*Summary:*

⦿ *For spacecraft near to the Earth this is routinely done with radar tracking using ground-based antenna.*

⦿ *Radar tracking is less accurate for deep space, and so is augmented by the Delta-Differential One-Way Ranging **(∆DOR) technique**. This uses two widely-spaced ground stations to receive a signal sent by the spacecraft.*

⦿ *Beyond the solar system, spacecraft will be too distant to rely on Earth-based tracking.*

⦿ *First is pulsar technique: By measuring the arrival time of a pulse from a single pulsar and comparing it to the expected arrival time, we can determine the spacecraft position relative to its expected position along the line-of-sight to the pulsar.*

⦿ *Another way** to determine spacecraft position in deep space is via direct triangulation of stars.*

⦿ *Recently, Coryn Bailer-Jones also developed and test a scheme for determining the 3D position and 3D velocity (6D coordinates) of a spacecraft from measurements of the angular positions and/or radial (Doppler) velocities of stars.*

*Article: *

How do we determine our position and velocity in space? For spacecraft near to the Earth this is routinely done with radar tracking using ground-based antenna. A signal sent to and then returned by a spacecraft can be used to determine the spacecraft’s distance via the time delay and its radial velocity via the Doppler effect. Accuracies of 1 m and 1 mm s¯^{1} respectively are routinely achieved in this way. The direction to the spacecraft (two-dimensional position) is also determined from the antenna pointing, albeit less accurately. By tracking the spacecraft in this way over time, its orbit may be computed. This can be done more quickly or more accurately by tracking simultaneously with two or more ground stations to enable a triangulation.

Radar tracking is less accurate for deep space, and so is augmented by the Delta-Differential One-Way Ranging (∆DOR) technique. This uses two widely-spaced ground stations to receive a signal sent by the spacecraft. The time-delay between the receipt of the signals establishes the angular position of the spacecraft, the accuracy of which is then improved by repeating this measurement for a quasar that lies within a few degrees of the line-of-sight. This provides the direction to the spacecraft relative to the quasar, the position of which is already known through earlier observations. An accuracy of the order 10 nrad (2 mas) may be achieved, which corresponds to a transverse positional accuracy of 1.5 km for a spacecraft 1 au (1.5 × 10^{8} km) from the Earth.

Beyond the solar system, spacecraft will be too distant to rely on Earth-based tracking. When travelling to the nearest stars, signals will be far too weak and light travel times will be of order years. An interstellar spacecraft will therefore have to navigate autonomously, and use this information to decide when to make course corrections or to switch on instruments. Such a spacecraft needs to be able to determine its position and velocity using only onboard measurements. In principle this can be done by integrating over internal measurements from clocks, gyroscopes, and accelerometers, but in practice this would be neither accurate nor reliable enough for mission durations of decades.

Pulsars are a well-studied proposed solution to the problem of deep space navigation. Pulsars are rapidly rotating neutron stars that emit narrow radio and Xray pulses at very stable and well-defined rates, with periods ranging from milliseconds to seconds. By measuring the arrival time of a pulse from a single pulsar and comparing it to the expected arrival time, we can determine the spacecraft position relative to its expected position along the line-of-sight to the pulsar. As pulses are indistinguishable, this only establishes the position as somewhere on one of an infinite number of planes that are separated by a distance cT, where c is the speed of light and T is the period of the pulsar. By repeating this measurement for multiple pulsars in different directions, we can break the degeneracies and determine the three-dimensional position of the spacecraft. This method is analogous (but not identical) to global positioning systems on Earth, which use satellites as opposed to pulsars as navigation beacons. The accuracy of pulsar navigation is set, among other things, by the accuracy with which the pulsars’ periods and directions have been determined in advance (on the Earth). Simulations by Shemar and colleagues predict that a spacecraft position could be determined to an accuracy of 2 km for a spacecraft up to 30 au from the Earth. Errors in the timing model due to dispersion by the interstellar medium may reduce this accuracy at light year distances, however.

