1906.08828.pdf

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METRICS AND MOTIVATIONS FOR EARTH-SPACE VLBI: TIME-RESOLVING SGR A* WITH THE EVENT HORIZON

TELESCOPE

ANIEL

C. M. P

ALUMBO

HEPERD

S. D

OELEMAN

ICHAEL

D. J

OHNSON

ATHERINE

L. B

OUMAN

AND

NDREW

A. C

HAEL

Center for Astrophysics

|

Harvard & Smithsonian, 60 Garden Street, Cambridge, MA 02138, USA

ABSTRACT

Very-long-baseline interferometry (VLBI) at frequencies above 230 GHz with Earth-diameter baselines gives spatial resolution

finer than the

∼

as “shadow” of the supermassive black hole at the Galactic Center, Sagittarius A* (Sgr A*). Imaging static and

dynamical structure near the “shadow” provides a test of general relativity and may allow measurement of black hole parameters.

However, traditional Earth-rotation synthesis is inapplicable for sources (such as Sgr A*) with intra-day variability. Expansions

of ground-based arrays to include space-VLBI stations may enable imaging capability on time scales comparable to the prograde

innermost stable circular orbit (ISCO) of Sgr A*, which is predicted to be 4-30 minutes, depending on black hole spin. We

examine the basic requirements for space-VLBI, and we develop tools for simulating observations with orbiting stations. We

also develop a metric to quantify the imaging capabilities of an array irrespective of detailed image morphology or reconstruction

method. We validate this metric on example reconstructions of simulations of Sgr A* at 230 and 345 GHz, and use these results to

motivate expanding the Event Horizon Telescope (EHT) to include small dishes in Low Earth Orbit (LEO). We demonstrate that

high-sensitivity sites such as the Atacama Large Millimeter/Submillimeter Array (ALMA) make it viable to add small orbiters to

existing ground arrays, as space-ALMA baselines would have sensitivity comparable to ground-based non-ALMA baselines. We

show that LEO-enhanced arrays sample half of the diffraction-limited Fourier plane of Sgr A* in less than 30 minutes, enabling

reconstructions of near-horizon structure with normalized root-mean-square error

.

3 on sub-ISCO timescales.

Keywords:

galaxies: individual: Sgr A* — Galaxy: center — space vehicles — techniques: interferometric

arXiv:1906.08828v1 [astro-ph.IM] 20 Jun 2019

ALUMBO ET AL

INTRODUCTION

A black hole leaves a dark imprint (the “shadow”) on

nearby emission with a boundary shape dependent on black

hole parameters (Bardeen et al. 1972; Falcke et al. 2000). An

image of the bright accreting material near the event hori-

zon provides an electromagnetic view of the local space-

time. Measuring the shadow size when the black hole mass is

known (e.g., by studying stellar orbits as in Ghez et al. 2008)

provides a null hypothesis test of general relativity (Psaltis

et al. 2015). However, the dynamics of the matter surround-

ing the event horizon provide a more direct probe of param-

eters such as the black hole spin, which are difficult to ex-

tract solely from the shadow geometry (Johannsen & Psaltis

2010). For instance, the innermost stable circular orbit, or

ISCO, is highly dependent upon spin, and can be studied by

resolving periodicity near the event horizon (Doeleman et al.

2009b; Fish et al. 2009).

Very-long-baseline interferometry (VLBI) enables angu-

lar resolution of the immediate vicinity of the largest known

black holes. The Event Horizon Telescope (EHT) aims to im-

age the immediate vicinity of the supermassive black holes in

Sagittarius A* (Sgr A*) and Messier 87 (M87) using a global

network of radio telescopes which together provide high an-

gular resolution through VLBI (Doeleman et al. 2009a). The

2018 configuration of the EHT observed at 230 GHz, provid-

ing an effective angular resolution on Sgr A* of 23

as. This

resolution is below the expected angular sizes of the black

hole shadows in both Sgr A* and M87. The mass to dis-

tance ratio is well-known for Sgr A* and yields an expected

shadow size of

∼

as (Gravity Collaboration et al. 2018a).

