# How would a space probe determine its distance from a black hole while orbiting around it?

Let's say that a space probe is sent to swing once around a black hole. Now, as long as our knowledge of physics goes, if the spaceship stays outside the event horizon it can, at least in principle, get away from the black hole when it wants to.

Assuming that the mass of the black hole is known, is there a way for the space probe to determine its distance from either the black hole or the event horizon (whichever of the two it's easier to measure) so that it can observe the space travel safety guidelines, which strictly forbids trespassing the event horizon of a black hole, while getting as close as possible to the black hole itself?

Note: I am neglecting the effect of tidal forces on the integrity of any object getting so close to a black hole

• The boundary of the event horizon should be filled with mass travelling at relativistic speeds. Wouldn't staying out put of the "halo" be good enough? Inside the halo your ship would have more immediate problems than being pulled into a black hole at some indeterminate point in time. Such as masses travelling at relativistic speeds Commented Feb 23, 2021 at 13:16
• Not a full fledged answer, but it's really exactly the same way you don't run into a star. Look at it, measure what you see with various instruments, derive the size and mass from readings and just avoid the bad bits - whether the event horizon or the surface of the star. Black holes aren't invisible, in fact they are typically ridiculously bright x-ray sources. Commented Feb 24, 2021 at 0:53
• Note that the event horizon is not the limit of where you can safely escape. The event horizon is the boundary where objects moving straight up at the speed of light marginally do/don't make it out. If you plot that line backwards, you'll see it intersects the black hole. For photons, the "photon sphere" is 1.5 times the schwarzschild radius, but material objects should stick to at least twice the radius if they want an orbit that doesn't spiral inward due to relativistic effeccts. Commented Feb 24, 2021 at 14:32
• Measuring the light deflection of known stars is probably the easiest way, but there’s a lot of alternatives. Commented Feb 24, 2021 at 17:57
• Are you also neglecting the impossibility of chemical rockets lifting the thing out again beyond a certain point? (Or indeed any known fuel/energy-source) Commented Feb 24, 2021 at 21:50

I am neglecting the effect of tidal forces on the integrity of any object

Well, don't be doing that! Those forces can be used to answer your question.

Given a couple of bars of known length and mass, and a black hole of known mass, you could measure the relative differences in length of a bar placed parallel to the local gravity vector and one placed perpendicular to it using eg. interferometry. The vertical bar will be stretched by tidal forces the closer you get, and the amount of stretch will be predictable and related to your distance from the hole.

As an alternative that uses boring old orbital mechanics instead of exciting space-mangling super-intense gravity fields, you could measure the period of your orbit by observing background stars, which you can throw into the usual equations to work out the orbital radius. This works when you are far enough away from the hole to avoid frame dragging effects, but as you get closer, more clever mathematics will be required.

• Would the tidal not stretch the space itself? Can you measure the difference in size, if your "ruler" or measurement device is stretched as well? Commented Feb 24, 2021 at 8:29
• @jnovacho nah, tidal forces aren't that subtle... its a physical stretching of the object by the force of gravity. Consider: we can measure tides on earth! This is a lot simpler than eg. detecting gravity waves. Commented Feb 24, 2021 at 8:41
• Correct me if I'm wrong, but isn't this exactly how we measure gravity waves? Just pick up and drop one of our gravity wave detectors on your craft and you've got a great way of measuring distance from any known mass Commented Feb 24, 2021 at 15:37
• @bendl gravity wave detectors are a bit more sophisticated... take a look at Advanced LIGO, for example. You could probably make this trick work, albeit crudely, using a tapemeasure with high tensile strength and a plasic rod. Commented Feb 24, 2021 at 16:38
• Understood... You could make the system less sophisticated, but it's still measuring the same thing. I guess there's some "Cool Factor" from literally measuring how much a physical object changes shape, but it seems that by the time we can take trips to a black hole, the technology in LIGO would be trivial Commented Feb 25, 2021 at 17:22

Determine the location of the photon sphere using laser beams.

https://en.wikipedia.org/wiki/Photon_sphere

The photon sphere is located farther from the center of a black hole than the event horizon. Within a photon sphere, it is possible to imagine a photon that's emitted from the back of one's head, orbiting the black hole, only then to be intercepted by the person's eyes, allowing one to see the back of the head.

https://profmattstrassler.com/2019/06/14/a-ring-of-controversy-around-a-black-hole-photo/

You here use the photon sphere as a proxy for the event horizon. You will swing your (shark-mounted) laser slowly from left to right such that it will trace out a diameter of the circular region of interest. There will be a point where you can detect your photons coming back to (your shark) as they loop around the black hole. That distance from the center is just outside the photon sphere which is just outside the event horizon. As your laser continues to the right its photons will be lost into the black hole. As it emerges on the left your will detect the photons coming around from the right. You have now marked the lateral edges of the photon sphere here used as a proxy for the black hole. Should you suspect the region is other than circular you can draw a few more diameters with your laser.

