Effects of Earth’s Oblateness on Black Hole Imaging through Earth–Space and Space–Space Very Long Baseline Interferometry

Earth-based very long baseline interferometry (VLBI) has made rapid advances in imaging black holes. However, due to the limitations imposed on terrestrial VLBI by the Earth’s finite size and turbulent atmosphere, it is imperative to have a space-based component in future VLBI missions. This paper investigates the effect of the Earth’s oblateness, also known as the J 2 effect, on orbiters in Earth–space and space–space VLBI. The paper provides an extensive discussion on how the J 2 effect can directly impact orbit selection for black hole observations and how, through informed choices of orbital parameters, the effect can be used to a mission’s advantage, a fact that has not been addressed in previous space VLBI investigations. We provide a comprehensive study of how the orbital parameters of several current space VLBI proposals will vary specifically due to the J 2 effect. For black hole accretion flow targets of interest, we demonstrate how the J 2 effect leads to a modest increase in shorter-baseline coverage, filling gaps in the (u, v) plane. Subsequently, we construct a simple analytical formalism that allows isolation of the impact of the J 2 effect on the (u, v) plane without requiring computationally intensive orbit propagation simulations. By directly constructing (u, v) coverage using J 2-affected and J 2-invariant equations of motion, we obtain distinct coverage patterns for M87* and Sgr A* that show extremely dense coverage on short baselines as well as long-term orbital stability on longer baselines.


Introduction
The field of radio astronomy, in particular very long baseline interferometry (VLBI), has made rapid advances in the field of black hole imaging, stimulated primarily by the observations of the Event Horizon Telescope (EHT) collaboration.The images of M87 * (Event Horizon Telescope Collaboration et al. 2019a) and Sgr A * (Event Horizon Telescope Collaboration et al. 2022) provide a novel means to access the strong-field regime of black holes and enable direct probes of the magnetic field structures in the inner accretion flow through the polarization of synchrotron radiation (Event Horizon Telescope Collaboration et al. 2021a, 2021b).Observations with lower-frequency arrays such as the GMVA continue to improve our understanding of black hole accretion and jet launching (Lu et al. 2023) as well.
However, it is important to note that the EHT and its immediate successor, the Next Generation Event Horizon Telescope (ngEHT; Ayzenberg et al. 2023;Doeleman et al. 2023;Johnson et al. 2023), are Earth-based networks of observing stations.This has direct physical implications on the scientific value of the observed data.For example, the maximum length of a baseline that can be obtained is constrained by the diameter of the Earth (Event Horizon Telescope Collaboration et al. 2019b) and the observing frequency has to be chosen so as to ameliorate the effects of the Earth's atmosphere (Event Horizon Telescope Collaboration et al. 2019c;Raymond et al. 2021).
It is therefore apparent that in order to continue improving the resolution and the Fourier plane (hereafter the (u, v)) coverage, future VLBI missions targeting black hole astrophysics must have a space-based component.Indeed, there have been several recent mission concept studies in this domain (Kudriashov et al. 2019;Andrianov et al. 2021;Johnson et al. 2021;Kurczynski et al. 2022;Likhachev et al. 2022;Hudson et al. 2023;Roelofs et al. 2023;Rudnitskiy et al. 2023;Trippe et al. 2023).The space missions can be a single orbiter working in tandem with a ground-based station to produce very long baselines, leading to very high angular resolution.However, in this case one would still have to deal with the effects of Earth's atmosphere.The other alternative is space-only VLBI that contains several orbiters forming baselines with one another, wherein deciding an appropriate formation geometry is a key task, encompassing both astrodynamics and VLBI site optimization.Here, one would eliminate any corrupting effects due to the Earth's atmosphere.
In this paper, we provide a detailed investigation of how one realistic astrodynamical effect, namely the effect of Earth's oblateness on an orbiter's motion (the so-called J 2 effect), impacts the (u, v) coverage for observing M87 * and Sgr A * using both Earth-space and space-space VLBI.The choice of these sources is based on the goal of developing a linear trajectory of improvement over the existing observations of the EHT.Now, it is a reasonable question to ask why specifically the J 2 effect is being studied when sophisticated orbit propagation models are already being used for space VLBI mission studies.These models include not just the J 2 effect, but also the effects of air drag, the dynamics of the upper atmosphere, and so on (Vallado 2007).Our reason for specifically investigating the J 2 effect is its peculiar nature, wherein by an informed choice of related orbital parameters, the effect can be used to one's advantage, as compared to being a "corrupting" influence on our observations.In particular, choices of orbital parameters informed by analyzing the J 2 effect will lead to a significant impact for calculations related to the fuel budget of the mission, a nontrivial factor in mission design.
Keeping all of these factors in mind, the paper studies the J 2 effect on the (u, v) coverage using analytical tools developed in astrodynamic literature that model this effect for the orbiter's motion.Since we are directly using analytic expressions, the framework is much less computationally expensive than sophisticated orbit propagation models, while also providing insights into how such physical effects can impact black hole observations through a space VLBI component.The layout of the paper is as follows: In Section 2 we provide an accessible introduction to the J 2 effect, its effect on orbital parameters, and how the effect can be used to one's advantage when selecting orbits to maximize scientific output from the coverage of (u, v) in observing M87 * and Sgr A * .In Section 3 we provide a detailed discussion of how the orbital parameters of existing VLBI missions with space-based components are affected by the J 2 effect.In addition, a brief discussion is provided on the effect's impact on the visibility domain signature of a photon ring modeled as an infinitesimally thin ring of unit flux.In Section 4 a novel analytic formalism is developed that describes the (u, v) coverage in terms of the equations of motion of two space-based orbiters in relative motion with respect to one another, namely a chief and a deputy.In Section 5 the relative motion is studied without incorporating the J 2 effect and the corresponding features of the baseline coverage are studied for both M87 * and Sgr A * .In Section 6, the J 2 effect is incorporated through a linearization scheme in the equations of motion and the corresponding (u, v) coverage for the same two black holes is studied.In Section 7, the equations of motion that produce orbits that are invariant under the J 2 effect are used and the bounded relative motion is observed in the (u, v) plane as well.Finally, Section 8 provides the conclusion, as well as some potential avenues for future work.For the benefit of the reader, a tabulation of some of the terminology used in the paper is provided in Table 9 in Appendix B.

