Astronomy & Celestial Mechanics
the sky, computed
Spaceflight

An angle measured against a quasar

A tracking station measures how fast a spacecraft is receding, which is one number where three are wanted. The two missing angles come from the Earth's rotation, slowly, and near a planetary encounter there is no time for slowly — so the position is instead measured directly, as a difference of arrival times between two antennas, referred to a quasar a few degrees away.

Assumes Radiometric navigation, Interferometry and Orbit determination.

15 min read 8 figures Measuring the unreachableThe sky as a clockTwo bodies, and no more

A spacecraft is a point source with no angular extent, at a distance where nothing resolves anything. What a deep-space antenna measures is the frequency of the returned carrier, which gives the rate of change of the distance to a fraction of a millimetre a second, and the round-trip time of a ranging code, which gives the distance itself to a metre or two.

Both of those are the same direction. Neither says anything about where the spacecraft is across the line of sight, which is the direction that decides whether an orbit insertion happens or a flyby misses.

The imbalance is structural rather than technological. A radial measurement is a comparison of two frequencies or two times, and frequencies and times are the best-measured quantities in physics. A transverse measurement is a comparison of two directions, and there is nothing on board that points.

A metre along the line of sight and 4 kilometres across it. Above: the round-trip light time of a ranging signal, against distance. It is a straight line of slope one, which is the point — the delay is the distance, with no model and no calibration between them, and a code correlated to a few nanoseconds gives the radial coordinate of a spacecraft at Mars to about 1 metre. Below: what the same tracking session gives across the line of sight. The angle is good to 50 nanoradians — the Doppler noise alone would give 12, and the difference is the troposphere, the station coordinates and the solar plasma, none of which average down — and an angle times a distance is a length that grows: at Mars's 0.52 au it is 4 kilometres, 3890 times the radial error. The error region of an interplanetary spacecraft is therefore not a sphere but a pancake, thin along the line of sight and enormously wide across it, and every operational consequence follows from that shape: a Mars arrival is aimed at a corridor in the plane of the sky rather than at a distance, the last approach manoeuvres are almost entirely lateral, and the observation that improves the fit most is never another day of Doppler but a single differenced measurement against a quasar. The cheapest observable measures the coordinate that matters least.
Fig. 1 The asymmetry, drawn. Radiometric tracking locates a spacecraft to about a metre along the line of sight and to kilometres across it — three orders of magnitude worse in the direction nobody can afford to be wrong about. The along-track information is essentially free and the plane-of-sky information has to be worked for.

How the angles come out of a frequency, and why it is slow

There is a route to the angles from Doppler alone, and it is the one that carried planetary navigation for decades.

An eight-hour pass, and a 351 m s⁻¹ sinusoid that is the whole of the angle. Above: the range rate a two-way Doppler measurement returns over one pass from Goldstone, for a spacecraft receding at 14.6 km s⁻¹. Nothing here is an angle. The measurement is the fractional shift of a carrier the spacecraft coherently turned around and sent back, and its interpretation is that the distance is changing at some rate. Below: the same data with the spacecraft's own smooth signature removed. What is left is a sinusoid of exactly one cycle per day — the station's own motion, carried east at 379 metres a second by the rotation of the Earth, projected onto the line of sight. Its amplitude is that speed times cos δ and returns a declination of 22.0°; its zero crossing is the moment the spacecraft passed the meridian and returns the right ascension. The Earth's rotation is the interferometer. With Doppler good to 0.05 mm s⁻¹ at a 60-second cadence, 480 samples fit that amplitude to 0.003 mm s⁻¹ and the declination to 23 nanoradians — which is 4.7 milliarcseconds, from an instrument with no image plane and no angular resolution of any kind. What the picture cannot show is the part that makes this hard in practice: the spacecraft's own signature is not a straight line but a trajectory with unmodelled accelerations in it, and separating a slow non-gravitational force from a slow drift in the angles is the whole art of the fit.
Fig. 2 An eight-hour tracking pass. The station is carried east at a few hundred metres a second by the Earth’s rotation, so its own velocity along the line of sight to the spacecraft varies sinusoidally through the day. That sinusoid rides on top of the spacecraft’s own range rate, and its amplitude and phase depend on where the spacecraft is on the sky — the whole angular measurement is in the daily sinusoid, and the spacecraft’s own motion is the part that has to be modelled away.

