By measuring the proper motions of the stars of an open star cluster, astronomers have found that they are all directed to one point in the sky, located 20 ° from the cluster itself. The proper motion is 0.1 “per year. The observers also measured the radial velocity of the stars, which turned out to be equal to 20 km / s, but due to absent-mindedness they forgot to indicate its sign, that is, they did not report whether the stars of the cluster were approaching or moving away from us. Restore this missing information and also determine the distance to the star cluster.
The solution to the problem is based on the assumption that all the stars of an open cluster have the same speed in outer space. For a young and non-massive open star cluster, this assumption is quite justified, since the relative speeds of the stars inside the cluster are much lower than the speed of the cluster in space. The stars of the cluster can be compared to a meteor shower, the particles of which also move along parallel lines, and from the Earth are seen flying out from one point – the radiant, indicating the direction in space from where the meteor particles fly. However, in our case, the stars, on the contrary, fly in the sky in the direction of one point (antiapex). This means that, unlike meteors, the stars of the cluster are moving away from the Earth, and the “+” sign must be assigned to the value of the line-of-sight velocity measured by astronomers.
Knowing the angular distance of the cluster from its antiapex, we know the angle γ between the cluster velocity vector and the direction from the Earth to the cluster (see figure). The tangential component of the cluster velocity vT is related to the radial velocity vL by the relation
vT = vLtgγ
and is 7.28 km / s or 1.53 AU / year. A segment of 1.53 au visible from Earth at an angle of 0.1 ″, hence the distance to the cluster is 15.3 pc