deflected inward. Indeed, for equatorial orbits, AGM is entirely in-plane because Bg is directed along the negative z axis; for corotating particles, it is radially
directed outward at the point of closest approach and decreases the
gravitational pull felt by the orbiter, while for counterrotating probes, it is
radially directed inward at the pericenter and
increases the net gravitational acceleration. The flyby epoch is left almost
unaffected. By taking the difference between the integrated perturbed and
unperturbed orbits sharing the same initial conditions, it can be shown that in
the corotating case the radial velocity, which is one of the direct observables
in real planet-spacecraft close encounters, experiences an increase with
respect to the unperturbed one just around the flyby epoch, while the radial
components of v∞i and v∞o are left unaffected. In the counterrotating
case vr decreases at the closest approach. Concerning
the body-centric range r,
it turns out that it suddenly increases (decreases) around the flyby epoch for
the co-(counter-)rotating case and remains about at that level also after the
flyby.Figure 2
Effect of the gravitomagnetic
force on the hyperbolic motion of a test particle around an astronomical
rotating body located at the origin of the depicted frame. The body spinS is directed along the positive z axis, that is, outside the figure. Red line:
unperturbed hyperbola. Blue dashed line: perturbed orbit. For illustrative
purposes, we choose the Earth as central body and rescaled the magnitude of its
gravitomagnetic force by 1010 so that AGM/AN=0.4 at perigee. We adopted the initial conditions x0=0,y0=−p=−a(e2−1),z0=0,vx0>0,vy0>0,vz0=0;
the particle moves in the equatorial plane of the spinning Earth in such a way
that the radius vector rotates in the same sense with respect to the Earth,
that is, anticlockwise. We used a=8,493.326 km, e=1.81.
The perturbed orbit is deflected outward with respect to the unperturbed
one.Figure 3
Effect of the gravitomagnetic force on the
hyperbolic motion of a test particle around an astronomical rotating body
located at the origin of the depicted frame. The body spinS is directed along the positive z axis, that is, outside the figure. Red line:
unperturbed hyperbola. Blue dashed line: perturbed orbit. For illustrative
purposes, we choose the Earth as central body and rescaled the magnitude of its
gravitomagnetic force by 1010 so that AGM/AN=0.4 at perigee. We adopted the initial conditions x0=0,y0=p=a(e2−1),z0=0,vx0>0,vy0<0,vz0=0;
the particle moves in the equatorial plane of the spinning Earth in such a way
that the radius vector rotates in the opposite sense with respect to the Earth,
that is, clockwise. We used a=8,493.326 km, e=1.81.
The perturbed orbit is deflected inward with respect to the unperturbed
one.Let us, now, consider the case in which the
unperturbed hyperbola entirely lies in an azimuthal plane, for example, the{yz} plane, containing the spin S of the central body. Now, since v is contained in the same plane of Bg,
the gravitomagnetic acceleration is out-of-plane, so that it can be expected
that the perturbed trajectory will be displaced along the x axis. This fact is confirmed by a numerical
integration shown in Figures 4 and 5   from which the displacement of the orbit
perpendicularly to the initial osculating plane is apparent.Figure 4
Effect of the gravitomagnetic
force on the hyperbolic motion of a test particle around an astronomical
rotating body located at the origin of the depicted frame. The body spinS is directed along the positive z axis, that is, outside the figure. For
illustrative purposes, we choose the Earth as central body and rescaled the
magnitude of its gravitomagnetic force by 1010 so that AGM/AN=0.4 at perigee. We adopted the initial conditions x0=0,y0=−p=−a(e2−1),z0=0,vx0=0,vy0>0,vz0>0 to have the spacecraft initially moving in the
osculating {yz} plane. The perturbed trajectory is displaced
along the x axis on the upper left corner.Figure 5
Effect of the gravitomagnetic force on the hyperbolic motion of a test
particle around an astronomical rotating body located at the origin of the
frame shown with its spin directed along the positivez axis, that is, outside the figure. For
illustrative purposes, we choose the Earth as central body and rescaled the
magnitude of its gravitomagnetic force by 1010 so that AGM/AN=0.4 at perigee. We adopted the initial conditions x0=0,y0=p=a(e2−1),z0=0,vx0=0,vy0<0,vz0>0 to have the spacecraft initially moving in the
osculating {yz} plane. The perturbed trajectory is displaced
along the x axis on the lower right corner.