The amplitude of the diurnal signature gives the declination and its phase gives the right ascension. The station’s own velocity is known exactly — it is the Earth’s rotation rate times the radius times the cosine of the latitude, a number good to many figures — so the sinusoid is a calibrated ruler laid across the sky once a day.

It works, and it has two structural weaknesses.

Two angles from one sinusoid, and each of them fails where the other is best. What a pass of 8 hours of 0.05 mm s⁻¹ Doppler from Goldstone is worth as an angle, against the declination of the spacecraft. The diurnal signature carries both angles and carries them differently. Its amplitude is the station's east velocity times cos δ, so inverting it for the declination divides by sin δ and the error runs away at the celestial equator: at 2° it is 244 nanoradians, thirty times worse than at 45°. Its phase is the right ascension directly, and recovering a phase from a sinusoid of amplitude proportional to cos δ divides by that cosine, so the right ascension is the one that fails near the pole. The two curves cross at exactly 45° because 1/sin and 1/cos do. Neither failure is a defect of the instrument, and neither can be reduced by tracking longer: at zero declination the station's motion is entirely along the line of sight all day, so no amount of it contains any information about how far north the spacecraft is. The geometry is degenerate rather than noisy. That is the whole reason a deep-space network also observes quasars: a differenced one-way range against a source of known position measures the plane-of-sky angle directly and does not care about the declination at all — and every Mars mission needs it, because a transfer to Mars spends most of its cruise within ten degrees of the equator.
Fig. 3 The first. The two angles are recovered from one sinusoid, and each fails where the other is best determined: the amplitude carries the declination and vanishes for a spacecraft near the celestial equator, so the declination becomes indeterminate exactly where the phase is most useful. Near zero declination the geometry degenerates and no amount of tracking helps.

The second is time. Extracting the sinusoid requires watching for hours, and interpreting it requires the spacecraft’s own acceleration to be modelled over the same hours — so a manoeuvre, an unbalanced thruster firing, or an unmodelled solar-radiation force appears as an apparent change of position. Approaching an encounter, when the trajectory is being corrected and the geometry is changing fastest, is precisely when both weaknesses bite.

There is a third weakness that is easy to miss and has bitten real missions. The diurnal sinusoid is being separated from the spacecraft’s own signature by their different time dependences, and anything on the spacecraft with a period near a day — a thermal cycle, an attitude-control duty cycle, a communications schedule — leaks straight into the angle. A tiny periodic acceleration that would be irrelevant to the trajectory is a large error in a quantity that is being read out of a daily modulation.

The measurement that is an angle directly

Two antennas separated by thousands of kilometres record the same signal. The wavefront reaches one before the other, by an amount that depends on the angle between the source direction and the baseline:

τ  =  Bccos⁡θ.\tau \;=\; \frac{B}{c}\cos\theta .

Differentiate and the angular sensitivity is σθ=c στ/B\sigma_\theta = c\,\sigma_\tau/B for a well-placed source. The Earth is eight to twelve thousand kilometres wide between the deep-space complexes, so a delay measured to a tenth of a nanosecond is a few nanoradians.

A nanosecond across the Earth is 35.69 nanoradians on the sky. The angular accuracy of a differenced-delay measurement against the length of the baseline it is measured on, both axes logarithmic, for three levels of delay precision. The relation is σ_θ = cσ_τ/B and nothing else, so every curve is a straight line of slope −1.00: the only two ways to measure an angle better are a better clock or a wider Earth, and only one of those is available. The three marked baselines are the ones that exist — the deep-space complexes in California, Spain and Australia, 8,400, 10,600, 11,700 kilometres apart. On the longest of them a delay good to 0.05 nanoseconds is 1.28 nanoradians, which at 0.52 astronomical units is 100 metres across the line of sight; a more typical 0.15-nanosecond measurement on the shortest baseline is 5.35 nanoradians. What makes any of this survivable is that the same pair of antennas observes a quasar a few degrees away immediately afterwards. The quasar is at infinity, its position is known better than the measurement, and subtracting its delay from the spacecraft's removes the clock offsets, the water vapour over each dish and the station coordinates in one step — so the number that comes out is not a delay at all but an angular separation from a fixed point in the sky.
Fig. 4 The scaling, and the baselines that exist. Angular accuracy falls with baseline as a slope of exactly minus one, so the only two ways to improve an angle are a better clock or a wider Earth, and only one is available. On the longest baseline a delay good to fifty picoseconds is a little over one nanoradian, which at half an astronomical unit is a hundred metres across the line of sight — three orders of magnitude better than the Doppler route, obtained in minutes rather than hours.

