The antenna phase center motion effect in spacebased experiments for fundamental physics and astronomy
Abstract
We consider the effect of phase center motion of mechanically steerable highgain parabolic antennas for groundbased and spacecraftmounted antennas. For spacecrafts on highly elliptic Earth orbits the magnitude of the effect is as large as several mm/s in terms of the required velocity correction, both for groundbased and spacecraftmounted antennas. We illustrate this with real data from the RadioAstron spacecraft and also provide results of our simulations for the concept of a possible followup space very long baseline radio astronomy mission. We also consider a specific configuration of satellite communication links, with simultaneously operating oneway down link and twoway loop link, pioneered by the Gravity Probe A experiment. We find that this configuration provides for complete compensation of the phase center motion effect due to the onboard antenna and significant compensation of that due to the ground antenna. This result is important for future spacebased fundamental physics experiments, primarily those concerned with studies of gravity.
keywords:
Antenna phase center motion, highgain antenna, Gravity Probe A, space VLBI, RadioAstron, gravitational redshift1 Introduction
The effect of phase center motion of steerable highgain antennas has long been recognized as an important part of timedelay models of very long baseline radio interferometry (VLBI) wade1970apj () and the source of a significant correction to computed values of observables in spacecraft orbit determination moyer1971techreport (); moyer2005book (). In both cases one is usually concerned with groundbased highgain mechanically steerable dish antennas, several tens of meters in diameter, and the correction is due to a nonzero offset between the rotation axes of the antenna mount. Spacecraftmounted antennas also exhibit this effect but, up until recently, steerable highgain antennas have been mostly used by deep space probes and other planet orbiters, in which case the magnitude of the effect is generally considered negligible due to large distance to the Earth lemoine2001jgr ().
The success of the first spaceVLBI missions, VSOP hirabayashi1998sci () and RadioAstron kardashev2013ar (), utilizing Earthorbiting satellites with highgain mechanically steerable dish antennas, makes it necessary to reconsider the importance of the spacecraft part of the antenna effect, which we attempt with this paper. In spaceVLBI the necessity for a highgain steerable onboard antenna comes from the requirement of high data transmission rate (up to several Gbits/sec) and high eccentricity of the orbit (). Frequency measurements of the data downlink signal are input into the orbit determination and thus should be corrected for the antenna motion effect. We present the results of real data processing from the RadioAstron spacecraft and demonstrate the importance of taking into account the phase center motion both for ground and onboard antenna. We find that the magnitude of the effect is as large as 6 mm/s in terms of the velocity correction and thus must be taken into account not only for precise but even ordinary orbit determination. These demonstrate that these results are relevant also for future spaceVLBI missions, such as kardashev2014ufn () and hong2014aa ().
We also consider a specific configuration of spacecraft radio communication links, namely, that of simultaneously operating oneway downlink and twoway loop link. Such configuration was first used by the Gravity Probe A mission vessotlevine1980prl () for firstorder Doppler and tropospheric frequency shift compensation and is now an integral part of mission concepts for spaceVLBI and spacebased fundamental physics experiments altschul2015asr (); jetzer2017ijmpd (). We show that in the output of this âcompensation schemeâ the antenna phase center effect of the oneway downlink and that of the loop link cancel each other, both for the ground and onboard antenna. The cancellation depends on the distance to the spacecraft, angular velocity of the spacecraft relative to the ground station and the rate of change of the angular velocity. This observation provides for effectively using highgain mechanically steerable dish antennas in highprecision and multidisciplinary spacebased physics missions.
The antenna phase center motion of nonsteerable antennas has recently gained much attention in the problem of precise orbit determination of satellites of global navigation satellite systems (GNSS) springer2017handbook (). In this case the phase center offset and its variation is routinely taken into account not only for groundbased receiving antennas but also for transmitting antennas of GNSS satellites and GNSSreceivers of other satellites. Although this is a similar effect, we do not consider it in this paper.
