Question

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An exciting area of astrophysical research is the discovery of exoplanets, also known as extrasolar planets, which are planets that orbit stars other than our sun. The least massive of the exoplanets is Draugr, its mass is 1.20x10- kg, and it has a radius of 2.05x10ºm Draugr orbits a star of mass 2.79x109 kg in a circular orbit of radius 2.85x10" m 14. What is the period of the motion of exoplanet Draugr about the star that it orbits? (A) (B) (C) (D) (E) 25.6 days 75.8 days 553 days 849 days 1520 days 15. What is the magnitude of the acceleration of exoplanet Draugr as it orbits the star? (A) (B) (C) (D) (E) 0.109 m/s2 0.144 m/s2 0.177 m/s2 0.202 m/s 0.229 m/s2 16. If exoplanet Draugt were an isolated object not orbiting a star, what would be the free-fall acceleration near its surface? (A) (B) C) (D) (E) 0.882 m/s2 1.03 m/s2 1.32 m/s2 1.60 m/s2 1.90 m/s2

Answer 1

Gravitational constant = G = 6.67 x 10^-11 N.m²/kg²

Mass of the star = M = 2.79 x 10³⁰ kg

Mass of the exoplanet Draugr = m = 1.2 x 10²³ kg

Radius of the orbit of the exoplanet Draugr = R = 2.85 x 10¹⁰ m

Radius of the exoplanet Draugr = R_p = 2.05 x 10⁶ m

Orbital speed of the exoplanet Draugr = V

The centripetal force for the circular motion of the exoplanet Draugr is provided by the gravitational force of the star on it.

$\frac{mV^{2}}{R} = \frac{GMm}{R^{2}}$

$V^{2} = \frac{GM}{R}$

$V =\sqrt{ \frac{GM}{R}}$

$V =\sqrt{ \frac{(6.67\ast 10^{-11})(2.79\ast 10^{30})}{2.85\ast 10^{10}}}$

V = 80805.81 m/s

Orbital period of the exoplanet Draugr = T

VT = 2$\pi$R

(80805.81)T = 2$\pi$(2.85x10¹⁰)

T = 2.216 x 10⁶ sec

Converting to days,

$T = \frac{2.216\ast 10^{6}}{24\times 60\times 60}$

T = 25.6 days

Magnitude of acceleration of the exoplanet Draugr as it orbits the star = a

$a = \frac{V^{2}}{R}$

$a = \frac{(80805.81)^{2}}{2.85\ast 10^{10}}$

a = 0.229 m/s²

Free fall acceleration near the surface of the exoplanet Draugr = g

$g = \frac{Gm}{R_{p}^{2}}$

$g = \frac{(6.67\ast 10^{-11})(1.2\ast 10^{23})}{(2.05\ast 10^{6})^{2}}$

g = 1.90 m/s²

14) Period of the motion of the exoplanet Draugr as it orbits the star = 25.6 days

15) Magnitude of acceleration of the exoplanet Draugr as it orbits the star = 0.229 m/s²

16) Free fall acceleration near the surface of the exoplanet Draugr = 1.90 m/s²