Adam Hibberd
I've been doing a little algebra. Let me state the problem.
Let us say we have a swarm of space sails flying edge on to the interstellar medium (ISM). This swarm lies in a plane at right angles to its velocity relative to this ISM.
Now lets bring in an element of the unknown, a small object, X, of non-negligible mass makes its presence felt, not in the form of an image taken by some of the space sails (thus we have no precise idea as to the location of X as the swarm flies by it), but its presence is felt by its gravitational influence on several nearest members of the swarm of space sails.
The question I have been asking is this: 'Can we work out from its gravitational influence on the space sails, the consequent mass of the unknown body X?'
The answer to this question requires some thought as to in WHAT WAYS could the trajectory of the space sails be altered by X, and therefore X make its presence felt.
Having contemplated this for some time, I discovered 3 measurable parameters of a space sail's trajectory which could help in this regard, these are listed below. Think about this for a while and then see if you too can come up with any of these (or possibly ones I hadn't considered).
(1) The change in speed of the space sails. Thus as the sails approach the body, they will accelerate as they are initially attracted by it and then decelerate back to their original speed, after they have flown past the point of closest approach.
(2) There will be a change in direction of motion of each of the swarm members affected, in other words they shall be deflected by a certain angle, which is measurable. I suspect that this would be more obvious than any perturbation in speed mentioned by (1) above and would provide the better estimate of the mass, particularly when the swarm's speed is high (20% of the speed of light for example).
(3) Finally the least obvious one. As all the space sails fly past the mysterious X, they will be deflected as mentioned in (2) by a certain angle in a CERTAIN DIRECTION, and this direction will be different for each one. Another way of stating this is that the velocity before the encounter and after encounter will occupy a common plane, and the position of the body X WILL ALSO LIE IN THIS PLANE. Thus by calculating this plane for each space sail in turn, and then determining the point at which all these planes intersect, we are able to determine the precise position of X and from this we can, with the help of (1) and (2), better calculate the mass of X.
I should say I've derived an algorithm which exploits (2) & (3) of the parameters mentioned above and uses least-squares fits to generate the mass of an object as a swarm of sails flyby it. It's working quite well.