Let's deal with some of Mr. Stojanowski's comments made on Getting Crankier.
Only the latitudinal movement is relevant to angular momentum.But again, the latitudinal movement of Pangea, and its remnants, during the past 350 myrs is what is important, not just the dispersal movement after its breakup.
No. Any change in I, a body's moment of inertia (the distribution of its mass about a rotation axis), is relevant to angular momentum.
…it was the displacement of the core elements, an action which would increase the Earth’s AM to balance the reduction of AM from Pangea’s center of mass moving closer to the spin axis as it moved to a higher latitude.… an action which would increase the Earth’s AM to balance the reduction of AM from Pangea’s center of mass moving closer to the spin axis as it moved to a higher latitude.
|CREDIT: J. Stojanowski|
Let's apply Mr. Stojanowski's hypothesis to the problem. In his work, he claims that the Earth's core was displaced by 1000 km in the equatorial plane. (see his Fig. 3 to the left). He also references a paper in Geophysical Research Letters* in which he claims they show movement of Pangaea's center of mass. More on the center of mass later. Figure 3(a) in this article (shown below) does show variation in a parameter the authors label D. The authors are attempting to quantitatively estimate the "asymmetry of continental surface." To calculate D, the authors use the first moment of the distance to the equator with respect to the surface area. This is not the center of mass.
|CREDIT: Petrelis et al.|
However, let's use Mr. Stojanowski's assumption that the two are the same and apply the conservation of angular momentum to his model. See the previous article for a brief introduction to this principle.
We need a few more numbers before we can calculate. The mass of the core is about 1/3 of the Earth's mass. We'll use Mr. Stojanowski's number of 1000 km for rc. The last quantity required is the distance from the axis to Pangaea. Here we'll use his assumption that D is the center of mass. From Figure 3(a) in the Geophys. Res. Lett. article, D is about 100 km south of the equator. We use a bit of elementary geometry and trigonometry to find the distance from the axis to Pangaea; rp = 6370 km.
Plug away. In the meantime, let's recap. I've used the conservation of angular momentum and Stojanowski's model and numbers. We find that the consequence of his model is that the Earth rotates once every 16.6 hours. This contradicts what we know the length of the day; then it was about 23 hours long.
Yes, I did assume in Getting Crankier that the center of mass stays on the equatorial plane. However, as I clearly showed in my first attempt, the moment of inertia is not the same over 250 million years. There I modeled Earth 250 mya as a sphere and a object (Pangaea) located 6371 km from the axis. And my calculation clearly demonstrates that the angular velocity must change as a result of conserving momentum.His assumptions in his calculation are:1. Pangea’s center of mass was located on the equator 250 myr.2. Today, the continental crust is distributed uniformly across the globe.Based on the above assumptions, in both cases, the center of mass of continental crust is on the equator. Therefore, the moment of inertia in both cases must be the same, meaning the angular velocity must also be the same to conserve AM.
I mentioned above that I would say something about the Earth's center of mass, but I ran across another blog that has dealt with this issue quite well. Ed Brayton** blogged at Dispatches from the Creation Wars, and on Dec. 27, 2010, guest blogger W. Kevin Vicklund wrote New Explanation for Dinosaur Extinction? Revisited. He shows how bad Mr Stojanowski's figure of 1000 km is for the core offset. The short version is that if the Earth is to rotate about its axis then the Earth's center of mass must remain at the Earth's geometric center. Mr. Stojanowski's model doesn't do that. If the core moved as he claims, the Earth would have precessed like crazy.
Yes, surface gravity at the poles is less than at the equator based on difference in distance to the Earth’s center of mass. This is evident from the inverse square law (relative to distance) postulated by Newton. It’s exactly why surface gravity on Pangea changed when the Earth’s core elements moved off-center and away from Pangea, moving the Earth’s center of mass further away from Pangea.No. The effect is due to the centrifugal term in Newton's 2nd law as I explained in the previous post. If you want to use Newton's law of universal gravitation to find the magnitude of the gravitational field near the Earth's surface, here's what you do.
The polar radius is 6357 km, and the equatorial one is 6378 km, only 0.3% larger. Use these to calculate the gravitational field, and you will quickly see that the equatorial field is only 0.6% smaller. Mr. Stojanowski may counter that his model yields a value 8% smaller for Pangaea, but as I and Mr Vicklund have shown, that core motion he postulates violates what we know about the Earth's rotation.
Mr Stojanowski. Should you reply once again, I ask that you back up any claims with physics, mathematics, and numbers.
*Plate tectonics may control geomagnetic reversal frequency, F. Pétrélis, J. Besse, J.-P.Valet, Geophysical Research Letters, vol. 38, issue 19, October 2011.
**I highly recommend Ed Brayton's Dispatches from the Culture Wars.