The outline of the paper is as follows. In Section 2 we summarize the equations for the antenna effect for groundbased steerable dish antennas and present the equations for taking into account the effect for onboard steerable dish antennas. In Section 3 we present the results of data processing from the RadioAstron satellite and demonstrate the importance of taking into account the ground and onboard antenna phase center motion for regular (not precise) orbit determination. In Section 4 we consider the Chinese mission concept of a spaceVLBI mission with two spacecrafts in highly elliptic Earth orbits and show the importance of taking into account the antenna effect for this case. In Section 5 we consider the Gravity Probe A configuration of radio communication links and show that in this case the antenna effects cancel. We conclude with a discussion in Section 6.
2 Basic Equations
The antenna phase center motion effect for mechanically steerable dish antennas is due to motion of the antenna phase center relative to a fixed reference point. In order to define this reference point it is necessary to consider the existing types of antenna mounts. Four types of steerable ground antenna mounts are in common use (Fig. 1). The most common is the altazimuth mount with a fixed primary azimuth axis and the secondary moving axis being the elevation axis (Fig. (a)a). Polar, or equatorial, mount antennas have their polar axes oriented in parallel to the Earthâs rotation axis (Fig. (b)b), which provides for easy tracking of celestial objects with just a single rotation of the polar axis. For XY mount antennas the fixed axis lies in the local horizontal plane, and is orients in the NorthSouth or EastWest direction (Fig. (c)c). For many antennas the primary and secondary axis do not intersect, with the offset varying from a few millimeters to a few meters. The most prominent example is the Green Bank 43meter polar mount radio telescope, for which the offset is 14.9 m.
For antennas with intersecting rotation axes of the mount the antenna reference point is defined as the intersection of its rotation axes. For antennas with nonintersecting axes the reference point is defined as the orthogonal projection of the secondary axis onto the primary one moyer2005book (). In order to obtain equations for the antenna phase center offset we follow the treatments of wade1970apj (), sovers1998rmp () and sovers1987techreport (). Consider an incoming plane wave, propagating in the direction of the dish along its symmetry axis (Fig. 2) (taking into account wavefront curvature gives a negligible correction sovers1998rmp ()) . This wave will reach the antenna phase center (point C) earlier than the Earthfixed antenna reference point R by
Now we define: 1) a unit vector along the primary axis (for altaz and polar mounts in the direction away from the Earth, for XY mounts in the direction from North to South or from East to West, depending on the orientation of the primary axis); 2) unit vector in the direction of the source; 3) axis separation distance . Then one can easily see (sovers1998rmp ()) that:
where the plus or minus sign is chosen to give the direction from R to A. The plus sign is used when the point A comes closer to the source than R. The particular expressions for for different antenna mounts are given in Table 1. The extra delay is obviously the same in sign and magnitude both for reception and transmission.
The corresponding correction to the frequency of the received or transmitted signal is, obviously,
(1) 
where is the angle between vector and , is angle derivative at the moment of signal transmit or receive , is the speed of light, which is also given in Table 1 for each mount.
Antenna mount  Secondary angle  Delay  Frequency correction 

altaz  elevation  
polar  declination  
XY  auxiliary angle 
Let us now consider the case of a spacecraftmounted steerable dish antenna.
The orbit of a spacecraft is described as the trajectory of the motion of a material point corresponding to the center of mass of the satellite. Highgain antenna phase center has a fixed position in spacecraft frame and denoted by . Unit vector in the direction of the source also defined in spacecraft frame, taking into account its orientation in inertial reference system. The delay of the received or transmitted signal equals
The plus sign for delay is used because point A is always closer to the source than point C from spacecraft construction. The corresponding correction to the frequency during tracking the source is
(2) 
where and .