Incorporating Astrodynamic Considerations into VLBI:
Effect of Earth's Gravitational Field via the J 2 Perturbation The Earth is essentially an oblate spheroid, and is flatter at the poles and bulges out at the equator (Schaub & Junkins 2003;Capderou 2014).As a consequence of this, the gravitational attraction toward a body orbiting the Earth is not directed toward the Earth's center of mass, as would be expected from the classical Newtonian theory of gravitation.A common approach to modeling this effect on an orbiter is by expressing the gravitational potential in terms of spherical harmonics.If r, f, and λ are the satellite's orbital radius, latitude, and longitude, respectively, then the potential can be written as where μ is the product of Newton's gravitational constant and the mass of Earth and a e is the semimajor axis of Earth's ellipsoid shape.The functions denoted by P lm (...) are the associated Legendre functions of degree l and order m and C lm and S lm are the spherical harmonic coefficients.Extremely accurate satellite measurements of these coefficients have been incorporated into the Joint Gravity Model 3 (JGM-3) developed by NASA (Tapley et al. 1996).
To understand the separate effects of the harmonics, one often splits this equation into three parts: where Here, V 0 is the classical potential when Earth is treated as a point mass.The term V 1 corresponds to the part of the potential that does not have longitudinal dependence and hence has m = 0.By defining the "zonal" part of the potential is now written as The remaining part V 2 is dependent on the longitude.
For our purposes, we focus on Equation (7).It is the degree 2 zonal term, denoted as J 2 in the equation, that models the contribution from Earth's oblateness and hence the change in dynamics arising from it is sometimes known as the J 2 perturbation effect (King-Heele 1958).For this paper, we shall use the standard value of the J 2 coefficient (Sengupta 2003, hereafter S03): As far as the effects of the J 2 perturbation are concerned, it is the dominant perturbing influence for orbiters in low Earth orbits (LEOs).Indeed, the orbiter-Earth system can be modeled as a Keplerian two-body problem and the dominant contribution of the J 2 terms can be investigated using Earth models such as the JGM-3.For LEO deployment, incorporation of J 2 effects via the JGM-3 has a relatively better performance than the more widely used Simplified General Perturbation 4 (SGP4) model (Morales et al. 2019).
In the context of Earth-space VLBI, the motivation for studying the impact of the J 2 perturbation arises from the fact that several recent discussions of future missions have considered placing stations in LEOs (Palumbo et al. 2019).At altitudes of around 800 km (which is in the LEO range), the J 2 perturbation is the dominant effect when compared to other realistic considerations such as atmospheric drag, solar radiation pressure (Alessi et al. 2018), and electromagnetic effects (Marsden et al. 2001).For related work on the nontrivial importance of J 2 in relative orbit motion, see Schweighart & Sedwick (2002) and for efforts on modeling orbits invariant to J 2 effects, see Schaub & Alfriend (2001), Schaub & Junkins (2003), and Lee (2022).

Effect of J 2 Perturbation on Orbital Parameters
For a given orbiter in space, there are six Keplerian orbital elements that can describe its behavior: the semimajor axis a, eccentricity e, inclination i, R.A. Ω, argument of perigee ω, and mean anomaly M. Due to the J 2 perturbation, over a large number of orbits, the time evolution of these elements is affected as follows: where and the overdot represents the time derivative of the corresponding parameters.Now the longitude Ω, inclination i, and latitude θ form a 3−1−3 Euler angle system (S03).
Moreover, when considering long-term variation in orbital parameters over a large number of orbits, only the effects of  w are felt so we approximate (S03) thereby ignoring the effects of the harmonics of the true anomaly f.This approximation is valid for mission cycles of space VLBI missions such as Millimetron, which has an expected 10 yr cycle of operation (Lazio et al. 2020).
To get a better understanding of how the orbital elements evolve over time, we can integrate Equations (10), (11), and (12) assuming the initial values Ω 0 , ω 0 , and M 0 , respectively.Note that due to Equation (9), the inclination i in this framework does not have time dependence.One obtains the following expressions: For future reference, we also note the approximate relation between the true anomaly ν and the mean anomaly: For the choice of orbits given in Table 1 (justification for which is provided in the following section), the variations of Ω and ω for a time period of 1 day and 1 month are tabulated in Table 2.
The J 2 effect is apparent here.For example, the choice of ω for the equatorial highly elliptical orbit (HEO) is made to maximize the coverage of M87 * but the parameter has variability even over a single day, and it only increases with time.This causes variation in the coverage of M87 * that can be achieved, as will be demonstrated in subsequent sections.As another example, the effect of the so-called "critical/magic inclination" of i = 63°.4 for the Molniya orbit is apparent since there is no time variation in ω.Such an inclination has been known in astrodynamic literature (for example, see Sabatini et al. 2008) for some time, with Molniya orbits being used for satellite orbits since the 1960s (Allan 1971).The numerical value of this inclination is obtained as follows: Therefore, suitable modifications/extensions of the Molniya orbit for maximizing the coverage of a particular source (just as we do for the equatorial HEO) can be used to counteract the variation in source coverage that might arise from the J 2 effect on orbital parameter ω.