You do not need to get close to do this. You can do this from some distance, which I recommend at least the first few times.

• Usually laser measurements happen fast... At the speed of light - but in this case your craft will have to be going an appreciable fraction of the speed of light to maintain orbit at low altitudes... I'm not certain, but it seems to me that as you get closer to the black hole it should take longer and longer for this beam to come back to you, which could make this solution work worse at exactly the time you want it to work best - when you're closest to the black hole. Commented Feb 24, 2021 at 15:41
• @bendl I think until it skives pretty close the beam won't come back to you but will be bent off to go somewhere else. But I am much digging the idea of using the time dilation itself to map a region of space near a black hole. Commented Feb 24, 2021 at 18:16

The probe is in communication with a relay station live streaming the event back to NASA and eager nerds like me. The communication is bidirectional, and the probe is constantly acknowledging packets received. (Think TCP protocol but tweaked for space.)

The constant stream back of acknowledgements keeps the communication channel constantly active, allowing the probe to measure the frequency shift of the data link.

Because it knows that the relay is far enough away from the black hole (and traveling slowly enough) that time dilation isn't an issue to N significant digits, it can calculate the exact time dilation from the frequency shift of the known communication frequency.

Eg the 5Ghz radio signal is arriving at 25.453Ghz - so that relay stations clocks are running 5.0906 times faster than mine. That's means my clocks are running at 19.644% of real time. You now know your time dilation factor to as accurately as your communication unit can measure the changing frequency.

Assuming you're not orbiting at relativistic speeds, from this, you can calculate your approximate distance from the black hole.

If you are orbiting at relativistic speeds, ie, you're getting in that close, you can measure the change in time dilation (via frequency shift) at regular intervals (in probe time). I'm not smart enough to be 100% certain this will work without coding a simulation, but my gut instinct is that there will only be one valid orbital ellipse possible for a given sequence of time dilation measurements.

After you consider:

• time dilation from the gravity of the blackhole and the probes distance to it.
• time dilation from the speed of the probe.
• doppler shifting based on the relative velocity of the probe and the relay.

That sequence of changing times should give only one ellipse and your phase on that ellipse. From that, you can do your orbital maneuvers.

• Not key to your answer, but TCP doesn't really do well over interplanetary-scale distances. nasa.gov/directorates/heo/scan/engineering/technology/… Commented Feb 23, 2021 at 20:49
• On relativistic orbits, you'll also need to account for the Doppler effect. It should still be solvable, though. Commented Feb 24, 2021 at 0:23

You'd need to know in advance: the mass of the singularity - from this determine its Schwartzschild metric and the curvature of the geodesic, have a star-map and reference points for the emission spectra of those stars.

You can determine how far into the gravity-well you are by looking at how much the spectral emission lines of hydrogen in the surrounding stars have blue-shifted.

By applying the Lorentz factor - a solution to Einstein's field equations, you can then deduce how far from the event-horizon you are - and it wouldn't be necessary to be orbiting the black hole, you can take a straight approach.

• Lorentz factor is misused here. The Lorentz factor is the velocity-dependent factor that appears in special-relativistic time dilation/redshift/blueshift, we want gravitational time dilation here Commented Feb 24, 2021 at 16:19
• I'm no expert, but I'm pretty sure the relation holds. Ever since your comment I've been trying to work-out how to plug the variables in, no luck yet. If I figure it out then I'll edit the answer to include the info. If anyone reading this knows, please comment. Thanks @Tristan Commented Feb 24, 2021 at 17:21
• @Tristan I realize that it may be a few weeks/months before my edit is added, as I need to go away and re-learn the math I've forgotten over the last few decades. I'm quite serious, I've a number of posts which need editing for the same reason. Commented Feb 24, 2021 at 23:24

If you’re close enough to the black hole to be in any danger, you’ll be orbiting it very fast. Just use the varying parallax to other nearby objects to determine the shape of your orbit, and the black hole will be at one focus of the ellipse.

Just look!. The appearance of a black hole in space is rather distinctive, and basic measurements with a radiation-resistant video camera should give a good impression.

Of course, measuring the deflection of each star by general relativity gives a better impression. The makers of the video linked above could do that, and so can your space probe.