### 2.2. The Flyby Anomaly: The near Case

Many
spacecrafts launched in interplanetary missions
make use of one or more Earth flybys in order to gain or lose the heliocentric
energy required to reach their far targets (planets, asteroids, comets) without
using huge and expensive amounts of propellant [11, 12]. In the case of GALILEO
(twice), NEAR, Cassini and MESSENGER unexplained variations in v∞ were detected [2, 3]; the largest one was
measured at the close encounter of NEAR with the Earth that occurred in 1998
and amounts to(8)Δv∞=13.46±0.01mmsec−1.The unperturbed hyperbola of
NEAR is depicted in Figure 6.Figure 6
Unperturbed hyperbola of NEAR; its osculating
plane is tilted by 108 deg to the{x,y} plane assumed coincident with the Earth
equator. The starting point is in the right upper corner  (x0>0,y0>0,z0>0). The Earth has to be imagined located at the
origin of the coordinate system with its spin S directed along the positive z axis.Anderson et al.  [3] derived an empirical formula
which satisfactorily fit all the six flyby anomalies measured so far. It is(9)Δv∞v∞=(2ωRc)(cosδi−cosδo)=(3.099×10−6)(cosδi−cosδo),where ω and R are the Earth angular speed and equatorial
radius, respectively, while δi and δo are the ingoing and ongoing geocentric
declinations, respectively. Concerning possible explanations in terms of known
physics, many dynamical effects (tides, atmospheric drag, Earth albedo, solar
wind, terrestrial magnetic field, spin-rotation coupling between
electromagnetic waves and spacecraft, and Earth rotations) were preliminarily
considered by Lämmerzahl et al. [13]; an order-of-magnitude approach was
followed by confronting the magnitude of the accelerations induced by standard
forces considered with the one which is assumed to be responsible of the flyby
anomaly, that is, ≈10−4 m sec−2.
As a consequence, all the effects investigated were discarded. However, we note
that detailed analyses are in order. Indeed, even if some dynamical effect,
standard or not, was found to produce an acceleration with the right order of
magnitude, it might happen that its signature on the observable quantities is
not correct; that is, it may induce, for instance, a decrease of the radial
velocity. Mbelek  [14] suggested that the special relativistic Doppler effect
may explain the formula by Anderson et al. [3]. Among various explanations in
terms of nonconventional physics put forth so far, McCulloch  [15] proposed a
mechanism based on the hypothesis that inertia is due to a form of Unruh
radiation and varies with acceleration due to a Hubble-scale Casimir effect. It
qualitatively reproduces the latitude-dependence of (9) and is quantitatively
in agreement with three of the six measured flybys.Since (9) contains a term including quantities like
the speed of light and the first power of the Earth angular speed which enters
general relativistic gravitomagnetic effects, it seems interesting to apply our
previous results concerning the influence of Bg on hyperbolic orbits to the NEAR flyby. Note  that
the gravitomagnetic force was not modeled in the software used to process the
NEAR data. By using the state vector of NEAR at the flyby epoch (J D Anderson,
private communication to the author, November 2008) referred to a geocentric
equatorial frame {x,y,z},
and (7), it turns out that at the point of closest approach to Earth along the
flyby trajectory(10)AxGM=3.3×10−10msec−2,AyGM=7.5×10−11msec−2,AzGM=−1.7×10−10msec−2,  so that(11)AGM=3.8×10−10msec−2.We will now use a numerical integration of the
equations of motion perturbed by (7). We look at a time span starting from the
flyby epoch and extending in the future after it. The chosen initial conditions
are in Table 1. We will
consider the changes of the velocity along the radial r^,
transverse τ^, and out-of-plane ν^ directions; ν^ is directed along the orbital angular
momentum, while τ^=ν^×r^ is not directed, in general, along v.Table 1
Initial
conditions used for NEAR obtained with the HORIZONS software by NASA, JPL at
1998-Jan-23 07:00:00 coordinate time (CT, defined as the uniform time scale and
independent variable of the ephemerideshttp://ssd.jpl.nasa.gov/?horizons_doc#timesys), that is, 1353 seconds before the flyby.