This ratio is not as well known for M87, as gas and stellar dy-

namical results provide different mass estimates with corre-

sponding shadow sizes of either

∼

20 or

∼

as (Gebhardt

et al. 2011; Walsh et al. 2013).

The combination of the EHT array and VLBI imaging al-

gorithms designed to address the EHT’s particular challenges

is expected to be capable of reconstructing static images of

Sgr A* at this resolution, and has done so for M87 (see, e.g.,

Honma et al. 2014; Bouman et al. 2016; Chael et al. 2016;

Johnson et al. 2017; Akiyama et al. 2017a,b; Bouman et al.

2018; Kuramochi et al. 2018; Chael et al. 2018a; Event Hori-

zon Telescope Collaboration et al. 2019b). However, imag-

ing time-variable structure around supermassive black holes

requires well-sampled spatial baseline coverage (convention-

ally described in the (

) plane) on timescales comparable

to the innermost stable circular orbit (or ISCO). Though the

current EHT provides sufficient angular resolution to image

the shadow of both Sgr A* and M87, the array does not pro-

vide sufficient instantaneous (or “snapshot”) coverage to re-

construct a rapidly time-varying source intensity distribution

at Sgr A*, as we explore later. The (

) sampling of ground-

based arrays is fundamentally limited by the speed of Earth

rotation; thus, many sites are required to attain comprehen-

sive “snapshot” coverage of rapidly evolving sources.

The EHT plans to observe at 345 GHz in the near fu-

ture. This higher frequency will provide several advantages

when observing Sgr A*: the magnitude of interstellar scat-

tering effects decreases with the square of the observing

wavelength

, and the diffraction-limited angular resolution

(

λ/

) improves (see, e.g., Harris et al. 1970; Narayan 1992,

see also Johnson 2016; Johnson & Narayan 2016; Psaltis

et al. 2018). However, observing at 345 GHz also introduces

new challenges: receiver sensitivity decreases due to higher

system temperature and atmospheric phase fluctuations in-

crease, thereby limiting the feasible coherent integration time

of VLBI observations before calibration (Thompson et al.

2017). Furthermore, dishes require higher surface accuracy

at high frequencies in accordance with Ruze’s Law, favor-

ing smaller dishes that more easily meet these specifications

(Ruze 1966).

In this paper, we develop a methodology for analyzing

space-VLBI arrays. We then explore a possible future de-

velopment of the EHT: expanding the array to include dishes

in Low Earth Orbit (LEO), enabling time-domain analysis

and dynamical imaging reconstructions of Sgr A*. Space

dishes in low-Earth orbit provide benefits to imaging due to

the rapid formation of baselines to ground dishes with many

different lengths and orientations. To match the next gener-

ation EHT, we generally use 345 GHz as the simulated fre-

quency of observation for our analysis, though the differ-

ences in imaging at 230 and 345 GHz are discussed. In Sec-

tion 2, we review prior work on Sgr A* with VLBI, and we

examine theoretical constraints and prior space-VLBI mis-

sions to inform our investigation of a LEO expansion to the

EHT. In Section 3, we develop a pre-imaging metric for array

performance, and we demonstrate the value of adding space

dishes for improving the angular and temporal resolution of

the EHT. In Section 4, we compare examples of static and dy-

namical reconstructions of simulated models observed with

ground and space-enabled arrays. We apply simple image-

domain feature extraction algorithms to reconstructions of a

general relativistic magnetohydrodynamic (GRMHD) simu-

lation of Sgr A* and demonstrate the necessity for algorith-

mic development focused on temporal observables in the im-

age domain. In Section 5, we briefly discuss the parameter

space of sensitivity that may inform a future hardware study,

and look to other concepts for space-VLBI as well as areas

in need of further examination.

