1) A comet has a very elliptical orbit with a period of 134.3 y. If the closest approach of the comet to the Sun is 0.09 AU, what is its greatest distance from the Sun?
2) What is the magnitude of the gravitational field at the surface of a neutron star whose mass is 1.39 times the mass of the Sun and whose radius is 8.6 km?
2) What is the magnitude of the gravitational field at the surface of a neutron star whose mass is 1.39 times the mass of the Sun and whose radius is 8.6 km?
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Hi!
1) You know that the semi-major axis a, is given by:
a = r_a + r_p,
where r_a is distance at apogee (furthest)
where r_p is distance at perigee (closest)
You can use this to solve for r_a from Kepler's 3rd law (because it is valid for elliptic orbits along a)
---> T² = Ks (a^3) where Ks = 2.97 x 10^-19 (SI units).
Solve for a and then from that, you can get:
r_a = a - r_p.
Be sure to get consistent units since you have AU and years.
2) By gravitational field do you mean:
---> potential? V = - GM / r (traditionally this is the "field")
---> potential energy? U = - GMm / r (you would need the mass of a body in this field)
---> acceleration due to gravity? a = g = - GM / r² (I think this is what you mean?)
Either way, you just need to plug in 1.39*M_sun for M in any of the above and use the radius.
I believe M_sun ~ 2 x 10^30 kg
Hope that helps!
EDIT: Oops, sorry... there is a factor of 2 in there for semi-major axis! As my profs used to say, "Im not responsible for factors of 2, pi and i."
1) You know that the semi-major axis a, is given by:
a = r_a + r_p,
where r_a is distance at apogee (furthest)
where r_p is distance at perigee (closest)
You can use this to solve for r_a from Kepler's 3rd law (because it is valid for elliptic orbits along a)
---> T² = Ks (a^3) where Ks = 2.97 x 10^-19 (SI units).
Solve for a and then from that, you can get:
r_a = a - r_p.
Be sure to get consistent units since you have AU and years.
2) By gravitational field do you mean:
---> potential? V = - GM / r (traditionally this is the "field")
---> potential energy? U = - GMm / r (you would need the mass of a body in this field)
---> acceleration due to gravity? a = g = - GM / r² (I think this is what you mean?)
Either way, you just need to plug in 1.39*M_sun for M in any of the above and use the radius.
I believe M_sun ~ 2 x 10^30 kg
Hope that helps!
EDIT: Oops, sorry... there is a factor of 2 in there for semi-major axis! As my profs used to say, "Im not responsible for factors of 2, pi and i."
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Use kepler's third law to find the semimajor axis of the orbit:
a = {G*M*T^2/(4*pi^2)}^(1/3)
Where G = Universal gravitation constant, M = mass of the sun, T = period of the orbit (in seconds)
You'll get a in meters. Once you know a, you can use teh relationship between it, the perihelion and apohelion radius:
a = (ra + rp)/2 where ra = apohelion, rp = perihelion and solve for ra
ra = 2a - rp ---> set rp = 0.09 AU, convert to meters, plug in an you're done.
For question 2, you can use
F = GMm/r^2 Set M = 1.39*mass of sun and r = 8.6*10^3 m. Compute F/m
F/m = GM/r^2 which is units of Nt/m
a = {G*M*T^2/(4*pi^2)}^(1/3)
Where G = Universal gravitation constant, M = mass of the sun, T = period of the orbit (in seconds)
You'll get a in meters. Once you know a, you can use teh relationship between it, the perihelion and apohelion radius:
a = (ra + rp)/2 where ra = apohelion, rp = perihelion and solve for ra
ra = 2a - rp ---> set rp = 0.09 AU, convert to meters, plug in an you're done.
For question 2, you can use
F = GMm/r^2 Set M = 1.39*mass of sun and r = 8.6*10^3 m. Compute F/m
F/m = GM/r^2 which is units of Nt/m