Orbit Selection
As with many other aspects of space system design, the orbit is often a trade-off between what is most desirable to meet the scientific objectives and the feasibility from a mission analysis perspective.Orbit selection will affect many aspects of the mission including power generation, thermal environment, communications, and radiation exposure, to name a few.The effects on these factors must be carefully balanced through orbit selection to ensure the system can successfully meet the primary objectives and operate safely in the space environment.For VLBI, orbit selection is primarily a geometrical consideration to achieve sufficient (u, v) coverage and resolution when observing the target astronomical sources.Considering a single space telescope observing in collaboration with ground-based stations, angular resolution will be driven by the maximum altitude of the orbit as increasing the baseline length is required to improve resolution.For VLBI imaging dense (u, v) coverage is required to increase the fidelity of generated images.This is achieved by varying the baseline length throughout observations, which means operating at a range of altitudes for a space-based system.Rapid (u, v) coverage is also desirable as the radio-emitting environment of black holes is highly variable over short timescales.To capture the dynamic behavior of supermassive black holes, coverage should be achieved in the shortest observation time possible (see analysis in Roelofs et al. 2023).For an orbiting antenna, this encourages the selection of an orbit with a short time period (i.e., a high orbital velocity) to sample a range of (u, v) points in as short a time as possible.
For this study a number of commonly used Earth orbits are evaluated in terms of their benefits from a VLBI and mission analysis perspective.The subset of orbits selected for analysis is by no means comprehensive but provides a good example of the trade-offs that must be resolved and of the effect of the J 2 perturbation on the mission and subsequently on the VLBI observations.Table 1 contains the three analyzed orbits with their respective Keplerian elements.Figure 1 depicts the configurations in the Earth-centered inertial (ECI) frame.The ECI frame ( ˆˆˆ) X Y Z , , is such that ( ˆX Y , ) spans the equatorial plane of the Earth, Ẑ is along the North Pole, and X is along the vernal equinox.The origin of the system is the Earth's center.This is analogous to the right-handed Cartesian coordinate system used in VLBI for specifying positions of antennas in an array (Thompson et al. 2017).Now, depending on the specific objectives of the mission, the J 2 perturbation on the R.A. and argument of perigee can be either beneficial or detrimental.For VLBI, careful selection of orbital elements could result in the nodal and apsidal precession being advantageous for the observation of supermassive black hole targets.
A Sun-synchronous orbit is a good example of a LEO that makes use of the J 2 perturbation on the R.A. of the ascending node (RAAN).Many Earth-observing satellites observing at optical wavelengths require consistent lighting of specific targets.Sun-synchronous orbits make use of the nodal precession due to the J 2 perturbation to rotate the R.A. of the orbit so that the position of the Sun with respect to the orbital plane is constant throughout the year.The inclination and semimajor axis of a Sun-synchronous orbit are selected such that the nodal precession is equal to the rate of the Earth's rotation about the Sun.For VLBI observations this is highly beneficial.Due to the stringent sensitivity requirements on VLBI interferometers, it is highly likely that space-based VLBI systems will require cooling of the receiver electronics to very low temperatures, depending on the frequency selected, as discussed in Gurvits et al. (2022).Like the case of many space telescopes (e.g., James Webb), this may require all observations to be conducted away from the solar direction to simplify thermal control on board.Maintaining the Sun in a constant position with respect to the orbital plane would simplify operations of the spacecraft drastically.A 700 km altitude has been selected for this orbit as this keeps the spacecraft outside of the inner Van Allen belt, which begins to affect spacecraft electronics at higher altitudes.One notes that the use of a Sunsynchronous orbit for VLBI observations is one example of how careful orbit selection can make use of the orbit precession due to the Earth's oblateness in a beneficial way.
An equatorial HEO has been selected to demonstrate the apsidal precession effects of the J 2 perturbation.Having an inclination of 0°maximizes the rate of change of the argument of perigee.The semimajor axis and eccentricity have been derived by selecting a perigee altitude of 700 km (the same as that of the Sun-sync orbit) and an apogee of 21,000 km (half the apogee of the Molniya orbit) to provide a variation in (u, v) coverage from the other two orbits.The argument of perigee has been calculated to maximize the coverage of M87 * .As with the Molniya orbit, the increased apogee altitude as compared to the Sun-sync orbit provides a longer baseline to a ground-based station and thus a finer angular resolution.Unlike that of the Molniya orbit, the period of this orbit is only ∼4.5 hr providing more rapid (u, v) coverage as the spacecraft completes a full revolution of the Earth in almost a third of the time.
Molniya orbits were first designed to provide long periods of coverage over high latitudes, as opposed to the coverage typically offered by traditional communications satellites in geostationary Earth orbits.In order to achieve this, the orbit has a high eccentricity to increase the period of time spent over the target region.As has been discussed, apsidal precession due to the J 2 perturbation results in the rotation of the argument of perigee over time.To combat this, the Molniya orbit makes use of a 63°.4 inclination, which results in zero apsidal precession.This is generally referred to as a frozen orbit as some perturbation effects are canceled out through the orbital element selection.This feature, along with the orbit's large altitude variation (and hence its large baseline variation), makes it a subject of particular interest for this study.
Figures 2, 3, and 4 depict the (u, v) coverage achieved by each configuration for 7 day observations of M87 * and Sgr A * .Observations are conducted at 345 GHz, with a single ground station, the Large Millimeter Telescope (LMT) in Mexico.Observation could however be conducted with any ground-based antenna and still produce the same high-level (u, v) coverage features that are described below.As stated in Roelofs et al. (2023), this is the target frequency of the ngEHT.For these  2019b).The instrument duty cycle is 10 minutes, providing 5 minutes between the end of one scan and the start of the next.The exact duty cycle of a space-based instrument will depend on many factors such as power requirements, data processing and storage, thermal conditions, and observation of calibration sources.However, the general pattern of the (u, v) coverage remains consistent regardless of the duty cycle and it is simply the density of the coverage that will vary.The variations in the orbits selected can be clearly seen in their respective (u, v) coverages of M87 * and Sgr A * , shown in Figures 2, 3, and 4. As expected, the finest resolution of 5.1 μas is achieved by the Molniya orbit configuration as it has the largest apogee of the three designs.However, due to the long period of this orbit (12 hr), the (u, v) coverage is very sparse compared to the Sun-synchronous and equatorial alternatives, which complete 7.5 and 2.5 more revolutions about the Earth, respectively, in the same time.As M87 * is at a fairly low decl., the (u, v) coverage of the equatorial HEO has less variation in the v-plane than that of the alternative, inclined orbits.
All three configurations clearly demonstrate the benefits of space-based VLBI stations.The improvement in angular resolution is considerable compared to the 25 μas achieved by the EHT in its M87 * and Sgr A * observations (see Event Horizon Telescope Collaboration et al. 2019bCollaboration et al. , 2022)).The Sunsynchronous orbiter produces a far denser (u, v) coverage and a resolution of 20.8 μas for M87 * .The increased altitude of the equatorial HEO exhibits a resolution of 6.9 μas, an improvement on the EHT of almost a factor of 5.These results are achieved by a simple interferometer of only two elements, the orbiter and the ground-based LMT.Denser coverage would be possible for observations with many more ground stations, even for the sparsely populated Molniya orbiter (u, v) plot.The long baselines achieved by the equatorial HEO and the Molniya orbit configurations also introduce the possibility of probing the first-order photon ring of M87 * .Johnson et al. (2020) showed that the interferometric signature of the black hole is dominated by the photon ring contribution beyond ∼20 Gλ and that the submillimeter interferometric signatures of Sgr A * and M87 * should be dominated by the photon ring contribution.Detection of the signature on very long baselines   can allow one to estimate the angular diameter of the photon rings, from which uniquely robust and accurate tests of strong gravity and general relativity can be performed.
As is demonstrated in Figures 2, 3, and 4, the (u, v) coverage of a radio source is dependent on the geometry of the orbit design.It is therefore possible to optimize the orbit configuration to maximize the visibility and coverage of a certain source.The investment required to fund a future space-based VLBI mission will be significant and therefore the system will need to be highly versatile to justify the cost.Optimization for observation of a single source will not be desirable.Changing the shape and orientation of an orbit can be an expensive orbit transfer.Conducting plane change maneuvers for varying inclination and R.A. is particularly demanding and often completely impractical in terms of the propellant requirement.The relative R.A. of the source with respect to the orbital plane will vary throughout the year due to the nodal and apsidal precession caused by J 2 .This natural variation can be used to change the visibility that the space-based element of the interferometer has of different sources.
Therefore, precession due to J 2 should be considered when designing orbit configurations to optimize the coverage of certain sources.The equatorial HEO is selected to maximize the rate of apsidal precession to illustrate this approach.Figure 5 depicts the precession of this orbit across a 12 month period.The argument of perigee changes by 183°due to the J 2 perturbation in a single year.The initial argument of perigee is selected to maximize the baseline length achieved for observations of M87 * .The resolution of Sgr A * that would be achieved is ∼70% of that for M87 * .In this configuration, 8 months after the initial conditions the effect of the J 2 perturbation results in the resolution achieved during the observations of Sgr A * improving from 9.8 to 8.0 μas.
This highlights one of the disadvantages of the Molniya configuration and other frozen orbit types.By canceling out the J 2 perturbation effect on the R.A. and argument of perigee, the natural precession effects cannot be used to vary the coverage of different sources.However, as a consequence, sometimes months would pass between the optimal observation times for different sources.It is likely that this would be acceptable for a mission that has a lifetime of at least 5 yr.Scheduling with ground-based arrays would need to be carefully conducted such that observations could take place at the optimal time.The orbit configuration of a spacebased station can therefore be designed such that the natural perturbations due to J 2 are advantageous for the observation of radio sources and the operation of the spacecraft platform itself.However, ignoring the apsidal and nodal precession effects would likely result in an unfavorable geometry for certain sources that could otherwise be avoided.