BACKGROUND

Though the EHT is already nominally capable of recon-

structing images of static structure at Sgr A*, the array likely

requires expansion to image the time-varying structure that

is expected to exist at the event horizon scale. Small space

dishes may efficiently address this requirement, but geomet-

rical restrictions on orbiting VLBI dish performance present

challenges that we now consider in detail. Here we present

the considerations of source evolution, existing ground sta-

tions, past space-VLBI missions, and analytic constraints

that motivate and inform a time-domain focused expansion

of the EHT to space.

2.1.

Sagittarius A*

Sgr A*, the radio source at the center of our galaxy, is coin-

cident with a 4

black hole at a distance of 8.1 kpc

from Earth (Ghez et al. 2008; Gravity Collaboration et al.

2018a). Sgr A* is expected to have a shadow that subtends

∼

as, making it the largest known black hole as seen from

Earth. In order to resolve the shadow, an observing instru-

ment must have a diffraction-limited resolution finer than the

shadow size.

Meanwhile, properties of the emission from Sgr A* limit

the observing frequency. Observations and theoretical pre-

dictions of Sgr A* indicate synchrotron radiation in near-

horizon emission (see, e.g., Yuan et al. 2003; Bower et al.

2015; Chael et al. 2018b). At long wavelengths, the local

plasma is optically thick to synchrotron radiation, leading

to synchrotron self-absorption that obscures event-horizon

scale structure (see, e.g. Blandford & Begelman 1999; Chan

et al. 2015; Davelaar et al. 2018). Thus, observations of the

black hole shadow must occur at higher radio frequencies at

which the accretion flow is optically thin.

Radio emission from the galactic center scatters predomi-

nantly off of cold plasma in the ionized interstellar medium

with a dispersion relation that depends on the local electron

density (Kulsrud 2005). Perturbations to the electron density

cause delays in the phase velocity of an emitted signal, lead-

ing to warped radio images (Johnson & Narayan 2016). The

characteristic angle of the associated refractive effect scales

with the square of the observing wavelength; at high radio

frequencies, there persists a small but non-negligible diffrac-

tive blurring effect with refractive substructure. Though the

blurring angle is smaller than the nominal 230 GHz beam

of the EHT, tools have been developed to mitigate this fun-

damental limit on VLBI images of objects in the Galactic

plane (Doeleman et al. 2009a; Johnson 2016). The 230 and

345 GHz observing bands considered by the EHT fall within

windows of transparency for Earth’s atmosphere, enabling

observation from the ground.

Sgr A* has been observed at many frequencies to be in-

tensely time-varying on timescales as short as 30 minutes

(see, e.g., at mm/sub-mm: Miyazaki et al. 2004; Yusef-Zadeh

et al. 2006; Marrone et al. 2008; Bower et al. 2015, see also

Near-Infrared/X-Ray results in Baganoff et al. 2001; Gen-

zel et al. 2003; Aschenbach et al. 2004; Ghez et al. 2004;

Bélanger et al. 2006; Meyer et al. 2006; Yusef-Zadeh et al.

−

East-West Baseline (G

)

−

North-South Baseline (G

)

EHT Coverage of Sgr A*

230 GHz

345 GHz

Figure 1.

Baseline coverage of Sgr A* provided by the 2019 EHT

at 230 GHz and the future 345 GHz EHT (or “EHTII”) considered

in this paper. The 2019 array includes PV, SMT, SMA, ALMA,

SPT, APEX, JCMT, and LMT. The EHTII array is simulated with

sites at Kitt Peak and in the French Alps.

2006; Hornstein et al. 2007; Dodds-Eden et al. 2011; Neilsen

et al. 2013; Ponti et al. 2015; Gravity Collaboration et al.

2018a). The rapid time variability of Sgr A* provides both

challenges for imaging and opportunities for science beyond

improving reconstruction of the black hole shadow (Lu et al.