Another way to determine spacecraft position in deep space is via direct triangulation of stars. If the 3D positions of a set of stars relative to some reference frame are known, then the observed parallactic shift of these positions can be used to compute the position of the spacecraft in the reference frame. In reality stars are moving relative to the spacecraft, and due to the finite speed of light this introduces an apparent shift of the stars’ positions due to aberration. The size of this aberrational shift depends on the relative velocity, and so we could exploit this to determine the velocity of the spacecraft. Indeed, if we use sources such as quasars that are effectively infinitely far away, and so exhibit negligible parallactic shifts, we could use the aberration to infer just the velocity.

Now, Coryn Bailer-Jones in his recent paper, develop and test a scheme for determining the 3D position and 3D velocity (6D coordinates) of a spacecraft from measurements of the angular positions and/or radial (Doppler) velocities of stars. These are the only plausible measurements of stellar positions and velocities that can be made both quasi-instantaneously and without reference to an external system. As it would be difficult in practice to establish the absolute 2D angular coordinates of stars onboard a spacecraft, positional measurements are limited to one-dimensional angular distances between pairs of stars, as can be made with a sextant, for example.

The navigation scheme uses a catalogue of 6D coordinates of the stars in some reference system. Astrometric catalogues provide five of these six coordinates – latitude, longitude, distance, and two transverse velocities – with the sixth provided by radial velocity measurements from optical spectrographs. Space astrometry and spectroscopy with the Gaia spacecraft now provide such catalogues at sub-milliarcsecond and km s¯^{1} accuracy for millions of stars. The principle of constructing these catalogues is in some sense the inverse of navigation: the positions and velocities of stars are determined in the reference system – usually the International Celestial Reference System (ICRS) – using the known position and velocity of the observer relative to this reference system. The scheme I presented in his paper takes into account the parallax, space motion, and aberration of the stars.

‘The scheme exploits the parallax and aberration of the stars, both of which depend on the position and velocity of the spacecraft, by making measurements only of the angular distances between stars. Using multiple stars we can untangle the aberrational and parallactic contributions to the observed angular shifts to infer the 3D position and 3D velocity of the spacecraft relative to the Solar System Barycenter(SSB).’— told Coryn Bailer-Jones, author of the study.

With 20 stars and an onboard measurement accuracy of 100, he showed via simulations that the position and velocity of the spacecraft can be determined to within 3 au and 2 km s¯^{1} respectively. Increasing this to 100 stars improves the accuracies to 1.3 au and 0.7 km s¯^{1} respectively. The accuracy improves approximately as the inverse of the square root of the number of stars. The navigational accuracies are found to be in direct proportion to the measurement accuracy: With 20 stars and measurement accuracies of 1”, 0.1”, and 0.001”, he achieved positional accuracies of 3 au, 0.3 au, and 0.003 au respectively, and velocity accuracies of 2 km s¯^{1}, 0.2 km s¯^{1}, and 0.002 km s¯^{1} respectively. As aberration is a large effect – more than 1″ for velocities above 1.5 km s¯^{1} – the accuracy of the velocity determination is essentially independent the spacecraft velocity, so is as good for relativistic as for non-relativistic spacecraft.

The method uses MCMC to sample the likelihood (formally the posterior with a uniform prior), and from the resulting set of samples we can estimate the uncertainties in the inferred parameters.

“We find these to be close to the amplitudes of the residuals in general, making them a useful measure of the accuracy of the inferred parameters in a real-world situation.”— told Coryn Bailer-Jones, author of the study.

He also manifested that we may also measure stellar radial velocities from the spacecraft, as these too encode information about both the position and velocity of a star when compared to the catalogue. Whether these are useful depends on their accuracies. Using only radial velocities with an accuracy of 10 km s¯^{1} gives poor results: with 20 stars the positional accuracy is 160 au and the velocity accuracy is 10 km s¯^{1}. Improving the measurement accuracy by some factor improves the accuracy of the inferred positions and velocities by the same factor, as was the case with angular distance measurements.