Reference frame: ICRF/J2000.0. Coordinate system: Earth Mean Equator and
Equinox of Reference Epoch.
x0 (km)y0 (km)z0 (km)vx0 (km sec−1)vy0 (km sec−1)vz0 (km sec−1)4,496.8855949093816,930.47715373354913,199.11503591246–1.712684317202157–8.679677119077454–4.455285829060190The results forΔvr, Δvτ, Δv,
and Δr are shown in Figures
7, 8, 9, and 10,
respectively. They have been obtained by subtracting the unperturbed orbit from
the perturbed one, both numerically integrated with the initial conditions of
Table 1.Figure 7
Change in the
radial velocityΔvr induced by the gravitomagnetic force on NEAR
at the Earth flyby (1998-Jan-23 07:22:56 coordinate time (CT)). It has been
obtained by taking the difference between the integrated perturbed and
unperturbed trajectories sharing the same initial conditions of Table 1.
They have been obtained with the HORIZONS software by NASA JPL at 1998-Jan-23
07:00:00 CT and correspond to an instant 1353 seconds before the flyby.
Reference frame: ICRF/J2000.0. Coordinate system: Earth Mean Equator and
Equinox of Reference Epoch. The maximum effect Δvrmax=−5×10−5 mm sec−1 occurred just at the flyby.Figure 8
Change in the transverse velocityΔvτ induced by the gravitomagnetic force on NEAR
at the Earth flyby (1998-Jan-23 07:22:56 CT). The maximum value reached is of
the order of 10−5 mm sec−1.Figure 9
Change in the speedΔv induced by the gravitomagnetic force on NEAR
at the Earth flyby (1998-Jan-23 07:22:56 CT). The maximum effect Δvmax=2×10−5 mm sec−1 occurred just at the flyby while v∞o is left almost unchanged.Figure 10
Change in the radial distanceΔr induced by the gravitomagnetic force on NEAR
at the Earth flyby (1998-Jan-23 07:22:56 CT). The maximum effect Δrmax=−6×10−2 mm took place after the flyby.The gravitomagnetic force of the Earth
decreased the radial velocity of NEAR by≈10−5 mm sec−1 just at the flyby, while the transverse
velocity was augmented after the flyby up to 10−5 mm sec−1 level; it turns out that the normal velocity
was affected at an even smaller level. The total speed v was increased up to 10−5 mm sec−1,
that is, six orders of magnitude smaller than the observed increment. The
geocentric range of the spacecraft was reduced by about 10−2 mm.


## 2.1. Qualitative Features for Equatorial and Polar Osculating Orbits

First, we will
consider a trajectory lying in the equatorial plane of the rotating body for
the cases of corotation (Figure2) and counterrotation (Figure 3) of the
particle radius vector r with respect to the diurnal rotation of M whose spin S is assumed to be directed along the positive z axis (anticlockwise diurnal rotation).
