Probing the Secrets of Gravity

Title: Testing gravity with interstellar precursor missions

Authors: Indranil Banik and Pavel Kroupa

First Author’s Institution: Helmholtz-Institut für Strahlen und Kernphysik (HISKP), University of Bonn, Germany

Status: Published in MNRAS (open access on ArXiV)

Humans have long dreamed of conquering the stars. Yet, even with the best of today’s technological capabilities, such an endeavour could take centuries. Since Newton’s incident with an apple, his theories of motion and universal gravitation have remained largely accepted save Einstein’s 1915 overhaul. Still, there remains observations that Newtonian dynamics cannot fully describe. One such observation is that of a galaxy’s rotation curve. We would expect a galaxy to rotate more slowly the further we move away from its center. It turns out that the rotation remains the same, and in some cases increases. The common solution to this problem is to invoke dark matter. This invisible matter accounts for the “missing mass” that enables galaxies to spin more than they otherwise should. MOdified Newtonian Dynamics (MOND) is an alternative theory of gravity that also attempts to account for these rotation curves. Instead of invoking dark matter, MOND alters the gravitational acceleration. This paper shows how this modified acceleration can alter the trajectory of interstellar probes.

To the MOND And Back

MOND modifies the gravitational acceleration by introducing an empirical constant a0. The distance from a star at which the modified acceleration and usual Newtonian acceleration differ by this constant a0 is called the MOND radius. For the Sun, the MOND radius is roughly 7kAU (i.e 7000 times the average distance from the Earth to the Sun). For comparison, the closest star to the Sun, Proxima Centauri, is roughly 268kAU away. In today’s paper, the authors consider a spacecraft travelling at 0.01 c where c is the speed of light. At this speed, it would take approximately 400 years to reach Proxima Centauri, but only 11 years to reach the MOND radius.

Figure 1 (also Figure 1 in the paper): The “boost factor” by which MOND increases the radial component of the Sun’s gravitational acceleration in units of the MOND radius. Here the positive x axis points towards the galactic center.

As expected, Figure 1 shows very little deviation between MOND and Newtonian mechanics for positions well within the MOND radius. However, there is a significant boost to the gravitational acceleration, particularly in the direction away from the galactic center (the negative x axis). Overall, gravitational acceleration in MOND is stronger than the expected Newtonian acceleration.

Modified Newtonian Trajectories

To examine the changes on spacecraft trajectories, the authors considered only the part of the journey beyond 2kAU from the sun. This limit corresponds to the point at which MOND starts to have non-negligible effects (see the change in the shade of blue in Figure 1).

Figure 2 (Figure 3 in the paper): The reduction in the radial velocity of an interstellar probe due to MOND (in centimeters per second), as a function of time after crossing the 2kAU distance from the sun, for different launch angles θ, where 0 degrees indicates a launch directly towards the galactic center.

Figure 2 shows how the stronger gravity that exists under MOND leads to a small but no doubt significant reduction in the velocity of an interstellar probe. Players of Kerbal Space Program would no doubt appreciate how small changes in velocity can sometimes often result in catastrophic consequences. In this case, the changes are extremely small. The reduction of 3 cm/sec after 20 years corresponds to a two-way light travel time of 0.1 seconds, or a distance of 10−4 AU. Furthermore, the velocity reduction also results in the probe changing direction over time (see Figure 5 in the paper). Observers can thus verify MOND by measuring both the change in the downrange distance (i.e how far away the probe is) and the change in its angular position (which can be determined by interferometry).

Touching the Sky

Before humans can feasibly travel to other stars, precursor probes will necessarily have to be sent at relatively low speeds. As such, these probes could not only chart our next stage in space-faring capability, but also reveal the inner workings of gravity. MOND is not alone in challenging accepted theories of gravity: many other theories, such as the five-dimensional Kaluza-Klein theory and the elusive theory of everything have attempted to incorporate gravity as part of an overarching picture of how the Universe works. This paper shows that MOND is testable for it will affect the trajectories of interstellar spacecraft. All that’s left now is to reach for the stars.

About Mitchell Cavanagh

Mitchell is a PhD student in astrophysics at the University of Western Australia. His research is focused on the applications of machine learning to the study of galaxy formation and evolution. Outside of research, he is an avid bookworm and enjoys gaming, languages and code jams.

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  1. However, the galaxy rotation curves are not the only evidence for dark matter. Galaxy velocities in galaxy clusters, gravitational lensing, matter distribution after galaxy clusters collisions and other measurements also requires dark matter to be explained. AFAIK, the MOND theory says nothing about results of this measurements. Are there other MONDs, each tailored for explanation of one specific measurement? And how many MONDs will be needed to explain all measurements? Infinite amount?

    • While MOND has had great empirical success at the galactic scale (without needing to account for dark matter), you are right that MOND fails at accounting for cluster interactions (where the presence of dark matter cannot be easily ignored). There are ongoing attempts to integrate MOND with cosmological simulations in order to reconcile these differences (see

      • On the other side, Witten showed several years ago that if dark matter consisted on ultra-light axions with wave length on galactic scale, than it would have the same effect on the rotation curves as MOND. So it may turn out that the MOND is only effective theory describing effects of dark matter on galaxy scales (and nowhere else).

  2. For all the great minds out there:

    Einstein’s theory of gravity posits that space-time is curved and that is what causes gravity. Rather than space-time curvature, I prefer to think of it as the flow of space into massive objects (actually any mass object). This is based on Einstein’s roof-top-worker thought experiment. Objects in flat space where space is not moving will either stay in the same place or move in a straight line. Objects in ‘curved’ space where space is flowing into massive objects, will move either toward the massive object (like the roof worker falling off the roof) or in a curved line around the object as space flows into the object (like a satellite around the Earth or a planet around the Sun).

    Now, here is my question:
    If space ‘curvature’ causes gravity and if we can define curvature as the flow of space-time into massive objects, and if space-time itself is an entity that can affect the motion of massive objects, maybe space time also can have rotational flow around mass concentrations, like galaxies and such rotational flow of space-time is the cause of the galaxy rotation curve discrepancy? Normally, objects at greater distances would rotate at lower speeds, but if they are embedded in space-time that is also rotating, wouldn’t normal rotation look like faster than normal rotation to us?


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