Physics 320 

Prof. Dale E. Gary
NJIT 
Venus, Earth, Moon
A. Venus
Venus is unusual in having a retrograde rotation, although the rotation is very slow (sidereal period 243.01 days). It has only a 3 degree inclination (tilt of its axis relative to its orbital plane), although due to the retrograde motion it should be considered 177 degrees.The bulk density is 5200 kg/m^{3}, which matches Earth well considering Venus' slightly smaller mass (82% of Earth). This implies that Venus may have an interior very like Earth's.
Hydrostatic Equilibrium
We now want to look at the internal pressure of Venus. It may seem that there is no way to know it, but if we think about the fact that the pressure on an element of mass is just due to the weight of all of the mass above it, we will see that it is not so hard to estimate. Pressure is force/unit area, F / A. Pressure in the interior of a planet arises from the inward force of gravitybalanced by the outward pressure due to the normal force of what is below it. Of course, Venus can be considered as static, meaning that it is not expanding or contracting. Such a state is called hydrostatic equilibrium, and like all equilibrium situations, it is characterized by force balance. Consider a column of crosssectional area A. The forces on a section (parcel) of the column are shown in the figure on the left:
The mass of the parcel is related to the mass density by r = m/V = m/A(r_{2}  r_{1}). From the freebody diagram it is clear that
F_{1} = F_{2} + mg = F_{2} + A(r_{2}  r_{1})rg
where g is the local acceleration of gravity (which obviously varies with depth in the planet). This force equation can be rearranged to
DF / A = DP = Drrg
where DP is the pressure difference (P_{1}  P_{2}) on the two ends of the column section, and Dr is the radius difference, or thickness of the parcel, (r_{1}  r_{2}). Taking the limit as Dr approaches zero, we obtain the equation of hydrostatic equilibrium:
dP/dr = rg = r(r)GM(r)/r^{2} (1)
where M(r) is the mass interior to the radius r. This is the equation of hydrostatic equilibrium. Now, we cannot go further unless we know how the mass depends on radius, in other words, unless we know M(r). However, let's make an approximation that the interior of the Venus is of constant density, with r =<r> = 5200 kg/m^{3}. Then the mass function will be:
M(r) = 4/3 pr^{3}<r>,
and we can solve equation (1) by integrating:
dP =  <r>G (4/3) pr^{3}<r>/r^{2} dr
from the center to the surface:
0 R dP =  <r>^{2 }(4/3)pG r dr P_{c} 0 so
P_{c} = (2/3) pG <r>^{2 }R^{2} = (1.4 x 10^{10}) <r>^{2 }R^{2}.
Using our bulk density of 5200 kg/m^{3}, and the Venus radius of 6052 km, we find that the pressure at the center of the Venus is about 1.38 x 10^{11} Pa = 1.38 x 10^{6} atm.
The SI unit for pressure is pascals, where 10^{5 }Pa = 1 atm = 1 atmosphere.
The actual pressure at the center of the Venus is about 3 x 10^{6} atm, due to the fact that density is not actually constant with radius. By the same approach as above, we find that Earth's central pressure is about 23% higher (3.7 x 10^{6} atm), due to its higher average density (5500 kg/m^{3}) and slightly larger radius.
Venus appears completely cloud covered, with a high albedo (0.67). Swirling wind patterns can be seen in the clouds, as shown in the photos from Galileo, below:
Images from Galileo 1 day apart, except the bottom two
images were taken 2 hours apart. Note that the clouds
are lagging behind at the equator, and the patterns move
right to left in the figure. The temperate zones have the
equivalent of Jet streams.The clouds trap allow some visible light through, but trap infrared light due to the presences of greenhouse gases in the atmosphere.
