# What is the minimum distance the star 61 Cygnus will approach us if the parallax of this star is 0.3 ″ and its proper motion is 5.2 ″

**What is the minimum distance the star 61 Cygnus will approach us if the parallax of this star is 0.3 ″ and its proper motion is 5.2 ″. The star is moving towards us with a radial velocity of 64 km / s**

We are at point A. The star we are observing is at point B and is moving relative to us at a speed V (blue vector). We decompose the velocity vector V into two perpendicular projections (green vectors) – the radial velocity Vl (directed strictly towards us) and the velocity of “proper” motion Vc. The radial velocity is set explicitly, and Vc must be found. From the definition of parallax, it follows that 0.3 ” of proper motion is 1 astronomical unit (that is, 150 million km), which means that it flies in the transverse direction 5.2 / 0.3 = 17.3 AU per year. i.e. = 2,600,000,000 km, divide by the number of seconds in a year and get

Vc = 2 600 000 000 / (3600 * 24 * 365.25) = 82.4 km / s

We can also calculate the total speed (104 km / s) from the Pythagorean theorem, but we do not really need this, but we need the angle ABC, which is equal to arctan (Vc / Vл) = arctan (82.4 / 64) = 52.2 °

From the parallax side AB = s = 1 / 0.3 = 3.33 parsecs (we don’t translate into kilometers, it makes no sense). From the angle ABC and side AB we find the minimum distance AC = AB * sin (52.2 °) = 2.63 parsec.