Investigating J 2 Effects in Existing Earth-Space and Space VLBI Proposals
Recently, there have been several mission concept studies in literature that investigated the scientific merits of having a space-based VLBI station (Fromm et al. 2021;Kurczynski et al. 2022;Likhachev et al. 2022;Rudnitskiy et al. 2022;Roelofs et al. 2023;Trippe et al. 2023).These merits are intimately tied to the "cartography" of the (u, v) plots, with larger baselines implying higher angular resolution and shorter baselines improving the density of the (u, v) plane.Now, existing literature has explored in detail the scientific payoff of  the increased coverage of the (u, v) plane with space VLBI.However, an extended discussion of how realistic astrodynamical effects acting on the orbiter can influence the (u, v) coverage is lacking in these studies.Therefore, we now investigate how the J 2 perturbation specifically affects the orbital elements for the said mission concept studies.The focus is on those studies in which the relevant orbital parameters affected by J 2 are explicitly provided.
First, the parameters in Fromm et al. (2021) are investigated, wherein the authors devised an optimization algorithm to observe Sgr A * over a 12 hr time period.The proposal involved a single space-based orbiter in a medium Earth orbit (MEO).The orbital parameters of the proposal, along with the change in values due to J 2 , are given in Table 3.
We next investigate the proposal of Rudnitskiy et al. (2023).The authors considered the deployment of two or more orbiters in near-Earth circular orbits so as to rapidly fill up the (u, v) plane.In echoing the results of Kudriashov et al. (2021b), they stated that the choice of RAAN of −43°can provide similar (u, v) coverage for M87 * and Sgr A * .Now, we know that the RAAN (which is Ω in our notation) is indeed affected by the J 2 perturbation and based on the reasoning discussed earlier, the effect is expected to be much more prominent for near-Earth orbits.The variation of specifically the RAAN is shown in Table 4.It is therefore quite evident that the J 2 perturbation has a nontrivial effect on the RAAN for LEO deployments.Since the choice of RAAN was to observe a specific astronomical source (both M87 * and Sgr A * for Ω 1 or just M87 * for Ω 2 ), if the J 2 effect is not taken into account, there will be significant, unexpected variation in the visibility of target sources.This would affect the times at which the source is in view of the interferometer and the achievable variation in baseline length.
Lastly, we consider the proposal of Andrianov et al. (2021) that considered the orbiter to be on an HEO.Since the orbiter is in an HEO, the effect of J 2 is less prominent here.The impact of J 2 on their orbital parameters is given in Table 5.Nevertheless, due to the fact that the proposal requires HEOs, mitigation of the J 2 effect through orbit adjustment would lead to a nontrivial contribution to the fuel budget of the orbiter (Lee 2022).A notable feature of the study by Andrianov et al. (2021) is that they used the EGM96 Earth gravity model, which does indeed take into account the noncentral nature of the Earth's gravitational field (including, of course, the J 2 effect).This is similar to an Earth-space VLBI mission with Radio-Astron being the space-based component that also took into account the noncentral gravitational field of the Earth, along with perturbations due to the Moon and the Sun, noting that the presence of such factors "substantially complicated determination of the spacecraft orbit" (Kardashev et al. 2013).
From investigating the tables, several general inferences can be obtained.First, the fact that the J 2 effect seems "negligible" in some cases is a consequence of the fact that the proposals discussed here considered orbiters in and around an MEO, where the J 2 effect is smaller when compared to the LEO case.This is straightforward to infer from a physical sense because the farther the orbiter goes from the Earth, the lesser the effect of Earth's gravitational field will be on it.Nevertheless, the J 2 effect does indeed contribute to the change in values of the orbital parameters, which in several cases have been obtained from optimization algorithms tailored to observing a particular source, for example Sgr A * in Fromm et al. (2021).The fact that the J 2 effects reduce as the altitude of the orbiter increases is apparent.Therefore, since the primary impetus of going to larger baselines (and therefore to MEOs and beyond) was to obtain sharpened angular resolution, it is important to investigate how specific perturbing effects can influence these science goals.As far as LEO deployments are concerned, the importance of mitigating the J 2 perturbation is quite evident from Table 4.
The lessons drawn here are analogous to the ones obtained from orbit determination studies for RadioAstron (Zakhvatkin et al. 2020).In addition to the evolution of the orbital parameters due to Earth's gravitational field, it was noted that the solar radiation pressure was the main perturbing force affecting the orbiter's motion.The modeling of the orbiter's dynamics specifically taking this effect into account was crucial to studying the orbiter's efficacy for conducting observations during the mission's observing cycle.Therefore, one can envision that the J 2 effect can play a similar role in the selection of orbits for LEO deployments of space VLBI missions.