This is interferometry, and it is the same measurement as resolving a star with two separated telescopes, with the difference that a spacecraft is a point and the quantity wanted is its position rather than its size.

The delay is measured as a group delay rather than a phase delay, and the distinction matters. A phase delay is ambiguous by whole cycles of the carrier, which at eight gigahertz is under four centimetres of path — precise and useless without an independent way to count the cycles. A group delay is obtained from how the phase varies across a spanned bandwidth, so its precision goes as the reciprocal of the span rather than of the carrier frequency, and it is unambiguous. Spacecraft transmit deliberately spaced tones for exactly this purpose, and the span of those tones is a design parameter chosen against the navigation requirement.

Why the delta is the whole technique

A raw delay is not measurable to fifty picoseconds. The two stations have independent clocks that drift; the signal passes through different amounts of troposphere and ionosphere above each dish; the station coordinates are known to centimetres rather than to nothing; and the Earth’s orientation wanders.

Every one of those errors is very nearly the same for two sources a few degrees apart on the sky. So the observation is made twice: the spacecraft, then immediately a quasar in almost the same direction, then the spacecraft again. Subtracting the two delays removes everything common, and what survives is the angular separation between the spacecraft and the quasar.

The quasar is the right reference for three reasons. It is at cosmological distance, so it has no measurable proper motion and no parallax — it is as close to a fixed point as the universe supplies. Its position is known to a fraction of a nanoradian, because thousands of them have been observed for decades to define the celestial reference frame. And it is a broadband noise source, so the same group-delay technique that works on a spacecraft’s tones works on it. The result is not a delay and it is not a position. It is an angular separation from a catalogued point in the sky, which means the measurement is delivered directly in the frame the ephemerides are expressed in — with no intermediate step in which the station’s own coordinates or the Earth’s orientation have to be believed.

An eight-hour pass, and a 189 m s⁻¹ sinusoid that is the whole of the angle. Above: the range rate a two-way Doppler measurement returns over one pass from Goldstone, for a spacecraft receding at 14.6 km s⁻¹. Nothing here is an angle. The measurement is the fractional shift of a carrier the spacecraft coherently turned around and sent back, and its interpretation is that the distance is changing at some rate. Below: the same data with the spacecraft's own smooth signature removed. What is left is a sinusoid of exactly one cycle per day — the station's own motion, carried east at 379 metres a second by the rotation of the Earth, projected onto the line of sight. Its amplitude is that speed times cos δ and returns a declination of 60.0°; its zero crossing is the moment the spacecraft passed the meridian and returns the right ascension. The Earth's rotation is the interferometer. With Doppler good to 0.05 mm s⁻¹ at a 60-second cadence, 480 samples fit that amplitude to 0.003 mm s⁻¹ and the declination to 10 nanoradians — which is 2.0 milliarcseconds, from an instrument with no image plane and no angular resolution of any kind. What the picture cannot show is the part that makes this hard in practice: the spacecraft's own signature is not a straight line but a trajectory with unmodelled accelerations in it, and separating a slow non-gravitational force from a slow drift in the angles is the whole art of the fit.
Fig. 5 The same pass for a spacecraft at sixty degrees declination rather than twenty-two. The diurnal sinusoid falls from its low-declination amplitude to 189 metres a second, because what modulates the range rate is the station’s east velocity times the cosine of the declination. The signal that carries the angle vanishes as the spacecraft approaches the pole, and that is not a defect of the technique but its geometry: a spacecraft directly over the Earth’s axis is not swept past by the station’s rotation at all.

