Semi-major axis changes due to Schwarzschild term
The first-order Gaussian perturbations can be used for deriving periodic perturbations and secular variations of the Keplerian elements. However, the constant offsets which are the integration constants cannot be derived based on Gaussian perturbation equations. Therefore, we will begin with the classical celestial mechanics equations for deriving the offsets of Keplerian parameters.
Considering Kepler’s law \(GM=a^3n^2\), where n is the mean motion, and assuming that GM is constant whereas n and a are subject to change due to perturbing accelerations, the effect of the relative change of semi-major axis a and mean motion n equals:
$$\begin{aligned} \frac{\varDelta n}{n} = -\frac{3}{2} \frac{\varDelta a}{a}. \end{aligned}$$
(3)
From the first-order orbit perturbations, any acceleration acting on a satellite in the radial direction \(R_0\) changes the mean motion n by (Beutler 2004):
$$\begin{aligned} \varDelta n = \frac{2}{na} R_0. \end{aligned}$$
(4)
Considering Eqs. (4) and (3), we may conclude that any radial acceleration \(R_0\) changes the semi-major axis by:
$$\begin{aligned} \varDelta a = -\frac{4}{3} \frac{a^3}{GM} R_0 \end{aligned}$$
(5)
which will be useful not only for the Schwarzschild effect but also for studying the impact of the Lense–Thirring and de Sitter effects on satellite orbits.
The Schwarzschild acceleration acting on a satellite in a circular orbit introduces a constant radial acceleration:
$$\begin{aligned} R_{Sch}=\mathbf {e}_{R} \cdot \varDelta \ddot{\mathbf {r}}_{\mathrm {Sch}}=3 \frac{(G M)^{2}}{c^{2} a^{3}}. \end{aligned}$$
(6)
Substituting \(R_0\) in Eq. (5) by the Schwarzschild term \(R_{Sch}\) from Eq. 6, one obtains the final equation of the change of the osculating semi-major axis for circular orbits due to the Schwarzschild term:
$$\begin{aligned} \varDelta a_{Sch} = -4 \frac{GM}{c^2} = - 17.74 \, mm . \end{aligned}$$
(7)
The change of the semi-major axis is thus independent of the satellite height for all circular orbits. Interestingly, the change of the semi-major axis due to the Schwarzschild effect is exactly twice larger than the Schwarzschild radius defining the event horizon for the Earth of a Schwarzschild black hole (Schwarzschild 1916).
Here, we assume that the GM value is the same in the classical theory of gravity as well as in GR. The constant offset would be different if we allow the gravitational constant G to change, as considered by Hugentobler (2008), who assumed that the mean motion n is unchanged, whereas the GM and a differ between Newtonian and post-Newtonian motion of Earth satellites. However, this assumption does not hold as we expect changes of the mean motion due to GR.
For noncircular orbits, the change of the semi-major axis consists of two parts—a constant offset and periodic variations:
$$\begin{aligned} \varDelta a_{Sch}^{Tot}= \varDelta a_{Sch} + \varDelta a _{Sch}. \end{aligned}$$
(8)
The periodical component can be derived on the basis of Gaussian first-order perturbations on the basis of radial and along-track accelerations and can be expressed as a function of the argument of latitude u:
$$\begin{aligned} \varDelta a _{Sch}= -14 \frac{ GM e }{c^2 (1-e^2)^2} \cos u , \end{aligned}$$
(9)
or, to a higher-order:
$$\begin{aligned} \varDelta a _{Sch}= \frac{ GM }{c^2 (1-e^2)^2} [(-14-6e^2)e \cos u -5e^2\cos 2u], \end{aligned}$$
(10)
which agrees with the formula derived by Kopeikin et al. (2011) for extragalactic pulsars, when neglecting the contribution of the second celestial object. The sum of the constant offset and periodical variations describes the evolution of osculating, that is short-term, semi-major axis variations. It is worth noting that the Schwarzschild acceleration changes as \(r^{-3}\) with the height (when assuming \(r=a\)), whereas the Newton’s acceleration changes as \(r^{-2}\), hence both effects can be easily separated for elliptical orbits. In contrast, for circular orbits, the Schwarzschild effect can be explained by a slight modification of the GM value. For elliptical orbits, the change as \(r^{-3}\) with the height is only an approximation because Schwarzschild introduced the along-track accelerations and cannot be considered as a central force.