3 Application to RadioAstron
The RadioAstron spacecraft, officially called SpektrR, is an Earthorbiting 10meter radio telescope used primarily for VLBI observations of distant radio sources kardashev2013ar (). The satellite is on a highly eccentric orbit around the Earth, evolving due to the gravitational influence of the Moon, as well as other factors, within a broad range of the orbital parameter space (perigee altitude 1,000–80,000 km, apogee altitude 270,000–370,000 km).
The satellite is equipped with a steerable 1.5 m highgain parabolic antenna, used both for downlink and uplink communications with ground tracking stations (there are also telemetry and control lowgain antennas which we do not consider here). The rotation axes of the spacecraft antenna are offset from each other by several centimeters (Fig. 3), which we ignore here. The intersection of the rotation axes (point A) is offset from the center of mass (point C) by vector:
with the Xcoordinate depending on the amount of fuel in the tank. The mission is served by two tracking stations kardashev2013ar (); ford2014procs (), with their parameters given in Table 2.
Tracking station  Location  Diameter  Mount  Axis offset 

Pushchino  Moscow region, Russia  22 m  altaz  0 m 
Green Bank  West Virginia, USA  43 m  polar  14.94 m 
One would expect the large axis offset of the Green Bank antenna, coupled with high angular velocity of the spacecraft near perigee passage, to produce significant effects on measured Doppler frequency. This is indeed the case, as seen in Fig. 4, which depicts the absolute value of the antenna correction to the downlink Doppler frequency for the period of January 2014. The magnitude of the effect near perigee reaches 6 mm/s, which is 2 orders of magnitude larger than 0.02–0.1 mm/s RMS of a typical 60saveraged velocity residuals zaslavsky2016la ().
Consider now the effect of the 3.43meter axis offset of the onboard antenna from the spacecraft center of mass. Fig. 5 shows the magnitude of the required Doppler correction for the lowperigee period of January 2014. Clearly, the magnitude of the onboard antenna effect is almost as large as that for the ground antenna, reaching 1.1 mm/s on January 19, 2014.
Let us now look on how much important it is to take these effects into account in the problem of orbit determination. The RadioAstron highgain antenna transmits signals at 8.4 and 15 GHz routinely measured at tracking stations. These frequencies make an important part of the input to orbit determination algorithms of the spacecraft zaslavsky2016la (). Frequency measurements are recorded with a time step of 0.04 s during each science observation with the space radio telescope. Now we pick a particular observation during the low perigee epoch, 19/01/2014 03:50–08:00 UTC (experiment code raks04c) and plot the frequency residuals with the antenna effects not taken into account (Fig. (a)a), only the ground antenna effect taken into account (Fig.(b)b) and both ground and onboard antenna phase center motion effects taken into account (Fig. (c)c). Clearly, taking into account the antenna phase center motion is very important for orbit determination.
4 Application to future spaceVLBI missions
Now we consider the importance of taking into account the antenna phase center motion effect for orbit determination of spacecrafts of possible future spaceVLBI missions, taking as an example the Chinese concept detailed in hong2014aa (). This mission concept implies using two spacebased radio telescopes, mounted on satellites with highly elliptic orbits with the perigee and apogee altitudes of 1,200 and 60,000 km, respectively, inclination of 28.5 at first phase of the program. Possible tracking stations include Kaishi, Miyun, Sanya and Shanghai 25 m radio telescope, which can also be functioned as a tracking station after update. For our simulations we take the satellite geometry identical to that of RadioAstron and choose the tracking station of Miyun. Other parameters of the simulation are given in Table 3. The Miyun antenna has a very small axis offset of 0.4 cm, thus we expect the effect due to the ground antenna to be negligible, while the effect due to the onboard antenna is expected to be comparable to that of RadioAstron. We see from Figs. 7 and 8 that it is indeed the case. The magnitude of the effect for the ground antenna is negligible for orbit determination (less than 0.003 mm/s), while the effect due to the satellite antenna is nonnegligible, reaching 3.7 mm/s near perigees.