What it buys at an encounter

The numbers are worth stating because they decide mission design.

Approaching Mars, a hundred metres of plane-of-sky knowledge translates into an entry-corridor placement of a few kilometres, which is the difference between landing inside an ellipse a few kilometres across and one a few tens of kilometres across. For a flyby the requirement is different and no less severe. Aiming at a plane rather than at a planet is how an encounter is targeted, and the aim point is specified as two coordinates in that plane — both of which are plane-of-sky quantities at the time of the last manoeuvre.

The measurement chain, stated as a chain

It is worth setting out what is assumed at each step, because the list is unusually short.

The two antennas record voltage against time, tagged with hydrogen-maser timestamps. The recordings are correlated to find the delay that maximises their agreement — the spacecraft’s, then the quasar’s. The two delays are subtracted. The difference is divided by the baseline length to give an angle, and added to the quasar’s catalogued position.

What is not in that list: no model of the spacecraft’s dynamics, no assumption about its acceleration, no solar-system ephemeris, no atmosphere model beyond the assumption that the atmosphere is similar in two nearby directions, and no absolute time standard beyond the requirement that the two stations’ clocks stay stable across the few minutes of the switching cycle.

A measurement with a short chain is not merely more accurate; it fails differently. A Doppler-derived angle degrades gracefully and misleadingly when the spacecraft’s acceleration model is wrong, producing a confident wrong answer. A delta-DOR measurement with a problem usually produces no correlation at all.

An eight-hour pass, and a 351 m s⁻¹ sinusoid that is the whole of the angle. Above: the range rate a two-way Doppler measurement returns over one pass from Goldstone, for a spacecraft receding at 14.6 km s⁻¹. Nothing here is an angle. The measurement is the fractional shift of a carrier the spacecraft coherently turned around and sent back, and its interpretation is that the distance is changing at some rate. Below: the same data with the spacecraft's own smooth signature removed. What is left is a sinusoid of exactly one cycle per day — the station's own motion, carried east at 379 metres a second by the rotation of the Earth, projected onto the line of sight. Its amplitude is that speed times cos δ and returns a declination of 22.0°; its zero crossing is the moment the spacecraft passed the meridian and returns the right ascension. The Earth's rotation is the interferometer. With Doppler good to 0.05 mm s⁻¹ at a 60-second cadence, 960 samples fit that amplitude to 0.002 mm s⁻¹ and the declination to 16 nanoradians — which is 3.3 milliarcseconds, from an instrument with no image plane and no angular resolution of any kind. What the picture cannot show is the part that makes this hard in practice: the spacecraft's own signature is not a straight line but a trajectory with unmodelled accelerations in it, and separating a slow non-gravitational force from a slow drift in the angles is the whole art of the fit.
Fig. 6 Sixteen hours of the same pass rather than eight. The sinusoid completes more of its cycle and its amplitude is fitted better for it — 351 metres a second here, read off a longer arc — and the smooth spacecraft signature underneath has had twice as long to accumulate. A longer pass buys the angle and costs the separation between the angle and everything else, which is the trade that decides how a tracking schedule is built.

What is left after the differencing

The residual error budget is short, which is the point of the technique, and each remaining term is instructive.

The quasar’s own position, which is a fraction of a nanoradian and is the ultimate floor. The imperfect cancellation of the troposphere, because the two sources are a few degrees apart and the atmosphere is not uniform on those scales — which is why the reference quasar is chosen as close to the spacecraft as one can be found, and why the technique degrades when none is nearby. The spacecraft’s own signal structure, since a group delay is measured across the spanned bandwidth of the transmitted tones and a wider span gives a better delay. And the switching interval, since anything that changes between the two observations does not cancel.

None of those terms grows with distance. The angular accuracy is the same at Jupiter as at Mars — but the transverse position error is the angle multiplied by the distance, so at Saturn one nanoradian is a kilometre rather than a hundred metres, and the technique’s usefulness at the outer planets is correspondingly reduced.