Figure 7 shows the change of the semi-major axis of E14 as a function of time derived from the numerical simulations and the first-order perturbations based on Eqs. (8) and 10. The discrepancies between the simulated and approximated values are caused by \(O(c^{-4})\) limitation and neglecting the higher-order perturbations of Keplerian elements, assuming that all other elements are constant. In numerical simulations, we allow all Keplerian elements to change simultaneously. The discrepancies disappear for circular orbits, because then, only the constant offset of the semi-major axis of \(- 17.74 \, mm \) occurs with no periodic variations dependent on e.
For Galileo E14 (and for E18, which is not shown here because the results are the same), the total change of a from the simulation is \(-29.02\) mm in perigee and \(-8.65\) mm in apogee, which gives the difference more than 20 mm, thus, fully detectable using current techniques of GNSS precise orbit determination (Bury et al. 2020). Please note that the orbit becomes smaller with the largest magnitude in the perigee and the smallest in the apogee due to the Schwarzschild term. For high-eccentric Earth satellites with a large value of e, such as the Laser Ranging Equipment (LRE) with \(e=0.73\), the offset of a could even be positive in the apogee.
Eccentricity changes due to Schwarzschild term
The eccentricity changes due to the Schwarzschild term described by Gaussian first-order perturbations. After integrating Eq. (2) over t-dependent values with \(u=n(t-t_0)=n \varDelta t\) where \(t_0\) is the reference epoch, the final equation for the eccentricity changes includes only periodical variations and reads as follows:
$$\begin{aligned} \varDelta e_{Sch}= -3\frac{GM}{c^2 a (1-e^2)}\cos u \end{aligned}$$
(11)
and when also considering higher-order \(e^2\) and \(e^4\) contributions (Kopeikin and Potapov 1994):
$$\begin{aligned} \varDelta e_{Sch}= -\frac{GM}{c^2 a e (1-e^2)}[(3+7e^2)e\cos u+\frac{5}{2}e^2 \cos 2u]. \end{aligned}$$
(12)
Figure 8 shows that the first-order approximations from Eq. (12) describe the majority of orbital eccentricity changes, however, for elliptical orbits of E14, the approximations are overestimated in apogee, whereas for the circular orbit of E08, a small shift in phase occurs with a proper value of the overall amplitude. The periodical changes of the orbital eccentricity are similar, even though the initial value of the E14 eccentricity was \(e=0.1612\) and for E08 \(e=0.0001\). The eccentricity changes in the perigee and apogee have opposite signs for both the eccentric and circular orbits.
For Galileo satellites in eccentric orbits, the change of the semi-major axis is negative in the perigee and the apogee, with the maximum change in the perigee. However, the size of the orbit is always reduced. The change of eccentricity is negative in the perigee, which implies a more circular orbit as the perigee goes higher, and positive in the apogee, which implies a more eccentric orbit, as the apogee also goes higher from the geocenter perspective. The Schwarzschild term changes the shape and the size of the orbit instantaneously in opposite directions. When using the chosen parameterization and the nonrotating geocentric frame, the Schwarzschild effect translates the circular orbits and elliptical orbits into irregular curves, as shown in Fig. 9.
