Question

The figures show a hypothetical planetary system at two different times. The system has a star S and three planets, labeled A, B, and C. The table provides the mass of each body in the system, as well as their spatial coordinates (2x, y) in their initial and final positions. The spatial coordinates of the bodies are given in Astronomical Units (AU). Body Mass (kg) Initital Position Final Position S ms = 2.6307 x 1030 (0,0) (as, ys) A ma = 2.4401 x 1040 (0.1119,0) (0, -0.2381) B mb = 6.1019 x 1026 (0.3751, 1.4241) (-1.9117,0) C mc = 8.6951 x 1027 (0, 1.0135) (-0.8865. -0.8445) The initial velocity of the center of mass of the system is zero. Find the magnitude ds of the star's displacement from the origin in its final position.

Answer 1

First, we find the x - component

2.6307e30 * 0 + 2.4401e28 * 0.1119 + 6.1019e26 * 0.3751 + 8.6951e27 * 0 = 2.6307e30 * x_s + 2.4401e28 *0 - 6.1019e26 * 1.9117 - 8.6951e27 * 0.8865

x_s = 0.004498 AU

___

we find the y - component

2.6307e30 * 0 + 2.4401e28 * 0 + 6.1019e26 * 1.4241 + 8.6951e27 * 1.0135 = 2.6307e30 * y_s - 2.4401e28 * 0.2381   - 6.1019e26 * 0 - 8.6951e27 * 0.8445

y_s = 0.008679 AU

so,

d_s = sqrt ( 0.004498² + 0.008679²)

d_s = 0.009776 AU