Two angles from one sinusoid, and each of them fails where the other is best. What a pass of 8 hours of 0.02 mm s⁻¹ Doppler from Goldstone is worth as an angle, against the declination of the spacecraft. The diurnal signature carries both angles and carries them differently. Its amplitude is the station's east velocity times cos δ, so inverting it for the declination divides by sin δ and the error runs away at the celestial equator: at 2° it is 98 nanoradians, thirty times worse than at 45°. Its phase is the right ascension directly, and recovering a phase from a sinusoid of amplitude proportional to cos δ divides by that cosine, so the right ascension is the one that fails near the pole. The two curves cross at exactly 45° because 1/sin and 1/cos do. Neither failure is a defect of the instrument, and neither can be reduced by tracking longer: at zero declination the station's motion is entirely along the line of sight all day, so no amount of it contains any information about how far north the spacecraft is. The geometry is degenerate rather than noisy. That is the whole reason a deep-space network also observes quasars: a differenced one-way range against a source of known position measures the plane-of-sky angle directly and does not care about the declination at all — and every Mars mission needs it, because a transfer to Mars spends most of its cruise within ten degrees of the equator.
Fig. 7 The same information content at 0.02 millimetres a second of Doppler precision rather than 0.05. Both curves improve by the ratio and neither changes shape: right ascension is best where declination is worst and the other way about, because one is carried by the sinusoid’s phase and the other by its amplitude. Improving the instrument moves both curves down and moves neither sideways, so the place where the technique is blind is a property of the geometry that no precision removes — which is the argument for the second technique below.

Why the two techniques are used together

Neither method replaces the other, and the reason is that they constrain different combinations and fail in different circumstances.

Doppler is continuous, cheap, available from a single station, and extraordinarily good along the line of sight. Delta-DOR is intermittent, expensive, needs two stations, and is extraordinarily good across it. A navigation solution is a weighted fit to both, and in practice the range and Doppler determine the orbit’s size and shape while the interferometric points pin its orientation on the sky.

There is a second reason to keep both, which is that they are sensitive to different systematic errors. A mismodelled solar-radiation pressure corrupts the Doppler-derived angles badly and the interferometric ones hardly at all; an error in the assumed station coordinates does the reverse. When the two disagree by more than their quoted uncertainties, something in the model is wrong, and the disagreement is the diagnostic. A navigation team’s confidence comes less from either measurement’s precision than from the two of them agreeing when they had every opportunity not to — the same logic that makes a cluster weighed three ways more convincing than any one of the three.

Where the picture stops

It requires two stations simultaneously, which is an expensive scheduling constraint on a network with more spacecraft than antennas, and it therefore tends to be used in the weeks around a critical event rather than routinely.

It requires a suitable quasar, and the sky is not uniformly supplied with strong compact ones. A spacecraft in a poor part of the sky gets a more distant reference and a worse cancellation.

The reference frame is maintained rather than given. The quasar positions are a catalogue produced by decades of geodetic observing, and it is revised: sources vary, some have structure that moves, and a source whose emitting region shifts by a fraction of a milliarcsecond shifts every measurement referred to it. Where a source is depends on the frame and the epoch, and the frame here is an artefact maintained by continuous work.

And it measures one component per baseline. A single pair of antennas gives the angle along their baseline and nothing across it, so a full two-dimensional position needs two baselines with different orientations — in practice, two passes on differently oriented station pairs, separated by hours.

A nanosecond across the Earth is 35.69 nanoradians on the sky. The angular accuracy of a differenced-delay measurement against the length of the baseline it is measured on, both axes logarithmic, for three levels of delay precision. The relation is σ_θ = cσ_τ/B and nothing else, so every curve is a straight line of slope −1.00: the only two ways to measure an angle better are a better clock or a wider Earth, and only one of those is available. The three marked baselines are the ones that exist — the deep-space complexes in California, Spain and Australia, 8,400, 10,600, 11,700 kilometres apart. On the longest of them a delay good to 0.05 nanoseconds is 1.28 nanoradians, which at 0.52 astronomical units is 100 metres across the line of sight; a more typical 0.15-nanosecond measurement on the shortest baseline is 5.35 nanoradians. What makes any of this survivable is that the same pair of antennas observes a quasar a few degrees away immediately afterwards. The quasar is at infinity, its position is known better than the measurement, and subtracting its delay from the spacecraft's removes the clock offsets, the water vapour over each dish and the station coordinates in one step — so the number that comes out is not a delay at all but an angular separation from a fixed point in the sky.
Fig. 8 The same relation with the systematic floor at twenty nanoradians rather than fifty. Each curve is still σθ=cστ/B\sigma_\theta = c\sigma_\tau/B and still a straight line of slope minus one, and the floor is still a floor — it just sits lower. Nothing about the baseline can beat a systematic, and the whole of the next section is about what has to be measured, modelled or differenced away to move that horizontal line at all.

