Determining spacecraft position & velocity using the trilateration of gravitational waves.
For more articles by Nathan Parrott, visit: www.nathanparrott.com
A pair of rotating white dwarves (J2322+0509) with a short orbital period of 1,201 seconds has been detected in a binary orbit and can act as a first set of “Verification Binaries” for the upcoming eLISA missions. They are the first gravitational wave sources of their kind ever to be identified. They should provide a characteristic wave source for 6–7 million years before coalescing into a single, more massive, white Dwarf.
“Theories predict that there are many double helium-core white dwarf binaries out there,” Warren Brown, CfA astronomer and lead author on the study, said in a statement. “This detection provides an anchor for those models, and for doing future experiments so that we can find more of these stars and determine their true numbers.”
Having a large catalogue of known binary white dwarves lends credence to the Galactic Navigation System described in detail below.
Current spacecraft positioning and navigation systems are well suited for operations within our own solar system. However, as mankind continues to push out into deep space and explore interstellar space, the traditional methods employed become far less practical. This paper explores the possibility of a spacecraft determining its position and velocity using the on-board detection of multiple, continuous, gravitational wave sources. This technique would allow spacecraft to determine their location over a much larger area than is currently possible (for example, within or between galaxies) all without any reliance upon the earth. The practicality of this system and its current and theoretical accuracy will be explored with the results given.
How could a space craft, tumbling through interstellar space, re-orientate itself? Or how can a spacecraft cruising towards another galaxy, make a mid-course trajectory correction to ensure it arrives accurately at its destination?
These questions may seem like distant problems set arbitrarily far into the future, however, interstellar travel has been deemed at least possible by authors like Forward  and projects like Breakthrough Starshot aim to send light propelled nanocraft to Alpha Centuri within our lifetime (breakthroughinitiatives.org).
Spacecraft navigation systems of today rely upon precisely timed radio signals sent back and forth between the craft and the earth in order to triangulate the craft’s exact location. However, as humans seek to explore beyond our own solar system, this form of positioning becomes impractical due to the large communication delays and the very weak signals being received due to the vast distances involved. The ability to self localise, therefore, provides a significant benefit for any future interstellar or intergalactic mission.
Gravitational Waves (GW) were first predicted by Albert Einstein as a consequence of his general theory of relativity  and have been indirectly observed through the measurements of the orbits of binary pulsar systems. The first of these binary pulsar systems was discovered in 1974 by Hulse and Taylor  who were able to show that the orbit of the pulsars degraded precisely as predicted as a result of the energy lost due to relativistic gravitational-wave emission. Direct measurement of gravitational waves occurred for the first time at the Laser Interferometer Gravitational-Wave Observatory (LIGO), on the 14th September 2015. The measured signal (GW150914) was the result of a binary black hole system coalescing to form a single black hole  and closely matched the wave function predicted by general relativity with a statistical significance greater than 5.1 Sigma  .
While the final stages of the merger event which resulted in the GW150914 measurement was short-lived (occurring over approximately 0.2s)  the emission of gravitational waves by this binary system had been occurring for millions of years prior, albeit, at much smaller amplitudes and frequencies. If the continuous emissions of gravitational waves can be detected at these lower amplitudes and frequencies, then their distance from the earth could be studied and their location in the sky determined.
It is therefore likely, that in time, a catalogue of gravitational wave emission sources will be created. If the location of these sources is well known (through years of study) then their signals could be used by spacecraft to determine their position relative to the various sources. This form of localization is advantageous because it can be done without any communication signals having to be sent back to the earth., which will be a requirement on any long-distance voyage. This form of localization is analogous to how vehicles can obtain their position through the trilateration of GPS radio signals here on Earth.
Issues with existing methods of spacecraft localisation
There are numerous methods for determining and updating the position of spacecraft within our solar system. The most common method is to determine the spacecraft’s relative position to beacons whose positions are well known.
For NASA probes, this is achieved using the Deep Space Network (DSN) which is comprised of three main communications dishes (and numerous smaller support dishes) placed approximately 120 degrees apart around the earth. These dishes allow for the spacecraft position and velocity to be determined using triangulation and Doppler measurements  out to a range of 250 million kilometres.