2016). The size and shape of the shadow is weakly de-

pendent on spin, yet the ISCO period of Sgr A* varies be-

tween 4 minutes for a maximal spin black hole and half an

hour for a black hole with zero spin (Bardeen 1973; Taka-

hashi 2004; Johannsen & Psaltis 2010). If there is variation

in the source intensity distribution on timescales similar to

the ISCO period, an observing VLBI array would need well-

sampled baseline coverage on

∼

30 minute timescales in or-

der to reconstruct instantaneous images of the source dynam-

ics (though non-imaging time-domain methods may have dif-

ferent sampling requirements as in Doeleman et al. 2009b;

Fish et al. 2009). Further, recent near-infrared astrometric

and polarization measurements of the Galactic Center sug-

gest orbital motion on ISCO timescales that is likely visible

in the angular region to which the EHT is sensitive (Grav-

ity Collaboration et al. 2018b). Temporally resolving Sgr A*

with an expanded EHT thus provides an opportunity to con-

nect measured variability from other frequency regimes with

imaged source dynamics at the event horizon scale.

2.2.

The Event Horizon Telescope

As of the April 2018 observing campaign, the EHT con-

tains 8 telescopes that observe Sgr A* from 6 geographic

ALUMBO ET AL

sites: the Atacama Large (sub)-Millimeter Array, or ALMA,

in Chile; the Atacama Pathfinder Experiment Telescope, or

APEX, also in Chile and very close to ALMA; the James

Clark Maxwell Telescope, or JCMT, near the summit of

Mauna Kea in Hawaii; the Large Millimeter Telescope, or

LMT, in Mexico; the 30-meter telescope on Pico Veleta

in Spain operated by the Institut de Radioastronomie Mil-

limétrique, or PV; the Submillimeter Array, or SMA, located

near the JCMT; the Submillimeter Telescope, or SMT, lo-

cated on Mount Graham in Arizona; and finally, the South

Pole Telescope, or SPT, operating at the National Science

Foundation’s South Pole research station. Two additional

sites are expected to join the Event Horizon Telescope array

in the near future: the Kitt Peak National Observatory, or KP,

and the Northern Extended Millimeter Array, or NOEMA,

in the French Alps. The EHT also includes the Greenland

Telescope, though it can not observe Sgr A*. The simulated

observations in this article include these dishes with realistic

hardware estimates to approximate the future EHT; hereafter

we refer to this array as “EHTII.”

EHT stations span a large range of antenna separations,

running from “trivially separated” dishes with

∼

100 k

base-

lines to distant telescopes with

∼

13 G

baselines at 345 GHz.

In a full day of observation, the EHT array has sufficiently

well-sampled baseline coverage of near-equatorial sources to

form static images (see, e.g., Chael et al. 2016), though cov-

erage along the northeast-southwest direction is particularly

sparse - see Figure 1 for the full-day (

) coverage of the

approximately

−

◦

declination of Sgr A*.

2.3.

Basic Requirements for Space-VLBI

VLBI baselines measure complex-valued spatial Fourier

components (“visibilities”) of the source brightness on the

sky by correlating co-temporal measurements of the elec-

tric field across large distances. As stations move in the

orthographically projected plane of the Earth as seen from

the source, different Fourier components are measured as

“tracks” are swept in the (

) plane, as in Figure 1. These

“tracks” are typically ellipses corresponding to the shift in

the displacement vector between two ground-based sites; for

space dishes, these tracks correspond to instantaneously el-

liptical paths with time-dependent semi-major axes.

Visibility measurements are corrupted by instrumental and

atmospheric gain variations discussed in detail in Thompson

et al. (2017). Orbiting VLBI stations face different obser-

vation parameter demands than ground-based stations. For

example, the integration time

is limited by the timescale of

phase coherence. Neglecting reference hardware coherence,

ground site phase coherence is dominated primarily by tur-

bulence in the atmosphere. However, for orbiting VLBI sta-

tions, the dominant constraints on the integration time arise

from thermal noise and the speed of the orbiter through the

(

) plane.