What a picosecond has to survive

Fifty picoseconds is one and a half centimetres of light travel, and the chain that delivers it is worth setting out because every link is a place the number could have been lost.

Each station has a hydrogen maser, whose fractional frequency stability over the few minutes of a switching cycle is about one part in 101510^{15}. Over a three-minute observation that is a few femtoseconds of accumulated timing error — four orders of magnitude below the requirement, which is why the clocks are not the limitation and are not permitted to become one.

The received signal is digitised and timestamped against that maser and recorded, and it is never demodulated in the way a communications link demodulates it. What is wanted is the raw voltage as a function of time at both stations, so that the two can be correlated afterwards; the delay is the lag at which their correlation peaks. That means the data volume is set by the bandwidth rather than by any information content, and both recordings have to reach the same place before the measurement exists.

The correlation itself is the step that turns two noisy voltage streams into a number. For a quasar the signal is broadband noise and there is nothing to lock onto, so the peak is found by trying every plausible lag — and the peak’s position can be located to a small fraction of the reciprocal bandwidth because it is fitted rather than merely detected.

Then the two delays are subtracted, and the subtraction is the load-bearing operation. The station coordinates are known to a couple of centimetres and drop out; the clock offset between the masers is a nanosecond or more and drops out; the dry troposphere contributes about two metres of excess path at zenith and mostly drops out. What is left is dominated by the thing that did not have time to cancel — the wet troposphere, which is variable on minutes and is why the switching cycle is short and why the two sources have to be close together on the sky.

The schedule an encounter imposes

A navigation measurement is only useful if there is time to act on it, and that constraint shapes the observing schedule more than the accuracy does.

Consider the sequence at an arrival. A tracking pass is recorded, the data are correlated and reduced, the orbit determination is rerun with the new points, a correction manoeuvre is designed, it is reviewed, it is uplinked, and the spacecraft executes it. Each of those steps takes hours, and the light time itself takes tens of minutes at Mars and hours at Saturn.

So there is a knowledge cutoff: a moment after which no further measurement can influence the trajectory, because the manoeuvre that would use it cannot be designed and executed in time. Everything after that point is monitoring rather than navigation.

The interferometric passes are therefore booked backwards from the cutoff rather than forwards from arrival, and their spacing is set by how long a reduction takes. A dense sequence of passes in the final week is worth much less than the same number spread over the preceding month, because the last few arrive after the trajectory has been frozen.

That schedule also decides which errors matter. An error that is constant across the campaign — a station coordinate, a catalogue position — biases every measurement identically and is partly absorbed by the fit. An error that varies from pass to pass adds noise, and the fit averages it down as the square root of the number of passes. The measurement that decides an arrival is a fit to a handful of points obtained weeks earlier, and the accuracy quoted for a single pass is not the accuracy the manoeuvre was designed with.

The same technique, pointed the other way

A footnote that is not really a footnote: the identical measurement is used to determine the Earth rather than the spacecraft.

The delay between two antennas observing a quasar depends on the baseline vector as well as on the source direction, and the baseline vector rotates with the Earth. Observing many quasars from many stations therefore solves for the Earth’s orientation — the length of the day, the position of the pole, the precession and nutation of the axis — at the level of tens of microarcseconds. That is the primary determination of those quantities, better than any other technique, and it is what makes the wobble inside the wobble an observed quantity rather than a computed one.

The two applications share their hardware, their correlators and their reference catalogue, and they differ only in which side of the equation is treated as unknown. In the geodetic case the sources are known and the baseline is solved for; in the navigation case the baseline is known and the source is solved for. It is the same measurement read in two directions, and each supplies the other’s calibration.