Just for illustrative purposes, we
assumed the Earth as source of the gravitational field and rescaled its
gravitomagnetic force by a factor 1010 in such a way to still keeping the condition AGM/AN<1 valid over the entire orbit. It turns out that
the perturbed trajectory remains confined in the equatorial plane of the central
body; for corotation, the path is deflected outward with respect to the
unperturbed hyperbola, while for counterrotation,
it is deflected inward. Indeed, for equatorial orbits, AGM is entirely in-plane because Bg is directed along the negative z axis; for corotating particles, it is radially
directed outward at the point of closest approach and decreases the
gravitational pull felt by the orbiter, while for counterrotating probes, it is
radially directed inward at the pericenter and
increases the net gravitational acceleration. The flyby epoch is left almost
unaffected. By taking the difference between the integrated perturbed and
unperturbed orbits sharing the same initial conditions, it can be shown that in
the corotating case the radial velocity, which is one of the direct observables
in real planet-spacecraft close encounters, experiences an increase with
respect to the unperturbed one just around the flyby epoch, while the radial
components of v∞i and v∞o are left unaffected. In the counterrotating
case vr decreases at the closest approach. Concerning
the body-centric range r,
it turns out that it suddenly increases (decreases) around the flyby epoch for
the co-(counter-)rotating case and remains about at that level also after the
flyby.Figure 2
Effect of the gravitomagnetic
force on the hyperbolic motion of a test particle around an astronomical
rotating body located at the origin of the depicted frame. The body spinS is directed along the positive z axis, that is, outside the figure. Red line:
unperturbed hyperbola. Blue dashed line: perturbed orbit. For illustrative
purposes, we choose the Earth as central body and rescaled the magnitude of its
gravitomagnetic force by 1010 so that AGM/AN=0.4 at perigee. We adopted the initial conditions x0=0,y0=−p=−a(e2−1),z0=0,vx0>0,vy0>0,vz0=0;
the particle moves in the equatorial plane of the spinning Earth in such a way
that the radius vector rotates in the same sense with respect to the Earth,
that is, anticlockwise. We used a=8,493.326 km, e=1.81.
The perturbed orbit is deflected outward with respect to the unperturbed
one.Figure 3
Effect of the gravitomagnetic force on the
hyperbolic motion of a test particle around an astronomical rotating body
located at the origin of the depicted frame. The body spinS is directed along the positive z axis, that is, outside the figure. Red line:
unperturbed hyperbola. Blue dashed line: perturbed orbit. For illustrative
purposes, we choose the Earth as central body and rescaled the magnitude of its
gravitomagnetic force by 1010 so that AGM/AN=0.4 at perigee. We adopted the initial conditions x0=0,y0=p=a(e2−1),z0=0,vx0>0,vy0<0,vz0=0;
the particle moves in the equatorial plane of the spinning Earth in such a way
that the radius vector rotates in the opposite sense with respect to the Earth,
that is, clockwise. We used a=8,493.326 km, e=1.81.
The perturbed orbit is deflected inward with respect to the unperturbed
one.Let us, now, consider the case in which the
unperturbed hyperbola entirely lies in an azimuthal plane, for example, the{yz} plane, containing the spin S of the central body. Now, since v is contained in the same plane of Bg,
the gravitomagnetic acceleration is out-of-plane, so that it can be expected
that the perturbed trajectory will be displaced along the x axis. This fact is confirmed by a numerical
integration shown in Figures 4 and 5   from which the displacement of the orbit
perpendicularly to the initial osculating plane is apparent.Figure 4
Effect of the gravitomagnetic
force on the hyperbolic motion of a test particle around an astronomical
rotating body located at the origin of the depicted frame. The body spinS is directed along the positive z axis, that is, outside the figure. For
illustrative purposes, we choose the Earth as central body and rescaled the
magnitude of its gravitomagnetic force by 1010 so that AGM/AN=0.4 at perigee. We adopted the initial conditions x0=0,y0=−p=−a(e2−1),z0=0,vx0=0,vy0>0,vz0>0 to have the spacecraft initially moving in the
osculating {yz} plane. The perturbed trajectory is displaced
along the x axis on the upper left corner.Figure 5
Effect of the gravitomagnetic force on the hyperbolic motion of a test
particle around an astronomical rotating body located at the origin of the
frame shown with its spin directed along the positivez axis, that is, outside the figure. For
illustrative purposes, we choose the Earth as central body and rescaled the
magnitude of its gravitomagnetic force by 1010 so that AGM/AN=0.4 at perigee. We adopted the initial conditions x0=0,y0=p=a(e2−1),z0=0,vx0=0,vy0<0,vz0>0 to have the spacecraft initially moving in the
osculating {yz} plane. The perturbed trajectory is displaced
along the x axis on the lower right corner.


