Escape Speed and Atmospheric Temperatures of Planets: Calculations and Analysis

[zferic28]

a) Show that the escape speed for a particle to leave the gravitational infuence of a planet is given by (2GM/R)^1/2, where M is the mass of the planet, R is its radius, and G is the gravitational constant.
b) The temperature near the top of Jupiter's multicolored cloud layer is about 140K. The temperature at the top of the Earth's troposphere, at an altitude of about 20km, is about 220k. Calculate the rms speed of huydrogen molecules (h2) in each of these environments. Give your answers in the m/s and as a fraction of the escape speed from the
respective planet.
c) Hydrogen gas is a rare element in the Earth's atm. In the atmosphere of Jupiter, by contrast, 89% of all molecules are H2. Explain why, using your results from the previous part.
d) Ceres is an asteroid with mass equal to .014 times the mass of the moon, a density of 2400kg/m^3 and a surface temp of about 200K. Suppose an astronomer claims to have iscovered an oxygen (O2) atmosphere on the asteroid Ceres. You are asked by a TV news reporter to comment on this claim. What would you say, and how would you support that?

a) I don't fully understand what I have to calculate or demonstrate in this, can someone please break this down for me? Thanks!




b) I need the formula Vrms = sqrt(3RT/M)

M of H2 = 2g/mol = .002kg/mol

Vrms Jupiter = sqrt(3x8.314x140k/.002kg) = 1.321x10^3 m/s
Vrms Earth = sqrt(3x8.314x220k/.002kg)= 1.656x10^3 m/s

I don't quite understand what I have to do to get the fraction of the escape speed relative to the planet.

I suppose use the given formula (2GM/R)^1/2 to calculate each plantes escape and then divide that number by the Vrms of Jupiter and Earth?? What is g, M, and R of jupiter?? Is 9.8m/s the gravitational force at 20km high in the Earth's atmosphere?? Please help!

c) I am guessing this will have to do with the escape velocity of each planet? Maybe Jupiter likes to hang on to it's H2 and Earth let's them go?

d) Lost me again on this one? Thanks for all the help..

 

[Farsight]

Once you've got the rest d) is dead easy zferic. You just plug a) and b) back in.

 

[kyle8921]

a) To calculate the escape speed, we need to use the formula (2GM/R)^1/2, where M is the mass of the planet, R is its radius, and G is the gravitational constant. This formula shows the minimum speed that a particle needs to escape the gravitational pull of a planet. To demonstrate this formula, we can use the principles of escape velocity and gravitational potential energy.

b) To calculate the rms speed of hydrogen molecules (H2) in each environment, we can use the formula Vrms = sqrt(3RT/M), where R is the gas constant, T is the temperature in Kelvin, and M is the molar mass of the gas. The molar mass of H2 is 2g/mol, so we can use this value to calculate the Vrms for both environments. To get the fraction of the escape speed, we can divide the Vrms by the escape speed calculated in part a.

c) The reason why there is a higher percentage of H2 in Jupiter's atmosphere compared to Earth's atmosphere is due to the escape velocity of each planet. Jupiter has a larger mass and radius compared to Earth, which means it has a higher escape velocity. This makes it more difficult for H2 molecules to escape Jupiter's gravitational pull, leading to a higher percentage of H2 in its atmosphere.

d) Based on the given information, it is highly unlikely that an oxygen (O2) atmosphere exists on the asteroid Ceres. Ceres is much smaller and has a lower mass and density compared to the Moon, which does not have enough gravitational pull to hold onto an atmosphere. Additionally, the surface temperature of 200K is not high enough to sustain an O2 atmosphere. Therefore, it is more likely that any O2 detected on Ceres is due to other factors such as surface contamination or instrumentation errors.