The J 2 Effect and Photon Ring Observations
High-resolution observations of a black hole photon ring are one of the prime targets for future Earth-space and space-only VLBI observations (Fish et al. 2020;Gralla et al. 2020;Johnson et al. 2020;Kurczynski et al. 2022;Hudson et al. 2023).Indeed, one of the primary motivations for putting an orbiter into space for VLBI observations is to obtain baselines that are much longer than the ones that are possible using solely Earth-based stations.Keeping this in mind, it is imperative to investigate whether the J 2 effect impacts observations of the photon ring.
To get a handle on these effects, we attempt to model the changes in the (u, v) plane due to the presence of the J 2 effect directly into photon ring observations in the visibility domain.In this domain, the interferometric signature V(u) of an infinitesimally thin, uniform circular ring with a diameter d (measured in radians) observed on (u, v) distances u (measured in wavelengths) is given by the zeroth-order Bessel function of the first kind: Note.
Here Ω 1 is the RAAN identified as giving similar (u, v) coverage for M87 * and Sgr A * , while Ω 2 is to make Sgr A * the primary target.
This ring has a unit total flux.Figure 6 depicts the variation in coverage of a model photon ring about M87 * due to the J 2 perturbation on the equatorial HEO interferometer configuration.On an equatorial HEO, the inclusion of the J 2 effect leads to a preferable increase in the maximum baseline length achieved, after only 4 months of precession.The density of coverage over the shorter baselines is however reduced.The precession of different orbit configurations would no doubt result in a more drastic variation in the baseline coverage of the interferometer over time.The equatorial HEO example does however show that an appreciable difference in the interferometric signature can be observed due to the precession of the orbit.Therefore, this must be considered when planning observations of such a mission as depending on the current rotation of the R.A. about the equator, the baseline range achievable will vary.

Incorporating Equations of Motion of Orbiters into the (u, v) Space
In radio interferometry and therefore in VLBI as well, the standard method to judge the efficacy of a baseline for various scientific goals is to investigate the coverage in the so-called (u, v) space (Thompson et al. 2017).Since astrodynamical effects, particularly the Earth's oblateness discussed here, directly impact the equations of motion of the orbiter, one needs a set of equations that directly map these equations to the coverage in the (u, v) plane.
Astrodynamic considerations become all the more important for space-only VLBI observations since astrodynamical effects would now affect each and every orbiter in a potentially distinct manner (based on the orbiter design, orbital parameters, etc.), as compared to a simple Earth-space VLBI deployment of a single orbiter.Therefore, it is even more important for space VLBI to have a mapping that allows for a suitable translation between astrodynamical effects and the associated consequences on the (u, v) plane.
For this paper, we shall consider the case of two orbiters in relative motion with each other, thereby forming a space-only baseline.Borrowing nomenclature from astrodynamics (S03), one orbiter is labeled as the chief and the other orbiter is the deputy.The equations of motion are modeled for the motion of the deputy relative to that of the chief.The analysis proceeds as follows.
We require two main coordinate systems, namely the ECI frame discussed earlier and the local vertical-local horizontal (LVLH) frame, which has the origin at the orbiting satellite (S03).Now, if (H, δ) are the hour angle and decl., respectively, of the astrophysical source under observation, and λ is the observing wavelength, the (u, v, w) coordinates used in interferometry can be obtained from the (X, Y, Z) components in the ECI frame by 1 sin cos 0 sin cos sin sin cos cos cos cos sin sin .
For applications in VLBI, we only require the (u, v) coordinates since the contributions from the w coordinate are considered negligible, which arises from the assumption that the field of the source being synthesized is not too large (see Chapter 3 of Thompson et al. 2017 for an extended discussion on this matter).Therefore, we shall focus only on (u, v) and the corresponding equations are Now, let r c be the position vector components of the chief in the ECI system, and let the corresponding components in the LVLH frame be (X c0 , Y c0 , Z c0 ).Three of the orbital elements Ω, i, and θ form a 3 −1−3 Euler system (Ω−i−θ) and so the aforementioned components of the position vector in the ECI frame can be written in terms of the LVLH components using a directional cosine matrix (S03): Here θ c , i c , and Ω c are the latitude, inclination, and R.A., respectively, for the chief.Expanding the equation, we get  If we now define (X d , Y d , Z d ) as the position vector components of the deputy in the ECI frame, we can define its (u, v) coordinates as in Equation (23).Then, using an analogous expression for the chief, we can define We now write this equation in the LVLH frame.In this frame, suppose the components of the separation vector between the chief and deputy are given by Then, using Equation (26) for the chief, and exactly the same construction for the deputy (since both are in the LVLH frame, and Ω, i, and θ transform any general vector from the ECI to the LVLH frame, the angles would be the same for both the chief and the deputy), substituting in Equation ( 27) along with Equation (28) for the difference in the position vector components, we get our main equations: Equation (29), explicitly derived to the best of our knowledge for the first time, will be the main tool used to incorporate astrodynamical effects into space-only VLBI.In particular, one notices that due to the mapping between the equation of motion of an orbiter, given by (x, y, z), and the (u, v) plane, realistic effects incorporated in the former can now be reflected in the latter.
Lastly, one can note that since the origin of the chief's LVLH frame has (x = y = z = 0), the (u, v) coordinates are (0, 0).Thus, the (u, v) plane is "centered" on the chief orbiter's LVLH frame.