Combining the two types of measurements may improve the accuracy attainable from either alone. For example, with 20 stars and 1″ angular distances, adding radial velocities measured to 1 km s¯^{1} improves the position and velocity accuracies on average by factors of 2.8 and 4.5 respectively. But if the radial velocities are only accurate to 10 km s¯^{1}, the positional accuracy only improves by 10% and the velocity accuracy not at all. As it may be difficult to achieve stellar radial velocities more accurate than a few km s¯^{1} from an interstellar spacecraft, but it is comparatively easy to measure angles to 1″, attaining higher navigation accuracies should focus on achieving more accurate angular measurements. Additional stars could be used, but the performance only improves as 1/ √ N, in line with expectations.

Due to the motion of the stars, and the assumption that a relativistic spacecraft could not keep track of SSB time, they were forced to formally include the measurement time as a seventh unknown parameter in the inference. This has only a small impact on the inference, however, and consequently cannot be inferred very accurately.

The results were computed for spacecraft placed randomly in space up to 10 ly from the Sun with velocities up to 0.5c. The first interstellar missions are likely to be local, so we can use just nearby stars or this navigation. For more distant sojourns, the best performance would be obtained by using those stars expected to be nearest to the spacecraft, and surrounding it reasonably isotropically in its rest frame. The navigation will continue to work just as well for spacecraft at least as far as the average distance of accurately-measured stars in the astrometric catalogue which, with Gaia, is hundreds of light years. The position and velocity accuracies are likewise independent of the velocity of the spacecraft (relative to the SSB), although at extreme relativistic velocities some degradation will eventually occur due to the tiny sky coverage of the stars as seen from the spacecraft.

This study is primarily conceptual. Although they only rely on measurements that could be made from a relativistic spacecraft, e.g. they have not assumed the availability of a fixed reference frame, they have not considered the instruments themselves. Angular distances could be measured using a highly accurate sextant, which is similar in principle to the astrometric instruments on Hipparcos (1 mas accuracy) and Gaia (a few µas accuracy), except that these have a fixed “basic angle” between the two fields-of-view; they then allow stars to drift over the observing field, essentially converting time differences between focal plane crossings into angular distances, similar to how ground-based meridian circles operate. As 1″ can easily be achieved by direct-imaging commercial star trackers, it seems reasonable to assume that a specially-designed space sextant could do a lot better. Random errors can easily be beaten down through multiple measurements: observation time (or photons) is hardly an issue for a decades-long mission. The limiting factor will be systematic errors.

“Whether we can get to Hipparcos or Gaia accuracies depends strongly on the size of the spacecraft, and so on what metrology can be introduced to determine the basic angle.”— told Coryn Bailer-Jones, author of the study.

Some other implementational issues have not been considered. An important one is binary stars.

“I have assumed that the star catalogue gives us whatever information we need to extrapolate the positions and velocities of stars over decades. The accuracy of Gaia parallaxes and proper motion, combined with ground-based radial velocities, is sufficiently high to ensure this is possible for single stars. But stars in compact binary systems can have large enough accelerations that the assumption of linear motion is inadequate. Either we need to model their accelerations, or we must exclude them from the catalogue.”— told Coryn Bailer-Jones, author of the study.

**Reference***: (1) Coryn A.L. Bailer-Jones, “Lost in space? Relativistic interstellar navigation using an astrometric star catalogue”, ArXiv, pp. 1-20, 2021. https://arxiv.org/abs/2103.10389 (2) James N., Abellob R., Lanucarab M., Mercolinob M., Madde R., 2009, “Implementation of an ESA delta-DOR capability”, Acta Astronautica, 64, 1041–1049. https://www.sciencedirect.com/science/article/abs/pii/S0094576509000101 (3) Iess L., Di Benedetto M., James N., Mercolino M., Simone L., Tortora P., 2014, Astra: Interdisciplinary study on enhancement of the end-to-end accuracy for spacecraft tracking techniques, Acta Astronautica, 94, 699–707. https://www.sciencedirect.com/science/article/abs/pii/S0094576513002014 (4) Hoag D.G., Wrigley W., 1975, Navigation and guidance in interstellar space, Acta Astronautica, 2, 513–533. **https://ui.adsabs.harvard.edu/abs/1975AcAau…2..513H/abstract*

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