## 2.2. The Flyby Anomaly: The near Case

Many
spacecrafts launched in interplanetary missions
make use of one or more Earth flybys in order to gain or lose the heliocentric
energy required to reach their far targets (planets, asteroids, comets) without
using huge and expensive amounts of propellant [11, 12]. In the case of GALILEO
(twice), NEAR, Cassini and MESSENGER unexplained variations in v∞ were detected [2, 3]; the largest one was
measured at the close encounter of NEAR with the Earth that occurred in 1998
and amounts to(8)Δv∞=13.46±0.01mmsec−1.The unperturbed hyperbola of
NEAR is depicted in Figure 6.Figure 6
Unperturbed hyperbola of NEAR; its osculating
plane is tilted by 108 deg to the{x,y} plane assumed coincident with the Earth
equator. The starting point is in the right upper corner  (x0>0,y0>0,z0>0). The Earth has to be imagined located at the
origin of the coordinate system with its spin S directed along the positive z axis.Anderson et al.  [3] derived an empirical formula
which satisfactorily fit all the six flyby anomalies measured so far. It is(9)Δv∞v∞=(2ωRc)(cosδi−cosδo)=(3.099×10−6)(cosδi−cosδo),where ω and R are the Earth angular speed and equatorial
radius, respectively, while δi and δo are the ingoing and ongoing geocentric
declinations, respectively. Concerning possible explanations in terms of known
physics, many dynamical effects (tides, atmospheric drag, Earth albedo, solar
wind, terrestrial magnetic field, spin-rotation coupling between
electromagnetic waves and spacecraft, and Earth rotations) were preliminarily
considered by Lämmerzahl et al. [13]; an order-of-magnitude approach was
followed by confronting the magnitude of the accelerations induced by standard
forces considered with the one which is assumed to be responsible of the flyby
anomaly, that is, ≈10−4 m sec−2.
As a consequence, all the effects investigated were discarded. However, we note
that detailed analyses are in order. Indeed, even if some dynamical effect,
standard or not, was found to produce an acceleration with the right order of
magnitude, it might happen that its signature on the observable quantities is
not correct; that is, it may induce, for instance, a decrease of the radial
velocity. Mbelek  [14] suggested that the special relativistic Doppler effect
may explain the formula by Anderson et al. [3]. Among various explanations in
terms of nonconventional physics put forth so far, McCulloch  [15] proposed a
mechanism based on the hypothesis that inertia is due to a form of Unruh
radiation and varies with acceleration due to a Hubble-scale Casimir effect. It
qualitatively reproduces the latitude-dependence of (9) and is quantitatively
in agreement with three of the six measured flybys.Since (9) contains a term including quantities like
the speed of light and the first power of the Earth angular speed which enters
general relativistic gravitomagnetic effects, it seems interesting to apply our
previous results concerning the influence of Bg on hyperbolic orbits to the NEAR flyby. Note  that
the gravitomagnetic force was not modeled in the software used to process the
NEAR data. By using the state vector of NEAR at the flyby epoch (J D Anderson,
private communication to the author, November 2008) referred to a geocentric
equatorial frame {x,y,z},
and (7), it turns out that at the point of closest approach to Earth along the
flyby trajectory(10)AxGM=3.3×10−10msec−2,AyGM=7.5×10−11msec−2,AzGM=−1.7×10−10msec−2,  so that(11)AGM=3.8×10−10msec−2.We will now use a numerical integration of the
equations of motion perturbed by (7). We look at a time span starting from the
flyby epoch and extending in the future after it. The chosen initial conditions
are in Table 1. We will
consider the changes of the velocity along the radial r^,
transverse τ^, and out-of-plane ν^ directions; ν^ is directed along the orbital angular
momentum, while τ^=ν^×r^ is not directed, in general, along v.Table 1
Initial
conditions used for NEAR obtained with the HORIZONS software by NASA, JPL at
1998-Jan-23 07:00:00 coordinate time (CT, defined as the uniform time scale and
independent variable of the ephemerideshttp://ssd.jpl.nasa.gov/?horizons_doc#timesys), that is, 1353 seconds before the flyby.
