Astronomy 101

return to Homework page

Homework Answer Key: Homework 5

1.  Calculate the "escape velocity" for Mars and Venus in units of km/sec.
  • [Answer]
  •  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?

  • [Answer]
    1. The necessary formula is: vave = (3kT/m)0.5
      where k = 1.38 x 10-16 gm cm2 sec-2 deg-1), T = temperature, and m = mass of the molecule or atom of gas.  Note that the mass of a hydrogen atom is 1.67 x 10-24 gm.  We need to use mass units of gm (since that is what the constant k uses and temperature in degrees K.  Our answer will be in units of cm/sec (again, since those are the units embedded in the constant k):

      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.

    3.  In the upper atmospheres of the terrestrial planets, the temperature increases with height.  Using the general rule of thumb (above), can hydrogen atoms that reach a height in the Earth's atmosphere such that T = 600 K escape? What about deuterium atoms (heavy hydrogen, with twice the mass as normal hydrogen)? What about helium atoms?  Which of these  three materials - hydrogen, deuterium, helium - will escape the fastest?  Why is the top of Earth's atmosphere called the 'exosphere.'
     
     
  • [Answer]
    1. vave (H) = [ 3 x 1.38 x 10-16 x 600) / (1 x 1.67 x 10-24) ]0.5
      vave (H) = 385,700 cm/sec = 3.86 km/sec
      6 x vave (H)= 23.2 km/sec > vesc (Earth) = 11.2 so "yes," these H atoms can escape.

      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.

    4. Can nitrogen atoms escape from Mars?
    [Answer]
    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:
    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.
    5. What are the dominant constituents of the hydro/atmospheres of Mars, Venus and
         Earth?
    [Answer]
    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
         heating* by the Sun of these three planets.  Now provide a range of values for these
         same parameters for the four giant planets.
    [Answer]
    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.

       

    Return to Homework Page | Astronomy 101 Home Page

    Vanderbilt Home Page | Vanderbilt Libraries | Physics & Astronomy Department | Dyer Observatory