Answer to Question #169166 in Astronomy | Astrophysics for Sam

Question

The following table lists several orbital properties of each planet in the solar system. Mean distances from Sun are expressed in terms of astronomical units (1 AU = 1.496 x 10¹¹ m).

Table: https://cutt.ly/6zq2BQm

You are to obtain some of the values in the table using Newton’s law of universal gravitation. In all calculations involving the mass of Sun, use M = 1.989 × 10³⁰ kg. Assume that all orbits are circular.

1.  Calculate the acceleration of each planet due to the gravity of Sun.

2.  Recall that an object in uniform circular motion experiences acceleration directed towards the midpoint of the circle according to the equation

• a=v²/r

where a is the acceleration of the object, v is the velocity of the object, and r is the radius of the circular path. Calculate the mean orbital velocity of each planet using the above equation. Express your answers in units of km/s.

Answer 1

1. The gravitational acceleration is

$a = \dfrac{F_g}{m} = \dfrac{GM_{\odot}}{R^2}$ , where a is the semi-major axis of the planet's orbit.

For Mercury: $a = \dfrac{6.67\cdot10^{-11}\cdot1.989\cdot10^{30}}{(0.39\cdot1.496\cdot10^{11})^2} = 0.039\,\mathrm{m/s^2}.$

For Venus: $a = \dfrac{6.67\cdot10^{-11}\cdot1.989\cdot10^{30}}{(0.72\cdot1.496\cdot10^{11})^2} = 0.011\,\mathrm{m/s^2}.$

For Earth: $a = \dfrac{6.67\cdot10^{-11}\cdot1.989\cdot10^{30}}{(1\cdot1.496\cdot10^{11})^2} = 5.9\cdot10^{-3}\,\mathrm{m/s^2}.$

For Mars: $a = \dfrac{6.67\cdot10^{-11}\cdot1.989\cdot10^{30}}{(1.52\cdot1.496\cdot10^{11})^2} = 2.6\cdot10^{-3}\,\mathrm{m/s^2}.$

For Jupiter: $a = \dfrac{6.67\cdot10^{-11}\cdot1.989\cdot10^{30}}{(5.20\cdot1.496\cdot10^{11})^2} = 2.2\cdot10^{-4}\,\mathrm{m/s^2}.$

For Saturn: $a = \dfrac{6.67\cdot10^{-11}\cdot1.989\cdot10^{30}}{(9.54\cdot1.496\cdot10^{11})^2} = 6.5\cdot10^{-5}\,\mathrm{m/s^2}.$

For Uranus: $a = \dfrac{6.67\cdot10^{-11}\cdot1.989\cdot10^{30}}{(19.18\cdot1.496\cdot10^{11})^2} = 1.6\cdot10^{-5}\,\mathrm{m/s^2}.$

For Neptune: $a = \dfrac{6.67\cdot10^{-11}\cdot1.989\cdot10^{30}}{(30.06\cdot1.496\cdot10^{11})^2} = 6.6\cdot10^{-6}\,\mathrm{m/s^2}.$

2. $a = \dfrac{V^2}{R}, \;\; \;\; V = \sqrt{aR},$

for Mercury $V = \sqrt{0.039\cdot 0.39\cdot 1.496\cdot10^{11}} = 47.7\,\mathrm{km/s},$

for Venus $V = \sqrt{0.011\cdot 0.72\cdot 1.496\cdot10^{11}} = 34.4\,\mathrm{km/s},$

for Earth $V = \sqrt{0.0059\cdot 1\cdot 1.496\cdot10^{11}} = 29.7\,\mathrm{km/s},$

for Mars $V = \sqrt{0.0026\cdot1.52\cdot 1.496\cdot10^{11}} = 24.3\,\mathrm{km/s},$

for Jupiter $V = \sqrt{0.00022\cdot 5.2\cdot 1.496\cdot10^{11}} = 13.1\,\mathrm{km/s},$

for Saturn $V = \sqrt{0.000065\cdot 9.54\cdot 1.496\cdot10^{11}} =9.6\,\mathrm{km/s},$

for Uranus $V = \sqrt{0.000016\cdot 19.18\cdot 1.496\cdot10^{11}} = 6.8\,\mathrm{km/s},$

for Neptune $V = \sqrt{0.0000066\cdot 30.06\cdot 1.496\cdot10^{11}} = 5.4\,\mathrm{km/s}.$