The motion of VLBI observing sites is crucial to Fourier

synthesis, but also introduces fundamental limitations on in-

tegration time. As the baseline vector

rotates, the phase of

the visibility measurement rotates, eventually picking up a

full phase wrap over the course of one averaged measure-

ment. Thompson et al. (2017) provide a bounding condition

on integration time to prevent a phase wrap, formalized for a

source confined to within an angle

FOV

τ <

FOV

(1)

Here,

is the angular velocity of the rotation of the observ-

ing site,

max

is the length of the longest baseline in

wavelengths, and

FOV

is in radians. For a nearly-circular

Low Earth Orbit (as we examine later), the rotation rate is

with

≈

5 hours. We are interested primarily in

filling in gaps in existing (

) coverage, so we focus on co-

herent averaging measurements out to the maximum baseline

of a LEO-enabled array, giving

≈

15 G

at 345 GHz. We

further assume that the source structure of interest is confined

to a circular angular extent of diameter 180

as, sufficient to

contain multiple shadow-scales, though likely not to image

extended structure, such as a jet. These values together yield

.

1 minute, giving a bound on coherent averaging of 30

seconds (Thompson et al. 2017).

To generalize the coherence time metric to satellites with

arbitrary orbital semi-major axis

orb

and eccentricity

, we

must find the maximum instantaneous angular velocity for

an eccentric orbit. Conservation of mechanical energy yields

the

vis-viva

equation for the orbital speed

orb

√

(

−

orb

)

(2)

where

is the instantaneous distance of a small mass from the

Earth center of mass and

is the gravitational param-

eter, simplified to the product of the gravitational constant

and the Earth mass

. The maximum instantaneous angu-

lar velocity occurs at periapsis, where

max

is given by

orb

when

orb

−

), yielding:

max

√

)

orb

−

)

(3)

Assuming that the integration time is held constant through-

out the orbit requires that the bound (Equation 3) hold for

the longest baseline in the orbital geometry, which occurs

approximately at apsis; for an orbit with apsis inclined at an

angle

relative to the source line-of-sight (with

π/

2 cor-

responding to the “face-on” orbit described later),

≈

orb

) sin

(4)

neglecting motion of ground sites. This relation holds only

if the longest baseline in the array is comparable to the base-

line from the orbiter to the center of the Earth, as would be

the case for a VLBI array with only one orbiter far from the

Earth. Otherwise, in the case of an array with, e.g., two di-

ametrically opposed orbiters, or one orbiter with

orb

com-

parable to the Earth radius (as is the case for the LEO orbits

we consider), this approximation should be increased by a

factor of 2 (denoted by brackets in the equation below). Sub-

stituting this approximation and our expression for

max

into

Equation 1 gives

max

≈

[2]

FOV

sin

√

orb

−

)

(5)

For the LEOs discussed in Section 3,

= 0 and

orb

is approx-

imately equal to the Earth radius. Taking the factor of 2 into

account and using

FOV

= 180

as,

= 0

87mm and

π/

recovers the

max

.

1 minute found earlier.

The sensitivity of an individual station is described by its

system equivalent flux density, or SEFD, which is given in

terms of the Boltzmann factor

, the system temperature

sys

, and the effective collecting area

eff

SEFD =

sys

eff

(6)

The sensitivity of a particular baseline is described by its

thermal noise, which depends on the SEFDs of its constituent

stations. The thermal noise is given by (Thompson et al.

2017)

√

SEFD

∆

ντ

(7)

where

∆

is the observing bandwidth and

is a digital cor-

rection factor due to finite quantization of the received ra-

dio emission. If 2-bit quanitization is used (as in the current

EHT),

= 0

88.