Other methods of estimating relative position include using visual cues such as stars or known asteroids. Once a spacecraft’s position is determined it can continuously be updated using onboard Inertial Measurement Unit’s and Newton’s laws of motion. This dead reckoning approach can provide a “real-time” estimate of location, in-between receiving known location updates from the DSN on a periodic basis.
These current forms of localisation become ineffective over the vast distances associated with interstellar travel. As distance increases, the accumulated errors associated with dead reckoning become prohibitively large. For example, a one arc-second inaccuracy of trajectory over the 4.663 light-year journey to Alpha Centauri would result in a positional inaccuracy of 2.1 light-years (or 45% of the total journey). Furthermore, the power required to communicate from earth to an interstellar spacecraft and the time delays inherent with this approach make triangulation using the DSN impractical.
For example, at the time of writing, a signal from Voyager 1 which has only just left the Heliosphere, takes approximately 17 hours to reach earth. The spacecraft has a relatively large transmitter with a transmit power of 22.4 Watts  however, this is diminished to just a billionth-billionth of a Watt by the time it is received on Earth. These approaches also assume a rough line of sight with the earth which may not be possible should the spacecraft enter the orbit of a planet or star or lose track of its orientation (for example, after undertaking orbital maneuvers around foreign planets and stars).
Other methods of navigation such as star tracking, rely on well-defined landmarks, however, as any sailor who has lost sight of land on a cloudy night can attest, these beacons are not always available.
Sources of Gravitational Waves
Coalescing black holes such as GW150914, generate strong gravitational waves but unfortunately are not very suitable for localisation due to their relative rarity (tens per Gpsc³ per year ) and their short duration (0.2s in the case of GW150914).
A more suitable source for triangulation, is a continuous gravitational wave source. For example, the two black holes of GW150914 were continuously emitting GW’s for millions of years prior to their merger event which we detected. The direct measurement of the gravitational waves they emitted in this lead up are many orders of magnitude smaller than those generated during the merger event and as such, are much more difficult to detect. Continuous sources are characterized by a wave with a fairly constant and well-defined amplitude & frequency, generating an almost monochromatic source of gravitational waves.
Sources of these waves can be binary star, black hole or neutron star systems, or caused by the rapid rotation around its axis of an eccentric single star (such as a neutron star) with a large mountain or other irregularity present on it.
Unfortunately, these sources of gravitational waves are predicted to be much weaker (h ~ x10–24) than those caused by more catastrophic one off events such as Supernova, or the merger of neutron stars and black holes (h ~ x10–20) , making their detection significantly more difficult.
The characteristic amplitude and frequency of a continuous wave source depend upon the orientations of the masses as described by Einstein’s general relativity . In the case of a binary black hole system, the mass of the two black holes, their spin and their orbital distance all impact on the final wave function of the generated gravity wave. Because it is unlikely that the masses involved in different binary systems will have identical characteristics it is likely that each system will, therefore, have its own characteristic wave function which can then be used to uniquely identify and catalogue it.
Determining the location of Gravitational Waves
Determining the location of a GW source is often one of the first tasks following a confirmed signal. This is done by measuring the signal from multiple GW detectors located on Earth and triangulating the location of the signal. The detection of GW150914 was measured by only two detectors and the distance to the event was measured as 410 (+160 -180) Mpc at 90% confidence. With four detectors, however, (Virgo, LIGO, India & KAGRA) the errors in the distance for relatively nearby events (a few hundreds of Mpc away) could decrease by around an order of magnitude to within a few tens of Mpc .
This still constitutes an error of 32 million light-years or about 10 times the distance from the Milky Way to the Andromeda Galaxy. For this reason detection of binary neutron stars which contain relatively high stellar masses is of great interest. These binary systems have masses necessary to generate measurable gravitational waves whilst also having an accompanying electromagnetic component which can be used to more accurately locate the system and determine its distance (using long baseline interferometry).
The first direct measurement of two binary neutron stars occurred on the 17th August 2017 (GW170817) by the LIGO scientific collaboration. This was followed by independent detection of the accompanying Gamma Ray Burst (GRB).