Glass transmits visible light, but blocks infrared radiation. This is how greenhouses work. CO_{2 } is an example of a greenhouse gas, which behaves in a similar way, letting solar radiation in but trapping heat. (from YouTube video https://www.youtube.com/watch?v=Fx49t4sv7f0
Venus' atmosphere is 96.5% carbon dioxide (CO_{2}). Russia landed several landers on the surface, as shown in the accompanying images. The temperature and pressure conditions were so high that the landers survived only a few hours, despite their diving bell construction.
Venera 13 Lander images
Venera 14 Lander images
Because of the clouds, scientists in the 18th century used to speculate that Venus was a garden planet of lush tropical vegetation. We now know that it is more like biblical version of hell, with the clouds made of sulfuric acid, the atmospheric pressure at 95 atm and the temperature at about 740 K, hot enough to melt lead!
Surface Features:
You might think that we could know very little about Venus' surface, since it is perpetually cloudcovered. However, thanks to the Magellan Venus Orbiter, which was able to use radar to map the entire surface, we now have extremely detailed images of the surface, which includes the usual craters, but also volcanoes and many features never seen on any other planet.Let's look at some images of Venus (photo gallery)
 Why do you think the rays of ejecta from craters look so strange compared to the Moon or Mercury?
 Why are there few craters on Venus?
 Why is there so much apparent volcanism?
B. Earth and Moon
The Earth is a planet, but one that we can study in great detail. We know a lot about the structure of the interior of the Earth, but we must remember that other planets may be different in fundamental ways. We must take what we learn about Earth and compare and contrast it with the other planets.The mean or bulk density of a planet is an easily measured quantity that can tell us a great deal about what the planet is made of. Earth's bulk density is
<r> = 3M_{E}/4pR_{E}^{3} = 5520 kg/m^{3}(compare with the density of water, 1000 kg/m^{3}). But the density of the Earth's surface (density of silicate rocks) is only 2800 kg/m^{3}, so the interior must be much more dense than the surface, a result of the pressure of overlying rocks due to hydrostatic equilibrium, as we discussed above. The structure of the interior of the Earth has been pieced together from a number of clues, such as what the surface is made of, what we see coming to the surface in volcanoes, and most of all what we learn from earthquakes.The earthquakes, adjustments of the Earth's crust due to internal stresses, launch two types of seismic waves, longitudinal compression (P) waves (sound waves), and transverse distortion (S) waves. The P waves travel faster, and so the letter P could stand for Pressure or Prompt waves. The slower S waves arrive later, and so the letter S could stand for Secondary, or Slow waves. An important difference between longitudinal and transverse waves is that longitudinal waves can travel through liquid, but transverse waves cannot. Both kinds of waves refract due to density gradients in the Earth's interior. Due to the refraction of both kinds of waves, and the lack of propagation of S waves, we learn that the interior has a liquid layer, and also can measure such quantities as temperature and density as a function of depth. The size of the liquid core can be determined by measuring "earthquake shadow zones." (movie)
The interior of the Moon is completely different from the interior of the Earth, as can be determined from the Moon's bulk density of 3370 kg/m^{3}, similar in density to the Earth's crust. So the density of the Moon cannot rise much in the center, as Earth's does. From seismic experiments from stations left on the Moon by the Apollo astronauts, the current model for the Moon's interior includes an offcenter, solid core, a fluid outer core, a mantle, and outer crust.
Cartoon of Moon's interior, from NASA JPL web site. Michael Thorne, U. Utah A recent mission to the Moon (GRAIL), launched on Sept. 10, 2011 as a pair of orbiters that flew in formation and precisely measured the gravity of the Moon as a way of mapping the interior.
Another mission has been returning highresolution images of the surface: Lunar Reconnaissance Orbiter.
What is the temperature of the Moon's surface? Recall our equation for subsolar temperature,T_{ss} = 394[1  A]^{1/4} (r_{p})^{1/2}.The Moon is very dark, so its albedo is only 0.07. Its distance from the Sun, of course, is the same as the Earth's, so the subsolar temperature is a whoppingT_{ss} = 387 K.