And yet, here he was, last Sunday night on Coast to Coast AM, going one better. Just like last time, he was only summarizing what is on his web site so you can get the whole thing without having to slog through two hours of audio.
He seeks to explain why so many Mars landers have crashed instead of soft-landing as intended (10 out of 18, according to him, and for all I know it may be the correct figure - But see Update 28 August). Here are the steps in what he laughingly calls his "logic":
- The gravitational attraction a planet has for an object in its vicinity is only partly described by the Newtonian force G.m1.m2/d2.
- There's an additional term to consider, the centrifugal force generated by the planet's rotation.
- This force acts in opposition to the Newtonian force.
- Since Mars' rotation rate is 2.5% slower than that of Earth (actually the firgure is 2.8%) it generates less centrifugal force, and therefore more effective gravity, in its vicinity than Earth.
- An incoming lander is subject to the sum of the forces of gravity and centrifugal.
- Therefore a terminal flight profile calculated to be correct in Earth conditions fails on Mars.
Is he right? No, of course he isn't
Well, first off let me say that centrifugal force acting in opposition to the force of gravity on the surface of a planet is, indeed, a reality. "It's a thing," in the slang of today. This force can readily be calculated; it is
-(mv2 cos L)/r
where m is the mass of an object on the surface of a rotating planet
v is the linear velocity of the planetary surface at the equator (465 m/sec for Earth)
r is the radius of the planet (6.378 x 106 m for Earth)
L is the latitude where the force is measured
For Earth, the (v2 cos L)/r term works out as 0.034 m/sec2 at the equator where cos L evaluates to 1. A body, such as a fat woman, of mass 100kg weighs 340 grams less at the equator than at the poles, where cos L, and the centrifugal force, are both zero.note 2
Statements 1 & 2 are therefore in general correct when considering an object on a planet's surface. Statement 3 is also correct—it's perfectly possible to imagine a planet that rotates so rapidly that anything not tied down at its equator would be flung off into space. We would say that centrifugal force exceeds the force of gravity, in such a case.note 3
Statement 4 is a problem although Cotterell is basically correct in writing that centrifugal force is less on Mars. It has as much to do with the smaller size of the planet as with its rotation rate. However, that small difference is swamped by the fact that Newtonian gravity is very much less. Here are the figures (at the equator in both cases):
Earth, acceleration due to gravity: 9.863 m/sec2
Earth, acceleration due to centrifugal force: -0.034 m/sec2
Net acceleration: 9.829 m/sec2
Mars, acceleration due to gravity: 3.721 m/sec2
Mars, acceleration due to centrifugal force: -0.0171 m/sec2
Net acceleration: 3.704 m/sec2
It's in writing Statement 5 that Cotterell has gone completely haywire. He writes "Newton failed to recognize, in his equation, that a falling body is also under the influence of 'centrifugal force' caused by the spinning of the Earth on its axis." He's taken the purely local and surface-based phenomenon of centrifugal force, and made it a property of the planet as a whole, extending beyond the surface into the region where incoming landers start feeling the effect of a planet's gravity. This is as preposterous as Cotterell's prior comments about gravity, and shows complete lack of understanding of physics. Of course a spacecraft having no physical contact with a planet cannot possibly be influenced by rotation of the planet. Neither can a falling apple, come to that, so Newton's equation describes that event accurately.note 4
Statement 6 suggests that engineers devising flight profiles for soft landings simply don't know about this, and therefore miscalculate. Last Sunday night, even that old softie George Noory demurred in the gentlest possible way. He reminded Cotterell that the landing of MSL and its rover Curiosity in Gale crater six years ago (almost to the day, actually) had been a brilliant success and by no means a miscalculation. Cotterell mumbled something about engineers having learnt that when they completed their calculations they should "add a little bit, just for luck."
Cotterell is kind-of entertaining I suppose, with his bluff manner and his soft Lancashire accent, but he should be permanently banned from the fields of physics and mathematics lest he do even more damage to them.
Update 28 August:
Cotterell kindly provided the list from which he derived 10 of 18 failures, or 55%. It turns out he was including Phobos 1 and Phobos 2 as failures, when of course they were never intended to reach the surface of Mars. He also counted the rover "Prop-M" carried by Mars 3, and the rover "Sojourner" carried by Mars Pathfinder, as separate missions. And by the way his list was only 17 missions, not 18, so the percentage ought to be 53%. Small point.
I now believe the true statistic is 5 of 13, or 38%. Since the failure of Mars 2 was attributed to a computer malfunction, that perhaps should be adjusted to 4 of 13 or 30%.
 "Matilda told such dreadful lies"
 In fact, there's another phenomenon that affects the acceleration due to gravity on the planet's surface. The equatorial radius is 6378 km but the polar radius is only 6357 km. Since the fat woman is 21 km further away from the center of the planet when she's on the equator than when she's at the poles, gravity has less of a hold on her. The effect amounts to 0.668%.
 I'm going to be in trouble with the purists for even talking in terms of "centrifugal force." To them, this force is merely the "equal and opposite" reaction to a centripetal force. They would prefer to say "The force of gravity is inadequate to provide the centripetal force needed to keep objects attached to the planet." See the difference? But Cotterell uses centrifugal, and it's intuitive, so I'm going to stick with it.
 Note that as long as the apple is attached to the tree, it is pulled upwards by the small amount attributable to centrifugal force. As soon as it detaches, however, that small upward force vanishes.