Revolution period and velocity
The revolution period T of an Earth-orbiting satellite can be described as:
$$\begin{aligned} T= \frac{2 \pi }{n}, \end{aligned}$$
(13)
where n denotes the mean motion. Estimating the mean motion for each epoch, one obtains the ‘osculating’ mean motion and the ‘osculating’ revolution period, whereas, from the first-order perturbations based on Eq. (4), one obtains the constant change of n:
$$\begin{aligned} n_{Sch}=\left( \frac{GM}{a^3} \right) ^{1/2} \left( 1+\frac{6 GM}{c^2a} \right) \end{aligned}$$
(14)
and when considering higher-order \(e^2\) and \(e^4\) contributions:
$$\begin{aligned} n_{Sch}=\left( \frac{GM}{a^3} \right) ^{1/2} \left( 1+\frac{GM}{2c^2a(1-e^2)^2}[12-21e^2-6e^4] \right) . \end{aligned}$$
(15)
The mean motion change translates into a change of \(\varDelta T=-44.552\) \(\upmu s\) of the revolution period with a change in a range between \(-72.449\) and \(-21.569\) \(\upmu s\) for Galileo E14, see Fig. 10. This means that the revolution period under the Schwarzschild acceleration is shorter than under the Keplerian motion with the maximum change in the perigee and the minimum change in the apogee.
According to the second Kepler’s law, the angular momentum and the areal velocity A are constant over time in the solution of the two-body problem. The areal velocity can be defined as
$$\begin{aligned} A= \frac{1}{2} |\mathbf {h}| = \frac{1}{2} | \mathbf {r} \times \dot{\mathbf {r}}|, \end{aligned}$$
(16)
where \(\mathbf {h}\) is the angular momentum of a satellite.
The length of the angular momentum vector, as well as the areal velocity vary in time. When considering the mean term, we obtain:
$$\begin{aligned} |\mathbf {h}_{Sch}|=\left[ GM a (1-e^2) \right] ^{1/2}\left[ 1-\frac{GM}{2c^2a} + \frac{4GM}{c^2a(1-e^2)} \right] . \end{aligned}$$
(17)
Figure 10 shows a comparison between the ‘osculating’ areal velocity derived from numerical simulations and the mean change based on Eq. (17). During the satellite revolution, the position and velocity vectors change their lengths, as well as the mutual orientation; however, the temporal variations of the product \(| \mathbf {r} \times \dot{\mathbf {r}}|\) are regular. For circular orbits, A is changed by a constant offset described by Eq. 17, but the angular momentum and energy are conserved. For elliptical orbits, the second Kepler’s law does not hold, and the temporal A variations are similar to those of the temporal a variations.
The Schwarzschild effect acts as a \(r^{-3}\) term for circular orbits, which is a modification to the Newtonian \(r^{-2}\) central force. Kepler’s second law is equivalent to the conservation of angular momentum, and angular momentum is conserved in central forces. Thus, for circular orbits, Kepler’s second law is hold. For noncircular orbits, the Schwarzschild effect has the radial and along-track components (see Fig. 1), thus cannot be considered a central force. The angular momentum shows periodic variations and the Kepler’s second law does not hold when considering short-term variations for noncircular orbits. No secular term is included in the angular momentum changes, thus, over long periods, the angular momentum and energy are conserved, however, they vary for elliptical orbits over short periods.
The temporal changes of the length of the velocity vector also depend on the variable orientation of the perigee which results in a complicated non-periodic curve from Fig. 10c for the nonrotating reference systems as in the case of this study.
Perigee changes due to Schwarzschild term
The argument of perigee \(\omega \) is the only angular Keplerian element describing the orientation of the orbit that changes due to the Schwarzschild term. To observe the secular drift of \(\omega \) properly, an elliptical orbit is needed because for circular orbits \(\omega \) is undefined.
