The necessary formula is: vesc = (2GM/R)0.5
where G = 6.67 x 10-11 N m2 / kg2, M is the mass of the planet and R is the radius of the planet
for Mars, M = 6.39 x 1023 kg, and R = 3.397 x 106 m. Thus
vesc (Mars) = (2 x 6.67 x 10-11 x 6.39 x 1023 / 3.397 x 106 )0.5
vesc (Mars) = 5009 m/sec = 5.00 km/sec
for Venus, M = 4.90 x 1024 kg, and R = 6.052 x 106 m. Thus
vesc (Venus) = (2 x 6.67 x 10-11 x 4.90 x 1024 / 6.052 x 106 )0.5
vesc (Venus) = 10,400 m/sec = 10.4 km/sec
2. Calculate the average speed, vave (in units of km/sec), of an oxygen molecule (O2) in the Earth's atmosphere, assuming T = 22 C (= 295 K). Are oxygen molecules bound to Earth?
For an oxygen molecule, the mass is that of two O atoms, and each atom has 16 times the mass of a H atom:
vave = [ 3 x 1.38 x 10-16 x 295) / (32 x 1.67 x 10-24) ]0.5
vave = 47,800 cm/sec = 0.48 km/sec
Clearly, since 6 x vave = 2.86 km/sec is much less than vesc (Earth) = 11.2 km/sec, oxygen atoms cannot escape from Earth.
vave (D) = [ 3 x 1.38 x 10-16 x 600) / (2 x 1.67
x 10-24) ]0.5
vave (D) = 272,700 cm/sec = 2.73 km/sec
6 x vave (D) = 16.4 km/sec > vesc (Earth) = 11.2 so "yes," these D atoms can escape.
vave (He) = [ 3 x 1.38 x 10-16 x 600) / (4 x 1.67
x 10-24) ]0.5
vave (He) = 192,800 cm/sec = 1.93 km/sec
6 x vave (He) = 11.6 km/sec > vesc (Earth) = 11.2 so "yes," these He atoms can escape, but only barely.
H will escape fastest, D next fastest, He slowest.
The top of the Earth's atmosphere is the 'exosphere' because it is only from this layer of the atmosphere that a fast moving atom or molecule can exit, or escape. Lower down where the air is denser, a fast moving molecule will collide with another molecule, thus preventing escape.
I forgot to assign a temperature for this calculation! As a follow-up to #3, I intended for this calculation to be done at T = 600 K:5. What are the dominant constituents of the hydro/atmospheres of Mars, Venus and
vave (N) = [ 3 x 1.38 x 10-16 x 600) / (14 x 1.67 x 10-24) ]0.5
vave (N) = 10,300 cm/sec = 1.03 km/sec
6 x vave (N) = 6.2 km/sec > vesc (Mars) = 5.0 km/sec so "yes," these N atoms can escape.
Whoops! We didn't get to this stuff yet, so it won't be tested on this exam. The answer is that Mars and Venus have atmospheres dominated by carbon dioxide while Earth's is dominated by nitrogen, oxygen and water. We'll get back to this in two weeks.6. Provide a range of values for the masses, diameters, compositions, and relative
The three terrestrial planets have masses ranging from 10% to 100% of the Earth, sizes ranging from 50% to 100% of Earth, compositions that are similar - rock and iron, and receive comparable amounts of heat (Venus gets about 50% more sunlight than Earth, being at only 0.7 AU while Mars gets a bit less than half that of Earth.
The giant planets have masses ranging from 15 (1500%) to 318 times that of Earth, sizes from 3.9 to 11.2 that of Earth, compositions that are dominantly H and He gas, and amounts of sunlight that range from 0.037 (3.7%) to 0.0011 (0.1%) that of Earth.
Vanderbilt Home Page | Vanderbilt Libraries | Physics & Astronomy Department | Dyer Observatory