Determining accurately the location of GW sources is an ongoing pursuit by Earth-based Laser Interferometers and is often the first question asked once a new GW source has been detected. The detection of binary neutron stars which provide an accompanying electromagnetic wave could provide additional measurement accuracy as both luminosity and redshift calculations can be used to crosscheck the distance of the GW source from the earth.
Position estimation using trilateration
Trilateration is the process of finding the centre of the area of intersection of three spheres. The location (centre point) and the radius of the spheres must both be known.
Figure 2 shows the concept in two dimensions for simplicity. Here, each base station comprises a binary black hole (or other massive body) pair, whose orbit and mass imbalance results in the continuous emission of a characteristic GW.
The barycenter of each binary system is labelled B1,B2 & B3. If the locations of B1, B2 & B3 are known with some accuracy and if the distance to B1,B2 & B3 from the spacecraft can be measured (as explained later) then the location of the spacecraft relative to B1,B2 & B3 can be calculated. The location of the spacecraft relative to the Earth can also be determined by translating B1,B2 & B3 such that the coordinates of B1 are located at the Earth, i.e (0,0). The location of the spacecraft can be determined then the translation can be reversed to yield the location relative to the Earth.
Determining the distance to Gravitational Wave sources.
The distance of the spacecraft to B1,B2 & B3 can be estimated using a luminosity distance calculation. It is known that the energy of a wave is proportional to its amplitude such that:
It is also well understood that the Energy from a wave emitted from a source in three dimensions dissipates as a function of one divided by the radius squared.
which can also be written as
or, the amplitude is proportional to 1 divided by the distance from the source. Therefore, the Amplitude of the GW depends only on the distance from the source.
Calculating spacecraft position relative to gravitational wave sources.
Assume for the moment, that three binary pulsar systems have been detected and their locations relative to the Earth are accurately catalogued.
Assume also, that these three systems are sufficiently different from one another in terms of the masses that comprise them and their orbital periods such that they can be accurately differentiated from one another through accurate measurement of their characteristic gravitational waves, and that the distance of each of these events can be measured by using the luminosity distance technique described above.
If this is the case, then the location of the spacecraft can be determined using the trilateration of these sources as described below. For this example B1, B2 & B3 are taken as the barycenters of the three binary pulsar systems. r1, r2 & r3 are the measured distances of each system from the spacecraft (as a magnitude not a vector).
If all of these are known, then the location of the spacecraft can be found by solving the three sphere equations for the spacecraft‘s three unknown coordinates (xs,ys&zs ). These calculations can be simplified by assuming the barycenter of all binary systems lie on the plane z=0(since it is highly likely that all three points will by non-collinear and any three non-collinear points lie on a unique plain).
Additionally, the B1,B2 & B3 systems can be translated such that B1 lies at (0,0) by subtracting B1’ coordinates (x1,y1,z1) from each of the base stations. The whole view can then be rotated until B2 lies on the x-axis (i.e at x2=0). The final configuration after this translation and rotation is shown in Figure 3.
The equation for each of the three circles is given by,
Expanding these equations yields:
Subtracting equation (3) from equation (2)
Subtracting equation (4) from equation (3)
This is an equation of the form,
Solving (4) for ys
Solving (5) also for ys
Solving the two equations above simultaneously yields a solution for
Substituting the values of E, C, B, F, A, D yields,
Noting that x1, y1,y2 are all 0, this equation simplifies down to:
Repeating the above steps, but this time solve for ys. Solving (4) for xs
Solving (5) also for xs
Solving the above two equations simultaneously yields a solution for ys
Substituting values for D, C, A, F, B, E yields
Noting that x1,y1,y2 all equal 0,
Refactoring and simplifying,
We can now find the z Coordinate by substituting (7) and (8) into (1).
Recall that z1 = x1 = y1 = 0
The position of the spacecraft relative to B1 can now be found by substituting the known values of (r1, r2, r3, x2, x3, y3).
An example calculation is given below using the following known values.
The spacecraft position can now be calculated using Equations 7,8 & 9.