Small dishes contribute effectively to VLBI when forming

baselines to highly sensitive stations such as ALMA because

the thermal noise depends on the geometric mean of the sen-

sitivities of the constituent dishes. The LMT may also be

suitable as an “anchor” station for small dishes, should it ob-

serve at 345 GHz. The recently coherently phased ALMA

now has an SEFD at millimeter wavelengths on the order of

∼

100 Jy (Matthews et al. 2018). For the purposes of our

small-dish sensitivity computations, we use an orbiter with a

diameter of 4m. We assume an aperture surface efficiency of

80%; other factors such as illumination, blockage, etc., can

also contribute to the total aperture efficiency.

The

∼

4m class of dish has been successfully launched in

a non-deployable architecture (see, e.g. the

Herschel

instru-

ment, Pilbratt et al. 2010). Deployable architectures may also

be suitable for high-frequency performance (Wild et al. 2009;

Datashvili et al. 2014). We note, however, that 4m is not an

optimized diameter, and is adopted simply as a benchmark

“small dish” for the example calculations and reconstructions

that follow.

We thus compute the 345 GHz SEFD of a 4m dish to be

∼

20000 Jy, where we estimate the atmosphere-free system

temperature to be 75 K at 345 GHz (found by assuming simi-

lar performance to ALMA receivers at band 7 as in Matthews

et al. 2018). Using a

∼

150 Jy estimated zenith SEFD of

phased ALMA at 345 GHz, we can compute a minimum in-

tegration time

min

based on a desired nominal thermal noise

nom

by rearranging Equation 7:

min

SEFD

∆

(

nom

)

(8)

We choose a desired thermal noise of 10 mJy based on long-

baseline (

∼

7 G

) correlated flux densities of tenths of Jan-

skys observed for Sgr A* (Lu et al. 2018). This approxi-

mate mean sensitivity over a full observing track yields a

required

min

≈

1 second for space-ALMA baselines. Be-

tween the same LEO dish and a more typical ground site

with SEFD

≈

10000 Jy,

min

≈

80 seconds. Space-ALMA

baselines are thus necessary to reach ground-comparable sig-

nal quality within the motion-based decoherence of the VLBI

signal. For the simulated observations presented in this arti-

cle, we maintain the integration time at the 30 second limit

from Equation 1, guaranteeing detections to ALMA with-

out exceeding the motion-based limit. Space-ALMA detec-

tions would then allow calibration of all other space-ground

baselines on timescales shorter than the 80 second thermal

noise bound (see, e.g., Event Horizon Telescope Collabora-

tion et al. 2019a, for examples of network calibration with

ALMA).

2.4.

Past Efforts in Orbiting VLBI

The first Earth-space fringe detection was in 1986, using

the Tracking and Data Relay Satellite System (or TDRSS)

system in geostationary orbit at observing frequencies of

2.3 and 15 GHz (Levy et al. 1989). Non-geostationary or-

bits sweep through much broader baseline coverage and are

not fundamentally limited in baseline length; in 1997, the

VLBI Space Observatory Programme, or VSOP, brought the

8-m diameter Highly Advanced Laboratory for Communica-

tions and Astronomy (HALCA) into an elliptical Earth or-

bit with a period of approximately 6.6 hours and an apogee

of 21,000 km (Hirabayashi et al. 2000). HALCA was fol-

lowed by the 10-m diameter RadioAstron (or Spektr-R) (Kar-

dashev et al. 2013), with a period of 8.6 days and an apogee

of approximately 300,000 km. These missions operated at

centimeter wavelengths and successfully detected fringes de-

spite the difficulties of space-ground VLBI. Though some

ALUMBO ET AL

of these projects had a planned angular resolution similar

to the EHT (see Table 1), none was operating in the high-

frequency regime required to overcome the interstellar scat-

tering of emission from Sgr A*, which obscures near-horizon

structure at wavelengths as low as 3 mm (Issaoun et al. 2019).

These projects provide partial guidance for future efforts in

space-VLBI.

