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As asteroids orbit the sun, they experience gravitational force exerted on them by the sun, and they in turn exert a very minute force back on the sun. Because of their small size, asteroids don't tug on the sun a measurable amount, so therefore you cannot determine the gravitational force acting upon them. However, there are asteroids with known masses, and known gravitational forces (such as Ida) which have been visited. Since asteroids are so small their gravitational forces are negligible, could the gravitational force of an average asteroid be extrapolated to the rest of the asteroids in its family? And using this average force you could then find the masses of the nearby asteroids with Newton's law of gravitation? Or would the gravitational forces be too different to produce a meaningful result?

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You ask:

could the gravitational force of an average asteroid be extrapolated to the rest of the asteroids in its family?

But the question is what properties the members of a family have in common. You could argue that all the asteroids in a family will be made up from the same material and will have the same density. That means as long as you can measure the size and shape with a telescope you can calculate the volume and thereby the mass. But this seems a risky approach as there may be cavities and voids in the asteroids that can't be seen, and your calculated masses would just be an upper bound.

Historically asteroid masses have been determined by observing their interaction with other asteroids. For example the first ever determination of an asteroid mass was in 1966 when the mass of Vesta was estimated by measuring how much it deflected the smaller asteroid 197 Arete. More recently we've been able to fly spaceships to asteroids and measure their mass that way, but the majority of known asteroid masses have been measured by their interactions with other asteroids.

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  • $\begingroup$ I had heard that asteroids were typically grouped by orbital elements, and not by their physical characteristics (because these properties are unknown), but I'm wondering whether you could treat the gravitational force the sun exerts on them as the same. $\endgroup$ – Sigismund Apr 20 '16 at 16:43
  • $\begingroup$ @Sigismund: the gravitational acceleration produced by the Sun would be the same, but we can't measure the force just from the orbit so we can't obtain the mass. $\endgroup$ – John Rennie Apr 20 '16 at 16:44
  • $\begingroup$ That's true, but if an asteroid with a very similar orbit has a known force could that force be used for other asteroids nearby? $\endgroup$ – Sigismund Apr 20 '16 at 16:45
  • $\begingroup$ @Sigismund: no, because the force depends on the mass. All we can say is that nearby asteroids will have the same (or very similar) acceleration. There is no connection between the proximity of other asteroids and the relative forces acting on them. $\endgroup$ – John Rennie Apr 20 '16 at 16:47
  • $\begingroup$ I see, and since you don't know the mass you can't use f=ma to find the force either, so you're back at square one... $\endgroup$ – Sigismund Apr 20 '16 at 16:48

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