Equations for Relative Motion of Chief and Deputy: No
J 2 Effect Let ρ be the relative orbit radius.Then, under the assumption of a circular reference orbit and no perturbing forces, the relative equations of motion are given by the Hill-Clohessy-Wiltshire (HCW) equations (Schaub & Junkins 2003): where n c is the mean motion for the chief obtained by substituting the orbital parameter r c of the chief for the semimajor axis in Equation (13).Notice that the (x, y) motion is decoupled from the z motion.The former can be modeled as a coupled harmonic oscillator while the latter can be modeled as a harmonic oscillator.
For bounded motion with a relative orbit radius ρ and initial conditions such that the orbits trace a circle in the (y−z) plane, and assuming no offset in the x-and y-direction, the solution to these equations is given by

(u, v) Plots
We now construct the (u, v) plots using the equations of motion given in Equation (31).The choices of parameters are )] } ( ) l l q q q q q q q q l l d q q q q d q q q q d q q given in Table 6.The numerical values are inspired by the ones given in S03 and the form of the solutions is obtained using the equations in Ginn (2006).
In addition, we consider the mean relative orbit radius ρ and the offset α to be Lastly, we consider the observations to be at 345 GHz.Substituting all of these parameters into Equation (29), we obtain the (u, v) tracks shown in Figure 7.It can be noticed that the baselines obtained through this formalism are not large enough for the high angular resolution observations required for resolving the photon ring, for example (Johnson et al. 2020).This should be expected because the HCW equations were obtained using a linearization argument, because of which the order of separation between the chief and the deputy (tens of kilometers) has to be much less than the semimajor axis of the chief (thousands of kilometers).Therefore, while changing the values of ρ to higher values by "brute force" would give "bigger" ellipses in the (u, v) plane, it would not be faithful to the physical basis of the HCW solutions.Nevertheless, one can still obtain such elliptical plots in the (u, v) plane using a modified set of equations that make the orbits invariant to certain effects of the J 2 perturbation.This will be discussed in later sections.

Equations for Relative Motion of Chief and Deputy:
Incorporating J 2 Corrections through Linearization We now discuss how the effects of the J 2 perturbation can be incorporated using linearization of the equations of motion.The formalism for the same has been provided in Schweighart (2001) and Ginn (2006) and it is provided in the appendix.In the subsequent sections, we discuss the applications of the formalism to obtain (u, v) plots of M87 * and Sgr A * .

(u, v) Plots
The (u, v) plots of M87 * are given in Figure 8 while those for Sgr A * are given in Figure 9. Several inferences can be drawn.
First, the plots show a remarkably rich and distinctive character based on the black hole that is being observed, namely M87 * and Sgr A * .In particular, these plots follow directly from the equations of motion without use of any "optimization" procedures, thereby allowing us to directly capture the impact of the J 2 effect on the (u, v) coverage.Second, as time passes, the plots become increasingly dense and thus provide dense (u, v) on short baselines, a desired feature of space VLBI missions (Fish et al. 2020).We note that in order to generate these plots, only a small sample of the data points have been used that were generated by the choice of the sampling rate.For time periods of 1 day, 15 days, and 1 yr, we used 700, 15,000, and 30,000 data points, respectively.This choice was governed largely by the requirement of a reasonable processing time to generate and process the plots in a higher resolution.It has been verified that utilization of all the plot points leads to a modest increase in filling (u, v) in the "northwest" and "south-east" parts of the plot for M87 * and in the "north-south" parts for Sgr A * .In addition, there is an overall increase in the coverage density.Nevertheless, the existing plots still demonstrate the key scientific takeaway of rapidly increasing density over medium-length baselines providing a rich geometric coverage pattern generated primarily by the J 2 effect.
The extremely dense coverage over shorter baselines introduces the possibility of having an Atacama Large Millimeter/submillimeter Array-like ("ALMA-like") flying formation of orbiters in space that can have much longer integration times and can observe every single day throughout the year without having to mitigate intraday variability in weather.As another potential avenue, one can consider a "hybrid" mission wherein on the one hand, the chief and/or the deputy can work in an "Earth-space" mode, forming long baselines allowing high angular resolution images of the photon ring, as discussed in the earlier sections on Earth-space VLBI.On the other hand, the orbiters can operate in a "spaceonly" mode wherein the relative baselines formed between them as part of a formation can allow for observations of largescale structure pertaining to black hole accretion.While providing a detailed description of a mission design based on these principles is beyond the scope of this paper, the results here indicate that in both of these considerations, the J 2 effect  can turn out to be advantageous not only in providing dense (u, v) coverage, but also in aiding suitable orbit selection.