Doing so yields a position of 2.28125 Mpc in the positive x-direction and 1.015625 Mpc in the positive y-direction from B1.
An excel spreadsheet/workbook containing these equations and a use case can be downloaded from here.
Factors impacting accuracy
I t is noted that only the accuracy of certain measurements is critical for determining the spacecraft position.
The spacecraft’s X coordinate is only reliant upon r1,r2 & x2. I.e, the luminosity distance measured from Base Station 1 and 2 and the absolute position of Base Station 2.
The spacecraft’s Y coordinate is reliant upon r1,r3,x3&y3. I.e, the luminosity distance measured from Base Station 1 & 3 and the absolute position of Base Station 3.
The spacecraft’s Z coordinate is reliant upon r1, xs & ys. I.e, the luminosity distance measured from Base Station 1 and the previously calculated X & Y coordinates of the space craft.
Miniaturisation and practicality
Earth based GW observatories such as LIGO are incredibly large. Ligo’s detector arms span 4km and are the size of a two-story building. Obviously, this paper assumes some form of miniaturisation of GW sensing technology will be possible and will take place over the ensuing decades. Who knows, eventually, we may have solid-state gravitational wave detectors that can be installed onboard every spacecraft for a couple of dollars each.
Before this happens, however, a more likely approach would be for a spacecraft to deploy its own Laser Interferometer network by deploying an array of satellites in formation as it travels (similar to the recently funded Lisa Pathfinder mission https://www.elisascience.org).
The GW detector would relay data back to the spacecraft with a minimal time lag, as the array could be located within close proximity to the spacecraft (e.g 1 km in front or behind). This would be a relatively cheap solution that utilises entirely known physics and engineering.
Velocity estimation using the stochastic gravitational-wave background
The stochastic gravitational-wave background is formed by the random and complex interactions of all gravitational wave sources. For every event like GW150914, there are numerous other events taking place that are too distant to be resolved individually. The gravitational waves from these events combine to create a random stochastic background that is potentially measurable by correlating the signals from two or more gravitational wave detectors or by increasing the sensitivity of gravitational wave detectors.
The relatively high masses of the two black holes in GW150914 combined with the higher coalescence rate (16 [+38,-13] per Gpc³ per year) of binary black hole mergers than originally thought, both imply a brighter and thus more easily measured stochastic background that will be within the design sensitivity of the LIGO/Virgo detectors operating at their final design sensitivity .
Space-based observatories like eLISA (Pathfinder) and its proposed successor the Big Bang Observatory (BBO) allow for much, much larger measurement baselines and therefore stronger signals. eLISA and BBO promise to have sensitivities within the range to detect the stochastic background.
If it turns out that measurement of the stochastic background has some particular orientation or polarity (due to lasting effects from events in the early universe), then it may be possible to measure the relative velocity of the spacecraft to the stochastic background by measuring the Doppler Shift of the signal from the at-rest baseline. Similarly measuring the Doppler shift of the known catalogue of beacons with characteristic GW signals will also suffice.
Some good additional resources and future work
Below are two comprehensive videos on Gravitational Waves.
 A. Einstein, Sitzungsber. K. Preuss. Akad. Wiss. 1, 688 (1916).
 R. A. Hulse and J. H. Taylor, Astrophys. J. 195, L51 (1975)
 J. H. Taylor and J. M. Weisberg, Astrophys. J. 253, 908 (1982).
 B.P Abott et al, American Physical Society (2016)
 B.P Abott et al, arXiv:1602.03839
 J. Taylor, The Deep Space Network: A functional description (2016)
 R. Ludwig, J. Taylor, Voyager Telecommunications (March 2002)
 B. Abbott et al., arXiv:1602.03847
 ] B. Abbott et al., (2016), https://dcc.ligo.org/ LIGO-P1500217/public/.
 Forward, Rovert L (1984), http://www.lunarsail.com/LightSail/rit-1.pdf
 K. D. Kokkotas Gravitational Wave Physics
 F. Pretorious (2017) Dept of Physics Princeton University NJ, Email discussion on Gravitational waves.