Where this ladder goes next

Later rungs on this anchor: the group-delay measurement itself, and why the spanned bandwidth rather than the carrier frequency sets the precision; the celestial reference frame the quasars define, and how it is maintained; the same-beam variant, in which two spacecraft at one body are observed together and almost every error cancels exactly; optical navigation and the fusion of camera and radio data; and the use of spacecraft tracking to improve planetary ephemerides, where the navigation product becomes the science product.

Where this goes next

An eight-hour pass, and a 351 m s⁻¹ sinusoid that is the whole of the angle. Above: the range rate a two-way Doppler measurement returns over one pass from Goldstone, for a spacecraft receding at 14.6 km s⁻¹. Nothing here is an angle. The measurement is the fractional shift of a carrier the spacecraft coherently turned around and sent back, and its interpretation is that the distance is changing at some rate. Below: the same data with the spacecraft's own smooth signature removed. What is left is a sinusoid of exactly one cycle per day — the station's own motion, carried east at 379 metres a second by the rotation of the Earth, projected onto the line of sight. Its amplitude is that speed times cos δ and returns a declination of 22.0°; its zero crossing is the moment the spacecraft passed the meridian and returns the right ascension. The Earth's rotation is the interferometer. With Doppler good to 0.05 mm s⁻¹ at a 60-second cadence, 480 samples fit that amplitude to 0.003 mm s⁻¹ and the declination to 23 nanoradians — which is 4.7 milliarcseconds, from an instrument with no image plane and no angular resolution of any kind. What the picture cannot show is the part that makes this hard in practice: the spacecraft's own signature is not a straight line but a trajectory with unmodelled accelerations in it, and separating a slow non-gravitational force from a slow drift in the angles is the whole art of the fit. A position measured from a frequency A spacecraft is unresolvable and unreachable, and everything known about where it is comes from two scalars — a round-trip light time and a Doppler shift. Neither is an angle. The orbit solution returns two angles anyway, because the antenna is bolted to a rotating planet. 20 closure phases, unmoved by an atmosphere that ruins every baseline. Closure phase measured against closure phase true, for all 20 triangles of a 6-antenna array observing a binary 3.2 mas apart with a flux ratio of 0.35. Each antenna has been given an independent atmospheric phase of 65° rms, which corrupts the individual baseline phases by 102° rms — several times the 31.2° the source itself produces, so no single visibility phase in this simulation carries usable information. Every point here nonetheless sits exactly on the diagonal: the per-antenna terms cancel identically round any triangle, and the largest departure over all 20 is 2.5e-14 degrees, which is round-off. The price is in the counting. 6 antennas give 15 baseline phases of which 5 are consumed by the unknowns, so of the 20 triangles only 10 closures are independent — a fraction (N−2)/N = 0.667 of the phase information. For two antennas that fraction is zero and there is no closure at all; the Event Horizon Telescope's image rests on quantities of this kind and on no absolute phase whatever. A phase that survives what corrupts it An atmosphere over each antenna adds an unknown to the phase of every baseline that antenna takes part in. Sum the phases round a triangle and every one of those unknowns cancels identically — which is the reason an image can be made across ten thousand kilometres, and the reason it has no position on the sky. One star, two coordinate systems, at latitude 52°. The equatorial grid and the horizon grid drawn on the same sphere for an observer at latitude 52°. The star marked has declination 20° and hour angle -40° in the first, and altitude 45.5° and azimuth 239.4° in the second. The two frames differ by a single rotation through the co-latitude 38°, which is why the celestial pole stands 52° above the northern horizon. Where a star is depends on who is asking The sky needs two coordinate systems because two different things stay still in it — the observer's horizon and the stars themselves. One rotation converts between them, and the angle of that rotation is the time.

What links here

The 8 of 9 essays linking to this one that name the most of the same objects.

The objects this essay names

Each one links to every other essay that touches it.

Angular accuracyBaselineClock offsetDeep-space networkDelta-DORDifferential measurementEncounter navigationGroup delayInternational celestial reference framePlane of sky positionTroposphereVery long-baseline interferometry