Equations for Relative Motion of Chief and Deputy: J 2 -invariant Orbits
There have been several investigations into obtaining suitable orbital parameters such that the motion of the orbiter is unaffected by the J 2 perturbation (Schaub & Alfriend 2001;Lee 2022).In this section, we shall largely be following the discussion given in Lee (2022).
The J 2 perturbation independently affects the latitude θ and the RAAN Ω and so finding J 2 -invariant orbits boils down to searching for orbital parameters that can minimize their secular change over time, as given in Equations (10), (11), and (12).Let us consider the motion of the chief.If one has to minimize the effect of the J 2 perturbation on the chief, then the following equation must be solved: Now, for the subsequent discussion, we shall consider the chief to be in a circular orbit, and so Upon substituting the expressions from Equations (11) and (12) into Equation (33), and using the circular orbit condition in Equation (34), we get (Lee 2022) Since the terms outside the bracket are constants, this equation can only be satisfied when the term in the bracket is 0: Thus, if one wishes to minimize the change in latitude for the chief in a circular orbit, that can be achieved by having it at an inclination angle of 60°.It is important to note that this result holds for all altitudes of the chief satellite.Note that this is analogous to our earlier calculation in Equation (20) on having an inclination of 63°.4 that minimizes  w for the Molniya orbit.Now, if the change in latitude due to the J 2 perturbation has to be minimized, we have Using once again Equations ( 11) and (12), we get (Lee 2022)  Thus for a given value of the eccentricity of the deputy, the inclination that minimizes the J 2 perturbation can be obtained by solving Equation (39).One can also do similar calculations for minimizing the RAAN, which as we mentioned earlier has an independent effect on the equations of motion, but in this paper we focus on the effects of minimizing the change in latitude, postponing the RAAN-based discussion to a future work.Now, minimizing the drift in θ has a direct impact on the fuel budget of the mission.Under an impulsive control scheme that aims to maintain constant relative motion between the deputy and the chief during the mission lifetime, the so-called ΔV requirement for the scheme for latitude correction, exercised after one orbit, is given by Lee (2022): and so additional fuel will not be dispensed to maintain relative motion.Similar considerations apply for minimizing the RAAN (Lee 2022).
7.1.(u, v) Plots for J 2 -invariant Orbits We now obtain the (u, v) plots for J 2 -invariant relative motion using suitable parameter substitution in Equation (29).First, for the chief we consider the parameters given in Table 7.The mean anomaly M c is not mentioned since it does not play a role in the subsequent discussions.
The choice of i c = 60°implies that there would be no change in the latitude θ c over time.We thus assume Since the latitude and RAAN drift independently affect the relative motion under the J 2 effect, one can only design a configuration that minimizes either one of them.Indeed, the equations to solve for inclinations that minimize the J 2 effect on the RAAN are different from those for minimizing the effect on the latitude (Lee 2022).
Lastly, we do allow for the mean anomaly of the deputy to have a time dependence as per Equation (17).Now, following Lee (2022), the equation of motion of the deputy in the chief's frame-for the case where both chief and deputy have the same altitude, with the chief having a small eccentricity, modeled so that there is zero relative latitude drift where M d (t) is governed by Equation (17).For the conditions discussed here, the motion along the z-direction is negligible (Lee 2022) and so we assume We now describe in detail the choices for the parameters of the deputy.First, since the chief and the deputy have to have the same altitude and hence the same semimajor axis, we have Next, we assume the deputy to have a small eccentricity of ( ) = e 0.08, 46 Upon substituting the numerical values for the parameters, we get Then, using Equation (50) and substituting the value for i d into Equation (43), we get the final equations of motion: x t t t y t t 7000 cos 0 .0618367 0.08 7000 cos 0 .00494693 , 14, 000 sin 0 .00494693 .51 These are the final equations of motion that shall now be substituted into the (u, v) expressions of Equation (29).

(u, v) Plots
The (u, v) plots for M87 * and Sgr A * are given in Figure 10.The observation time is chosen to be 1 day at a sampling rate of 500 s.The choice of observing frequency is f = 345 GHz.The plots have distinctly different elliptical shapes as compared to the ones obtained without the J 2 effect from the HCW equations in Figure 7. Thus this at the very least establishes the fact that the J 2 effect in and of itself does impact the (u, v) coverage of the sources under consideration.More generally, the (u, v) plots look deceptively straightforward as standard VLBI plots for large baselines.However, the point we wish to convey with the plots is precisely that, which is to say that a suitable choice of parameters that are specifically aimed at minimizing the J 2 effect can give (u, v) plots that are suitable for VLBI investigations.As a particular example, if a spacebased VLBI mission that deploys antennas in LEOs takes a simplified approach of just specifying the orbital parameters of the antenna without taking into account realistic effects such as the J 2 effect, there would be a significant drift in the orbiter's motion, further affecting fuel budgets that would impact the mission cycle.While it is true that rigidity in the choice of orbital parameters might lead to certain compromises on the astronomical sources that can be observed, such considerations provide an impetus to investigate space-only VLBI optimization software that does take into account realistic astrodynamic considerations.We wish to investigate these avenues in the near future.

Conclusion
This paper attempts to investigate in detail the effects of the Earth's oblateness, also known as the J 2 effect, on an Earthspace and a space-space VLBI mission that aims to provide black hole images of the sources M87 * and Sgr A * .An extensive study has been performed of several existing proposals in literature that aim to investigate observing these sources using VLBI with only a space-based component.In a similar vein, for Earth-space missions, a detailed discussion is provided on how an informed choice of orbital parameters based on isolating the J 2 effect can be used to one's advantage in space mission design.For the space-space VLBI regime, the impact of the J 2 effect is studied using a simple, computationally accessible analytic framework that studies the relative motion of a chief orbiter and a deputy orbiter in relative motion.It is found that the J 2 effect leads to distinct patterns on the (u, v) plane for both M87 * and Sgr A * that are extremely dense over short baselines, thereby potentially opening up an avenue of providing a dense filling of the (u, v) plane in space VLBI missions using just a couple of orbiters.Moreover, it is demonstrated that an informed choice of orbital parameters can lead to bounded relative motion over longer baselines that is invariant under the J 2 perturbation.