Reference frame: ICRF/J2000.0. Coordinate system: Earth Mean Equator and
Equinox of Reference Epoch.
x0 (km)y0 (km)z0 (km)vx0 (km sec−1)vy0 (km sec−1)vz0 (km sec−1)4,496.8855949093816,930.47715373354913,199.11503591246–1.712684317202157–8.679677119077454–4.455285829060190The results forΔvr, Δvτ, Δv,
and Δr are shown in Figures
7, 8, 9, and 10,
respectively. They have been obtained by subtracting the unperturbed orbit from
the perturbed one, both numerically integrated with the initial conditions of
Table 1.Figure 7
Change in the
radial velocityΔvr induced by the gravitomagnetic force on NEAR
at the Earth flyby (1998-Jan-23 07:22:56 coordinate time (CT)). It has been
obtained by taking the difference between the integrated perturbed and
unperturbed trajectories sharing the same initial conditions of Table 1.
They have been obtained with the HORIZONS software by NASA JPL at 1998-Jan-23
07:00:00 CT and correspond to an instant 1353 seconds before the flyby.
Reference frame: ICRF/J2000.0. Coordinate system: Earth Mean Equator and
Equinox of Reference Epoch. The maximum effect Δvrmax=−5×10−5 mm sec−1 occurred just at the flyby.Figure 8
Change in the transverse velocityΔvτ induced by the gravitomagnetic force on NEAR
at the Earth flyby (1998-Jan-23 07:22:56 CT). The maximum value reached is of
the order of 10−5 mm sec−1.Figure 9
Change in the speedΔv induced by the gravitomagnetic force on NEAR
at the Earth flyby (1998-Jan-23 07:22:56 CT). The maximum effect Δvmax=2×10−5 mm sec−1 occurred just at the flyby while v∞o is left almost unchanged.Figure 10
Change in the radial distanceΔr induced by the gravitomagnetic force on NEAR
at the Earth flyby (1998-Jan-23 07:22:56 CT). The maximum effect Δrmax=−6×10−2 mm took place after the flyby.The gravitomagnetic force of the Earth
decreased the radial velocity of NEAR by≈10−5 mm sec−1 just at the flyby, while the transverse
velocity was augmented after the flyby up to 10−5 mm sec−1 level; it turns out that the normal velocity
was affected at an even smaller level. The total speed v was increased up to 10−5 mm sec−1,
that is, six orders of magnitude smaller than the observed increment. The
geocentric range of the spacecraft was reduced by about 10−2 mm.


## 3. The Gravitoelectric Force

In the
post-Newtonian approximation of order𝒪(c−2),
the acceleration induced by the gravitoelectric component of the field of a
static mass M is, in standard isotropic coordinates, [10](12)AGE=GMc2[4GMr4r−v2r3r+4(r⋅v)r3v].Concerning the influence of (12)
on the hyperbolic motion, it has only radial and transverse components, so that
no departures from the osculating plane occur. From
Figure 11, it turns out that the test particle is deflected inward with
respect to the unperturbed hyperbola.Figure 11
Effect of the gravitoelectric force on the
hyperbolic motion of a test particle around an astronomical body located at the
origin of the frame shown. Red line: unperturbed hyperbola. Blue dashed line:
perturbed orbit. For illustrative purposes, we choose the Earth as central body
and rescaled the magnitude of its gravitoelectric force by108 so that AGE/AN=0.1 at perigee. We adopted the initial conditions x0=0,y0=−p=−a(e2−1),z0=0,vx0>0,vy0>0,vz0=0.