The approximated value of the secular drift of perigee was derived by Einstein and equaled to:
$$\begin{aligned} \varDelta \omega _{Sch}=\frac{3GM}{c^2a(1-e^2)}n \varDelta t . \end{aligned}$$
(18)
The component \(n \varDelta t = 2 \pi \frac{\varDelta t}{T} \) contains the multiple of the full angle \(2 \pi \) in the radian measure. Kopeikin (2020) derived equations for the 1PN and 2PN precession of the orbital pericenter
$$\begin{aligned} \varDelta \omega _{Sch}=\frac{3GM}{c^2a(1-e^2)} n \varDelta t \left[ 1+\frac{3GM}{4c^2a(1-e^2)}-\frac{GM}{4c^2a} \right] . \end{aligned}$$
(19)
The equation gives a difference of \(8.2 \cdot 10^{-11}\) with respect to the Einstein’s equation. Figure 11 compares the secular drift of perigee derived from the Einstein’s term, first-order approximation, and the calculated osculating argument of perigee from the numerical simulation. The osculating argument of perigee is subject to periodic variations because of the temporal variations of the semi-major axis, eccentricity, and the revolution period, which introduce small changes of these quantities in the same epochs when the orientation of the lines of apsides changes.
Interestingly, the secular rate of perigee assumes similar values for the elliptical orbit of E14 and the circular orbit of E08, see Fig. 11. For E14, the rate is 0.6327 and 0.5825 mas over one revolution period for E14 and E08, respectively, which corresponds to a change of 1.2034 and 1.0471 mas after 24 h (please note that the revolution periods for E14 and E08 are different). The perigee change of E14 due to the Schwarzschild effect over one day is 163.2 mm, which is fully detectable when using the Galileo orbit determination techniques with the accuracy at several millimeter level. However, for near-circular orbits, the argument of perigee is hardly detectable, which leads to substantial formal errors of the determined parameter. For Galileo E14, the mean formal error of perigee determination is 0.2232 mas from 1-day solutions, whereas for E08, the perigee error is 95.4684 mas (Bury et al. 2020). Thus, the formal error of E14 perigee is almost 500 more accurate than the formal error of E08 perigee. For Galileo E14, the mean formal error of perigee determination after 1 day is over 5 times smaller than the expected drift due to the Schwarzschild effect. Thus, perigee observations for eccentric Galileo can be used for the verification of GR effects. For deriving GR effects based on short arcs, the periodical variations of the perigee should also be considered, because the amplitudes have a similar order of magnitude to the observed rate.
For E14, the perigee derived from numerical simulation is also about 500 times more accurate than that for E08 due to the ambiguities in the perigee realization in near-circular orbits, which causes differences in Fig. 11 between the theoretical and simulated values for E08. Therefore, the perigee observations of eccentric orbits are much more suited for the verification of the Schwarzschild effect than near-circular orbits.
Iorio (2020) derived a secular drift of the perigee up to the second-order with periodical terms. A modified version of Iorio (2020)’s equation 23, which is multiplied by missing coefficient \(n^{-3}\), reads as:
$$\begin{aligned} \begin{aligned} \varDelta \omega _{Sch}&=\frac{(GM)^2 [5(23+20e^2-4e^4)]}{2n^2c^4a^2(1-e^2)^3}\\&\quad +\frac{(GM)^2 6e[(34+26e^2)\cos u + 15e \cos (2u)] }{2n^2c^4a^2(1-e^2)^3} + O(c^{-4}). \end{aligned} \end{aligned}$$
(20)
A comparison between a modified version of Iorio’s equation and Einstein’s term for E14 is shown in Fig. 11a. For the near-circular orbit of E08, the equation does not describe well the simulated perturbations. The periodic variations can only be slightly better described when based on Iorio (2020) than the static Einstein’s term. The equation proposed by Iorio (2020) cannot be used for elliptical orbits, because it causes issues with the proper description of the secular drift, which is different than that from Eq. (19).
We can also derive, based on Gaussian equations, a more concise version of the first-order perturbations that includes the periodical perturbations:
$$\begin{aligned} \varDelta \omega _{Sch}=\frac{3GM}{c^2a(1-e^2)} \left( n \varDelta t+ \sin 2u \right) . \end{aligned}$$
(21)
The first-order perturbations are shown in Fig. 11b and compared to the perigee drift of Galileo E08 in a circular orbit. The periodic variations are well captured; however, a small shift in the rate occurs due to the uncertainties in the determination of the perigee for near-circular orbits. Nevertheless, the simple first-order perturbation equation describes well the short-term perturbations of the perigee.