Apogee (km)  60,000 

Perigee (km)  1,200 
28.5  
15  
220  
0  
01/01/2020 00:00:00 UTC 
It should be noted that although the ground antenna effect is small for ordinary orbit determination, it cannot be considered small if one intends to perform highprecision experiments, such as measuring the gravitational redshift of an onboard atomic clock to a high accuracy, which is a natural extension of the scientific output of a spaceVLBI mission litvinov2017pla (). This case is considered in the next section.
5 Compensation of the antenna phase center motion effect
Consider the phase center motion effect for the 2way mode of the satellite link. For this case the correction should be applied for the up leg of the link at the time of transmission and for the down leg at the time of reception of the signal:
(3) 
where – signal propagation time in up leg, – signal propagation time in down leg. For a simultaneously received signal of the oneway link the correction is:
(4) 
The computed value of the frequency of the oneway signal received from the spacecraft is:
(5) 
where is the gravitational redshift, is kinematic frequency shift, is media correction, is correction due to the phase center motion in oneway link. The computed value of the twoway frequency of the signal transmitted by the tracking station, received and coherently retransmitted by the spacecraft, and finally received by the tracking station is:
(6) 
where is correction due to the phase center motion in twoway link . Now consider the following observable: . This observable is important in that its computed value:
(7) 
contains the contribution of the gravitational frequency shift of the oneway downlink but cancels, up to nonreciprocity of the up and down legs, the nonrelativistic Doppler effect, which is orders of magnitude larger than all other terms (). Thus this observable is very useful in highprecision gravitational experiments, which was first demonstrated by the Gravity Probe A experiment vessotlevine1980prl () and then exploited in the gravitational redshift experiment with RadioAstron litvinov2017pla (). Alternatively, the left side of Eq. (7) can be used for studying the gravitation field at the location of the spacecraft. In the first case one can searches for violations of the Einstein formula for the gravitational frequency shift by comparing the measured value of the observable (6) with the one computed according to general relativity. In the second application one uses the measured value of the observable to determine the gravitational potential at the location of the spacecraft.
Note that, although the observables (5) and (6) include the contributions of the ground and onboard antenna effects, the observable (7) does not:
(8) 
The terms due to the onboard antenna is completely cancelled (neglecting the time delay of the onboard transponder), while the terms due to the ground antenna are reduced, with the residual term proportional to the signal light travel time, , between the spacecraft and the tracking station, and depends on the rate of angular motion, , of the spacecraft relative to the tracking station and its rate of change, . Thus, even though mechanically steerable spacecraft dish antennas are naturally not ideal for highprecision measurements, they can be used together with the compensation scheme of the left side of Eq. (7).
6 Conclusions
The phase center motion effect for steerable highgain antennas gives a significant contribution to computed values of observables in orbit determination and VLBI. Although the effect has been known for quite a while, we showed some novel aspects of it. In particular, we demonstrated, based on our experience in orbit determination of the RadioAstron spacecraft, that not only the effect of the ground parabolic antenna phase center motion needs to be taken into account for regular orbit determination but also the effect due to the onboard antenna is almost as large. This conclusion is important for planning of the future spaceVLBI missions, an example of which we considered in detail.
We also considered the prospects of performing highprecision spacebased experiments with spacecrafts equipped with mechanically steerable highgain parabolic antennas. We showed that a specific kind of experiments, i.e. those aimed at testing the fundamentals of gravity and also those for chronometric studies of the gravitational field, can safely be performed using mechanically steerable antennas by exploiting a specific configuration of the satellite communication links.
Acknowledgements
Research for the RadioAstron gravitational redshift experiment is supported by the Russian Science Foundation grant 171201488. The RadioAstron project is led by the Astro Space Center of the Lebedev Physical Institute of the Russian Academy of Sciences and the Lavochkin Scientific and Production Association under a contract with the Russian Federal Space Agency, in collaboration with partner organizations in Russia and other countries.
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