We used a=8,493.326 km, e=1.81.
The perturbed orbit is deflected inward with respect to the unperturbed one.The gravitoelectric acceleration experienced by NEAR
at the point of closest approach to earth along the flyby trajectory was(13)AxGE=9.5×10−10msec−2,AyGE=−5.26×10−9msec−2,AzGE=3.42×10−9msec−2,  so that(14)AGE=6.35×10−9msec−2.The impact of the gravitoelectric force onvr, vτ, v,
and r of NEAR at its flyby are depicted in Figures
12, 13, 14, and 15, respectively. Their patterns are quite different from
the gravitomagnetic ones; the  outgoing asymptotic velocity is changed by an
amount of the order of 10−3 mm sec−1.
(Contrary to the  gravitomagnetic one, the gravitoelectric
force was
modeled in the software used for processing the data.) The maximum variations of the range rate, the transverse velocity, the speed,
and the range are of the order of 10−2 mm sec−1 and 10 mm, respectively.Figure 12
Change in the radial  velocityΔvr induced by the gravitoelectric force on NEAR
at the Earth flyby (1998-Jan-23 07:22:56 CT). It is the difference between the
integrated perturbed and unperturbed trajectories sharing the same initial
conditions of Table 1. They have been obtained with the HORIZONS
software by NASA JPL at 1998-Jan-23 07:00:00 CT and correspond to an instant
1353 seconds before the flyby. Reference frame:
ICRF/J2000.0. Coordinate system: Earth Mean Equator and Equinox of Reference
Epoch.Figure 13
Change in the transverse velocityΔvτ induced by the gravitoelectric force on NEAR
at the Earth flyby (1998-Jan-23 07:22:56 CT).Figure 14
Change in the speedΔv induced by the gravitoelectric force on NEAR
at the Earth flyby (1998-Jan-23 07:22:56 CT).Figure 15
Change in the radial
distanceΔr induced by the gravitoelectric force on NEAR
at the Earth flyby (1998-Jan-23 07:22:56 CT).


## 4. Discussions and Conclusions

We investigated
qualitatively and quantitatively the impact of general relativity, in its
weak-field and slow-motion approximation, on unbound hyperbolic orbits around a
massive spinning body. We considered both the gravitomagnetic and the
gravitoelectric terms; the first one is responsible of the Lense-Thirring
precessions of elliptic orbits, while the second one causes the well-known
Einstein precession of the perihelion of Mercury of
43.98 arcseccty−1.
The gravitomagnetic force deflects an equatorial trajectory inward or outward
with respect to the unperturbed hyperbola according to the mutual orientation
of the orbital angular momentum L of the particle with respect to the spin S of the central body. For osculating orbits
lying in a plane which contains S, there is also a displacement in the
out-of-plane direction. The gravitoelectric force is not sensitive to the L−S orientation and deflects the trajectory inward
with respect to the unperturbed hyperbola.We applied our results to the flyby anomaly
experienced by the NEAR spacecraft at its close encounter with the Earth on
January 1998 when its asymptotic outgoing velocityv∞o was found larger than the ingoing one by 13.46±0.01 mm sec−1;
contrary to the gravitoelectric force, the gravitomagnetic one was not modeled
in the software used to process the NEAR data. From numerical integrations of
the perturbed equations of motion in a geocentric equatorial frame with
rectangular Cartesian coordinates over a time span
extending in the future after the flyby epoch, we quantitatively investigated
the changes in the radial and transverse components of the
velocities vr and vτ,
the speed v,
and the range r of NEAR induced by the general relativistic
gravitoelectromagnetic forces. Concerning the range, its variations are at the 10−2 level for the gravitomagnetic force and 101 mm level for the gravitoelectric one. The
radial and transverse velocities and the speed are affected at 10−2(gravitoelectric) to 10−5(gravitomagnetic) mm sec−1  levels.



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*Source: 807695-2009-02-16.xml*