Future Work
There is a smorgasbord of avenues that can be explored that cultivate a strong synergy between astrodynamic considerations and observations using space VLBI.For example, the impact/advantage of the J 2 effect in making polarimetric observations of the photon ring using Earth-space VLBI (Palumbo et al. 2023) can be explored to highlight whether an informed choice of orbital parameters can lead to more robust observables in the presence of astrodynamical perturbations.In a similar vein, one can investigate the similar impact on closure quantities in space VLBI operations, which in our nomenclature would correspond to one chief and at least two deputy satellites for closure amplitudes and three deputies for closure phases.In such an investigation, one can look at how a cluster of satellites (three, for example), while operating under the J 2 effect, can lead to robust interferometric closure observables; the appropriate orbit selection for these has been studied in Marsden et al. (2001) and the formation of orbits of such clusters including the J 2 perturbation effect has also been investigated in Schweighart & Sedwick (2002).We note that the J 2 effect is even more important here since the most significant error when modeling the relative orbits of a chief and a deputy is due to the assumption that the Earth is spherical (Schweighart & Sedwick 2002).
In terms of developments focusing on an astrodynamic framework tailored to VLBI, one can envision a "dynamic" space VLBI mission in which one utilizes a suite of astrodynamical maneuvers that can adjust the orbit parameters of the orbiter so as to optimize its coverage for several sources over a given mission cycle.For example, when an orbiter forms a baseline with an Earth-based station, the semimajor axis of the orbiter can be increased or decreased so as to investigate the photon ring science or jet-related structures, respectively.Maneuvers such as changing individual orbital parameters such as inclination, RAAN, etc. can also be performed to optimize the pointing at a particular source of interest.A comprehensive discussion of such maneuvers is given in Vallado (2007) and one can investigate whether taking advantage of the J 2 effect in the spirit of the discussion provided in the paper can help reduce the fuel budget for such missions.We expect to explore these avenues in the near future.
Lastly, the results of this paper suggest that an ambitious mission in the distant future having an ALMA-like configuration in space can be considered, where the orbit selection informed by the J 2 effect can provide extremely dense (u, v) coverage over shorter baselines.Such orbiters can also work in conjunction with other stations at faraway points like L2 to have very long baselines and thereby cater to observations of the black hole photon ring.This deployment can provide a VLBI network in space that is in principle similar to how ALMA currently operates with Earth-based stations to cater to a variety of scientific goals.We hope that the results of this paper would stimulate investigations into expanding the scientific scope of radio astronomy missions from space.where the dots indicate the dot product between the vectors.The gradient terms in the (r−θ−i) system are given by Now, taking the motion with respect to the reference orbit, we have Finally, using Equation (A4) in Equation (A7), we get Next, we take into account further considerations that arise due to the presence of the J 2 -dependent terms in Equation (A9).The equation is a linearized equation of motion in which the term ∇J 2 (r ref ) is not constant except for equatorial orbits.An approximate solution to take this into account is to time-average the term: Next, the effect of the J 2 force is that the perturbed satellite has a different period when compared to the case where the J 2 effect is absent.Due to this variation, the deputy drifts away from the reference orbit and the linearized equations break down.To fix this, we take the time average of the "J 2 force" ( ) [( ) ˆ( ) ˆ( ) ] ( ) m q q q q = --+ + r J J R r i x i y i i z 3 2 1 3 sin sin 2 sin sin cos 2 sin cos sin .A2 q q q q q q q q  = -   Argument of Perigee (ω) Orientation of the ellipse in the orbital plane (measured from the ascending node to the periapsis)

Baseline
Vector drawn between two telescopes observing the same source orthographically projected to the source

Chief Orbiter
The primary orbiter in reference to which the motion of other orbiters is studied Deputy Orbiter An orbiter moving relative to the primary orbiter.There can be several deputy orbiters moving in reference to a primary orbiter.
Eccentricity (e) Elliptical shape of the orbit (0 < e < 1) ECI Frame Earth-centered coordinate system fixed with respect to the celestial sphere LVLH Frame Spacecraft-centered coordinate system with respect to the nadir direction and the perpendicular local horizontal Hour Angle and Decl.(H, δ) The standard coordinates used to locate the position of an astronomical source on the celestial sphere Inclination (i) Orientation of the orbit with respect to the equator The effect on the dynamical motion of a body arising from the oblateness of the Earth Mean Anomaly (M) Fraction of an elliptical orbit's period that has elapsed since the periapsis RAAN (Ω) Intersection of the ascending direction of the orbit and the equator, with respect to the vernal equinox Semimajor Axis (a) Size of the orbit (average of the apoapsis and periapsis radii) The set of image-conjugate Fourier coefficients sampled by the baselines swept by a set of telescopes during a given observation (3-1-3) Euler Angle System Orientation of a rigid body with respect to an inertial coordinate system, described by three rotation angles such that the rotation is done first on the "third" axis of the original system, then on the "first" axis of the intermediate system, and finally on the "third" axis of the final transformed system

Figure 1 .
Figure 1.Analyzed orbit configurations plotted in the ECI reference frame.

Figure 2 .
Figure 2. (u, v) coverage achieved by Sun-synchronous orbit configuration with LMT observation at 345 GHz.

Figure 3 .
Figure 3. (u, v) coverage achieved by Molniya orbit configuration with LMT observation at 345 GHz.

Figure 4 .
Figure 4. (u, v) coverage achieved by equatorial HEO configuration with LMT observation at 345 GHz.

Figure 5 .
Figure5.Equatorial HEO precession due to the J 2 perturbation across a 12 month period.

Figure 6 .
Figure6.Interferometric signature of an infinitesimally thin, circular ring, using parameters consistent with GRMHD simulations of M87 * , demonstrating the variation in coverage caused by J 2 -induced orbit precession.Observations are conducted over 15 days with the same duty cycle and integration time parameters used for Figures2, 3, and 4.

Figure 8 .
Figure 8. (u, v) plots for M87 * J 2 effects and cross-track drift corrections.The time periods are, from left to right, 1 day, 15 days, and 1 yr.

Figure 9 .
Figure 9. (u, v) plots for Sgr A * including J 2 effects and cross-track drift corrections.The time periods are, from left to right, 1 day, 15 days, and 1 yr.

d
and using Equation (39) (taking only the positive value), we get a value for the inclination of

Figure 10 .
Figure10.(u, v) coverage of M87 * and Sgr A * using the J 2 -invariant equations of motion ofLee (2022) for a time period of 1 day.
Note that there is an error in Schweighart's Equation (3.27)(Schweighart 2001) in that the term in the denominator of  y 0 has to be 8k instead of 8c.

Figure 11 .
Figure 11.The drift in the z-direction from Schweighart's equations.

Table 1
Keplerian Elements of Three Prospective Earth Orbits for a Space-based Interferometer

Table 2
Variation of Orbital Elements Ω and ω due to the J 2 Effect after 1 day and 1 month simulations, each scan is conducted for 5 minutes, the average length of measurements conducted by the EHT in the observations of M87 * (see Event Horizon Telescope Collaboration et al.

Table 3
Variation of Orbital Elements in Fromm et al. (2021) due to J 2 Effect

Table 4
Variation of Orbital Elements in Rudnitskiy et al. (2023) due to J 2 Effect Figure 7. (u, v) coverage of M87 * and Sgr A * at f = 345 GHz using HCW equations for a time period of 6 months.

Table 5
Variation of Orbital Elements in Orbit Type 1 of Andrianov et al. (2021) due to

Table 7
Orbital Parameters of the Chief for J 2 -invariant Orbits r , A 6 ref and since the reference orbit is rotating with angular velocity ω, for example, the equation of motion, obtained by taking the double derivative of x, will have contributions from ω. From the known transformation in classical dynamics to a rotating coordinate system, we get

Table 9
Terminology Used in the Paper