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If a spacecraft's standard radar and optical metrology were lost, could a sextant be used to determine range and relative position of another spacecraft in order to execute a docking maneuver?

Let's assume the docking is similar to those that have taken place in orbit around a known, mapped body, like the Earth or the Moon, and at least one astronaut has a watch, paper, pencil, map, manual, etc. Not that those are required for the answer, but consider basic tools like this to be available, as they normally have been on missions involving docking.

edit: To follow up on comments, the paper and pencil can be used for long hand calculations if necessary, but not a computer or electronic calculator.

Mechanical calculators however, such as slide rules are of course allowed. Source: Numberphile See also Apollo 13 - Math Saves the Day.

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    $\begingroup$ A sextant is very much an optical metrology instrument :) $\endgroup$
    – Antzi
    Commented Nov 16, 2018 at 5:54
  • $\begingroup$ @Antzi indeed you are right, edited. $\endgroup$
    – uhoh
    Commented Nov 16, 2018 at 5:55
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    $\begingroup$ @DrSheldon Earth or Moon orbit is already addressed and is affirmative. I thought it would be pretty clear that the paper and pencil would be used to do the calculation tasks. But these days younger people may not know about that, so I've allowed for it explicitly. Per the edit, it's a no-go on computers. Luckily, perturbation theory has been around a century or two. Calculations Apollo astronauts were trained to do by hand if loss of guidance computer and/or data & coms link to Earth? $\endgroup$
    – uhoh
    Commented Nov 16, 2018 at 7:22
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    $\begingroup$ Shuttle had a COAS (Crew Optical Alignment Sight) telescope-like-instrument that could be used for rendezvous. $\endgroup$
    – Organic Marble
    Commented Nov 16, 2018 at 15:45
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    $\begingroup$ Put a born mathematician as the pilot with more windows. $\endgroup$
    – Muze
    Commented Nov 16, 2018 at 23:39

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Absolutely !

To be able to perform a rendez vous, you need to know your current position, heading, orientation, and velocity, as well as that of the target craft.

  • 2D Coordinates: The sextant can give the first information, provided the altitude (dip) is known.
  • 3D Coordinates: The altitude could be extrapolated from an angular measurement of the horizon of the orbiting body (I assume the size of the body is known).
  • Velocity+heading: If one makes a second timed measurement; the velocity and heading can be calculated.
  • Orientation: The stars positions can give this informations

Once visual contact is established; manual docking is possible.

In practice, I'm unsure wether you can get enough accuracy and perform all calculations fast enough. Please note that given enough time and propellant, you should be able to refine your orbit; visual contact can be established fairly far away from your destination.

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    $\begingroup$ How do you measure the orientation of the spacecraft with a handheld sextant? You need a sextant mounted to the spacecraft to measure angles between stars and the axes of the spacecraft. $\endgroup$
    – Uwe
    Commented Nov 16, 2018 at 10:56
  • $\begingroup$ @Uwe you are correct. Notice I didn’t use the word “sextant” 1st: eyeball to the desired orientation; 2nd using a scope in a predetermined location, orient the craft until the desired star is in the correct position. (Remember we only want the ability to orient the spacecraft). An other solution would be to use several compas (if applicable). $\endgroup$
    – Antzi
    Commented Nov 16, 2018 at 12:05
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The use of a sextant by itself isn't the limitation, it's having to do calculations by hand instead of by a computer.

I agree with uhoh's comment that the final approach can be done manually, without either the computer or the sextant. For example, the extraction of the Apollo lunar module from the 3rd stage could easily be done manually. I saw an article that stated that manual docking can be done within a distance of 0.1 km, but I can't find that article again.

The real problem is getting close enough in the first place. Consider the launch of a lunar module from the moon, trying to rendezvous with a command/service module. The CSM has an orbit with a particular altitude, eccentricity, and inclination. The lunar module needs to get into a matching trajectory, all while under the influence of the moon's gravity. This article explains the details of this process. Yes, you do need the position and velocities of both vehicles, and a sextant may help you get those. But there's far more to the calculation than those parameters. Most importantly, note the following near the bottom of that article:

Calculating the velocity needed for this maneuver is more than simply raising the pericynthion, as other orbital parameters, plus maintaining the delicate relationship between the CSM and LM need to be taken into account. These complexities are beyond the capacity of the limited memory of the LM guidance computer. Rather than selecting a specific program to perform the Boost maneuver, the calculations will be performed in the Real Time Computing Center (RTCC) and relayed to the LM crew. The crew, in turn, will use one of the "External Delta V" programs to perform the maneuver and manage this burn. Most of the other programs used in the LM (such as descent, ascent and rendezvous), have the guidance and navigation solutions calculated internally; that is, using only the resources available on the spacecraft.

If such calculation is too difficult for the on-board guidance computer, it's unlikely that the astronauts will do any better with pencil and paper. And a launch from Earth is even worse, as you also have to account for atmospheric effects.

Suppose you are lucky enough to get into the same orbit as the other vehicle, you're "behind" the other vehicle, and you want to catch up. If this were a car on a highway, you'd just speed up until you catch the other car. However, the article points out why this won't work:

If you speed up, it will push you out into a higher orbit. By Keppler's laws, this higher orbit will have a longer period than that of the other vehicle. You will therefore actually fall further behind the other vehicle!

Perhaps an astronaut could practice in a simulator, but it's hard to make a realistic simulator without a computer.

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  • $\begingroup$ Thanks for the answer and the link, both make for interesting read. $\endgroup$
    – uhoh
    Commented Nov 17, 2018 at 1:06

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