BASELINE COVERAGE

Orbiting VLBI elements are not bound by the surface or

rotation rate of the Earth, and can thus form a broader range

of baselines to stations on the ground on shorter timescales

than afforded by Earth-rotation Fourier synthesis. In particu-

lar, the orientation and period of the orbit can be chosen to fill

in gaps in the existing coverage of the array with greater flex-

ibility than is possible for a ground site, for which one must

account for such factors as altitude, weather, and infrastruc-

tural support. Here we present a simple example orbit for

rapid filling of (

) coverage of Sgr A* with space-ground

and space-space baselines when observing with the “EHTII”

array.

3.1.

Orbit Design and Simulation

We consider expanding the EHT to space in order to im-

prove instantaneous baseline coverage for dynamical imag-

ing of Sgr A*. This particular hypothetical space-enabled

EHT differs from previous space-VLBI missions such as

VSOP and Spektr-R (and its upcoming follow-up, Spektr-

M) and from other possible EHT expansion paradigms in

that we assume the ground-based EHT already provides suf-

ficient angular resolution to resolve the black hole shadow of

Sgr A*, and do not pursue major improvements to angular

resolution with longer space-ground baselines.

Instead, we utilize orbiting components of the EHT ar-

ray to fill in gaps in existing coverage over short timescales.

In the current EHT, large regions of missing (

) coverage

(Figure 1) limit the fidelity and dynamic range of recon-

structed images. Filling holes in the sampled (

) plane

reduces the magnitude of sidelobes in the Fourier trans-

form of the synthesized visibility measurements (or “dirty

beam”), generically improving image reconstructions across

algorithms.

To model space dishes operating in concert with the EHT,

we developed software to manipulate Two-Line Element sets

(or TLEs) and simulate VLBI observations with space dishes.

This software creates synthetic TLEs for arbitrary orbital el-

ements that are compatible with any Simplified Perturbation

Model-based orbit calculator (see, e.g., Wei & Zhao 2010).

Further, we can time-delay existing TLEs to precisely shift

orbital phase to any time relative to an EHT observing win-

dow, though we do not perform such an optimization in this

study. Instead, we choose an observation time in Greenwich

Figure 2.

Schematic diagram of a possible expansion to the EHT,

pictured as 4 Low Earth Orbiters always in view of Sgr A*. Thick

red baselines are shown at 0 GMST for the EHT’s expected ground-

based 345 GHz configuration, “EHTII.” Blue arrows correspond to

orbiter positions; cyan lines are space-space baselines. Other lines

are space-ground baselines grouped by color for each orbiter. Over

the course of a full 90 minute orbit, each orbiter contributes base-

lines across a wide range of (

) separations.

Mean Sidereal Time (or GMST) at which most EHT ground

stations can see the source; for an observation longer than ap-

proximately half of an orbital period, most baselines of inter-

est will be sampled, meaning that the initial phase is largely

irrelevant.

By rotating, time-delaying, and combining simple circular

orbits or orbits of existing space installations, constellations

of various orbiting configurations can be created and tested

in VLBI simulation environments such as

eht-imaging

(Chael et al. 2016, 2018a). For this initial examination of

LEO imaging capabilities, we use a constellation of orbiters

that can always see Sgr A*, and refer to such orbits as “face-

on.” Such orbits can be generated for a general source with

right ascension

and declination

by taking the right ascen-

sion of the ascending node

Ω

∓

◦

and the inclination

◦

, where signs are determined by the handedness

of the orbit relative to the source line of sight. We give our

orbiters a period of 90 minutes. A diagram of the relative

positions and baselines of such an orbit is shown in Figure 2.

This choice is useful for an initial examination of Earth-space

VLBI due to its continuous coverage over time. The total

additional (

) coverage provided by the orbiter is mostly

insensitive to the particular time window used to evaluate

coverage; thus, improvements to a dynamical reconstruction