8. The main objective of NASA's Kepler exploration is to find habitable planets in other planetary systems in our galaxy, the Milky Way. A large number of planets which orbit around stars have been detected by the exploration. One such observation was a planetary system consisting of two planets, planet A and planet B of orbital periods B = 300 earth days and T, = 50 earth days, respectively. Assume that the planets are uniform spheres and moving in circular orbits around a star 3 of mass M as shown in the figure. Neglect the interaction between the planets. (a) (b) (i) Derive an expression for the orbital speed (v) of the planet B in terms of M, the orbital radius R, of the planet B and universal gravitational constant G. (ii) Write down an expression for the period T, of the planet B in terms of R, and v (iii) Derive an expression for the mass M of the star at the centre in terms of T., R, and G. (iv) If R=0.3AU (1AU=1.5x 10" m), calculate the mass M of the star. Take G=6.7x10" m³ kg¹s² and² = 10. (i) Using the expression obtained in (a)(iii) above, derive an expression relating orbital radii R., R, and periods T and T of planets A and B. (ii) Calculate the orbital radius R, of planet A using given values. (c) The mass and the radius of the outer planet A are found to be 23 m, and 4.6%, respectively, where m TE are the mass and the radius of the earth respectively. and (i) Derive an expression for the gravitational acceleration, g, at a point on the surface of planet A, in terms of me and G. (ii) Obtain an expression for the 8, in terms of the gravitational acceleration g at a point on the surface of the earth. (iii) If a space landing module of mass 100 kg is landed on planet A, calculate the weight of the landing module after landing. (iv) The outer planet A is located within the habitable zone when compared with our solar system. Obtain an expression for the average density d of the planet A in terms of the average density d of earth.
8. The main objective of NASA's Kepler exploration is to find habitable planets in other planetary systems in our galaxy, the Milky Way. A large number of planets which orbit around stars have been detected by the exploration. One such observation was a planetary system consisting of two planets, planet A and planet B of orbital periods B = 300 earth days and T, = 50 earth days, respectively. Assume that the planets are uniform spheres and moving in circular orbits around a star 3 of mass M as shown in the figure. Neglect the interaction between the planets. (a) (b) (i) Derive an expression for the orbital speed (v) of the planet B in terms of M, the orbital radius R, of the planet B and universal gravitational constant G. (ii) Write down an expression for the period T, of the planet B in terms of R, and v (iii) Derive an expression for the mass M of the star at the centre in terms of T., R, and G. (iv) If R=0.3AU (1AU=1.5x 10" m), calculate the mass M of the star. Take G=6.7x10" m³ kg¹s² and² = 10. (i) Using the expression obtained in (a)(iii) above, derive an expression relating orbital radii R., R, and periods T and T of planets A and B. (ii) Calculate the orbital radius R, of planet A using given values. (c) The mass and the radius of the outer planet A are found to be 23 m, and 4.6%, respectively, where m TE are the mass and the radius of the earth respectively. and (i) Derive an expression for the gravitational acceleration, g, at a point on the surface of planet A, in terms of me and G. (ii) Obtain an expression for the 8, in terms of the gravitational acceleration g at a point on the surface of the earth. (iii) If a space landing module of mass 100 kg is landed on planet A, calculate the weight of the landing module after landing. (iv) The outer planet A is located within the habitable zone when compared with our solar system. Obtain an expression for the average density d of the planet A in terms of the average density d of earth.
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Transcribed Image Text:8. The main objective of NASA's Kepler exploration is to find habitable
planets in other planetary systems in our galaxy, the Milky Way. A
large number of planets which orbit around stars have been detected
by the exploration. One such observation was a planetary system
consisting of two planets, planet A and planet B of orbital periods
B
= 300 earth days and T, = 50 earth days, respectively. Assume that the planets are uniform spheres and
moving in circular orbits around a star 3 of mass M as shown in the figure. Neglect the interaction between
the planets.
(a)
(b)
(i) Derive an expression for the orbital speed (v) of the planet B in terms of M, the orbital radius R, of
the planet B and universal gravitational constant G.
(ii) Write down an expression for the period T, of the planet B in terms of R, and v
(iii) Derive an expression for the mass M of the star at the centre in terms of T., R, and G.
(iv) If R=0.3AU (1AU=1.5x 10" m), calculate the mass M of the star. Take G=6.7x10" m³ kg¹s²
and² = 10.
(i) Using the expression obtained in (a)(iii) above, derive an expression relating orbital radii R., R, and
periods T and T of planets A and B.
(ii) Calculate the orbital radius R, of planet A using given values.
(c) The mass and the radius of the outer planet A are found to be 23 m, and 4.6%, respectively, where m
TE
are the mass and the radius of the earth respectively.
and
(i) Derive an expression for the gravitational acceleration, g, at a point on the surface of planet A, in
terms of me and G.
(ii) Obtain an expression for the 8, in terms of the gravitational acceleration g at a point on the surface
of the earth.
(iii) If a space landing module of mass 100 kg is landed on planet A, calculate the weight of the landing
module after landing.
(iv) The outer planet A is located within the habitable zone when compared with our solar system. Obtain
an expression for the average density d of the planet A in terms of the average density d of earth.
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