This Geophysics Problem of the Week is based on a true(ish) story. You know how it is when you have a crush on someone. Flirtation leads to kissing and only later do you get around to really talking and it turns out that the object of your affection believes that the moon landing was an elaborate government hoax! Oh no! Moral of the story: Don’t kiss anyone who hasn’t done their geophysics homework!
To make sure you are on neither end of this kind of occurrence, here is a homework problem about the moon, about gravity, and to show that it is possible to use a pencil and paper to make the types of calculations that are required to send a rocket to the moon.
To really land people on the moon, there are many complicated calculations involved, but the physics of the problem—mostly using Newton’s laws—is well understood. But if you have never played around with these types of problems, perhaps you might be inclined to think that it is like magic—or succumb to ignorance-based conspiracy theories.
Q1. Calculate the gravity field of the moon, assuming it is a homogeneous uniform sphere. Calculate the acceleration due to gravity on the surface of the moon. How much force would you weigh on the moon?(note strange sentence construction—this is to emphasize units.)
One of the major efforts of the Apollo program was to apply local gravity corrections to the approximation of q.1 to ensure better accuracy in the landing site. Lunar Orbiter tracking data yielded evidence of gravity anomalies due to buried mass concentrations—MASCONS—on the surface of the moon. These were described in a 1968 Science paper by PM Muller and WL Sjorgen. Here is the abstract; the rest is behind a paywall:
“Lunar Orbiter tracking data have been processed to supply a qualitatively consistent gravimetric map of the lunar nearside. While a simplified model was employed, the results indicate that there are large mass concentrations under the lunar ringed maria. These mass concentrations may have important implications for the various theories regarding lunar history.”
|Image Credit: NASA|
|GRAIL Gravity Map of the Moon. NASA/MIT|
Quantifying the Mascons helped pave the way for the Apollo 11 landing (picture at left).
The GRAIL mission has just published its new gravity map for the surface of the Moon.
Q2. Find the sea of tranquility and the Apollo 11 landing site on the GRAIL map. This was not so easy for me.
Q3. How significant are the gravity perturbations? Compare the magnitude of GRAIL’s gravity perturbations with the total gravity on the surface of the Moon.
Q4. Imagine you are landing a spacecraft on the moon, and have not accounted for the gravity in your original calculations. How far off your target have you landed? Find a way to make this an order-of-magnitude problem to be solved with pencil-and-paper and not a problem requiring spherical harmonics out past the 100th degree. One way to do this may be to pretend your landing site is adjacent to one of the larger gravity anomalies. Calculate the force exerted by that anomaly on your passively landing spacecraft. Ignore all other anomalies.
|Bouger Gravity Map of the Moon NASA/MIT|
Q5. At left is a series of images of the Moon’s Bouger Anomaly. Find a friend or family member who is not a geophysicist, and describe why the gravity image above looks different from the gravity image below. Together, try to correlate the images.
|Another great GRAIL image of the Moon's gravity NASA/MIT|
For more great GRAIL information and images see:NASA's GRAIL site