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While the exact date of the origin of life is uncertain, but it widely viewed as occurring very soon after the Earth become even slightly hospitable to life. Lance, you completely ignored the article about the inevitability of life. Science is by no means a pseudoscience rag. That period starts rather than ends about 2.5 billion years ago. The oldest stromatolites fossils are about 2.7 billion years old. Up until that time, life was strictly anaerobic. By the time those ancient blue-green algae had finished their job, a good chunk of that ancient life was extinct because free oxygen was highly toxic to that ancient life. Photosynthesis appeared after life had been around for some time. Marking 2.7 billion years as the start of life is, IMHO, erroneous. 2.7 billion years ago demarcates the onset of non-primitive life. This is more likely a result of the fact that it is much easier to see such systems than is it indicative of the relative abundance of systems like the solar system versus systems in which the dominant gas giant has migrated from the gas giant region formation region sunward. (Note well: A gas giant planet cannot form in the vicinity of Mercury's orbit.) These systems with close-in gas giants would be inimical to the development of inner planets, period. The type II migration that takes the gas giants sunward clears most of the remnant junk from the inner star system, including any nascent planets. All extant theories of planetary formation agree on one thing: Planetary orbits around single star systems should be nearly circular. The observed deviation might well be yet another selection effect. It might also indicate that most stars systems have hidden brown dwarf companions, and it might even indicate that all of our theories on planetary formation are wrong. Regardless of which is the answer, you are placing star systems with hot Jupiters and planets with highly elliptical orbit in the wrong bucket. The prevalence of brown dwarfs will reduce R*. The prevalence of hot Jupiters will reduce ne. The fraction fl The is fraction of suitable planets on which life actually appears. The key here is "suitable planets." Neither a hot Jupiter nor a planet with a highly elliptical orbit is suitable. Double-counting will falsely reduce the final value for N.

Since I doubt the OP can explain what he is talking about, I will do so. First, the "neutral point". This is the point in space at which the gravitational acceleration toward the Earth is exactly equal but opposite to the gravitatinal acceleration toward the Moon. This point is obviously located along the line between the Earth and Moon. Denoting the distance from the Earth to the Moon as [math]r_m[/math] and the distance from the Earth to this neutral point as [math]r_p[/math], [math]-\,\frac{GM_e}{{r_p}^2} + \frac{GM_m}{(r_m-r_p)^2} = 0[/math] or [math]r_p = r_m\,\frac 1 {1 + \sqrt{\frac{M_m}{M_e}}}[/math] or about 90% of the way to the Moon. Second, the Earth-Moon L1 point. This is the point in space along the Earth-Moon line that is stationary in the rotating Earth-Moon frame. Denoting this point as [math]r_1[/math], [math]-\,\frac{GM_e}{{r_1}^2} + \frac{GM_m}{(r_m-r_1)^2} + r_1\,\frac{G(M_e+M_m)}{{r_m}^3}= 0[/math] or [math]M_e(r_m-r_1)^2({r_m}^3-{r_1}^3) = M_m{r_1}^2({r_m}^3+r_1(r_m-r_1)^2))[/math] This is a fifth order equation. The Earth-Moon L1 point lies about 80% of the way to the Moon, or about 200,000 miles from the Earth. The OP conflated the L1 point with the neutral point and as a result obtained goofy results.

Mainstream science says that the Moon's mass is about 0.0123 Earth masses based on vehicles that have been placed in orbit around the Moon since the 1960s. Based on assessments of the Moon's size, this means the Moon has a density of about 3.4 gm/cm3 and a surface gravity of about 1/6 that of Earth's surface gravity. In short, you are bass-ackwards here. What's wrong here is that you are using the location of the Earth-Moon L1 point, not the "neutral point". The latter, the point where the gravitational acceleration toward the Earth is equal to the gravitational acceleration toward the Moon, is rather meaningless. The former, the L1 point, is quite important but it is not the same as what are calling the neutral point. The Lunar Surface Gravimeter was flown on Apollo 17 but failed. Are you claiming this is part of some vast conspiracy? What is this bunch of non-sequiturs? Mass is quantized, but the size of the individual particles is so incredibly tiny that quantum mechanics simply doesn't come into play on a planetary scale. Wrong. We can determine the Moon's gravity field by observing how the Moon affects vehicles placed in orbit about the Moon behave. For example, see http://www.sciencemag.org/cgi/content/full/281/5382/1476. All you have done is to demonstrate a lack of understanding. Newton's law of gravity, at least so far as the Earth and Moon are concerned, is quite fine. We have of course discovered that Newton's law of gravity is wrong in the sense that it does not account for the precession of Mercury or behavior around incredibly large, small masses (e.g., black holes). One needs general relativity to describe gravity for fast moving objects and very large masses. General relativity is equivalent to Newtonian gravity for smallish velocities and smallish masses.

Pure unadulterated garbage, and that is putting it nicely.

Why is this thread still alive? It's rather obvious that the OP is not paying attention to the critiques.

Drag was a factor in the very early solar system. This is what accounts for the extra-solar jovian planets that orbit at incredibly close distances to their parent sun. Our solar system escaped this problem; the planets cleared almost all of the junk from there paths. There is nothing left in the solar system to cause such drag today. There is a tiny amount of drag with the solar wind, but that is immeasurably small. Solar radiation pressure (which is directed outward) is a much, much bigger effect, and even that is incredibly small. There is no need to invoke dark matter in explaining the Moon's recession from the Earth. Conservation of angular momentum using simple Newtonian gravity does the trick. Invoking dark matter begs the question: Why then isn't the Earth receding from the Sun? Newtonian gravity is broken? Newton gravity does fail in regimes of very high velocity (Mercury's anomalistic precession) and very large masses (black holes). NASA still uses Newtonian gravity to plan and operate space missions. It works quite fine in regimes of non-relativistic velocities and smallish masses. The Moon's velocity with respect to the Earth (1/300,000 c) and the Earth's gravity field are both quite small.

Caveat: making conclusions from a sample size of one is a bit recklessness. I think we should all admit we are engaging in pseudoscience/speculation. Given that, there are some signs that life on Earth originated 4.25 billion years ago and very strong signs that life originated more than 3.8 billion years ago. This has led some to believe that life is inevitable. While extremely simple life might be inevitable, even less simple single celled life forms are not. It took quite some time to make the jump from primitive extremophiles to more modern single-celled life. About 40% of the totality of the existence of life is in the form of primitive extremophiles. Some planets are likely to be stuck in this mode, not even developing photosynthesis. Even with the aid of photosynthesis it took life even longer to clear the next hurdle: complex, multicellular life. About 85% of the totality of the existence of life is in the form of single-celled organisms. The "too elliptical an orbit" is quite unlikely. All of the planets in our own solar system (Pluto isn't a planet) have nearly circular orbits. Models of planetary formation indicate that this isn't a fluke. The early solar system had a density gradient that increased sunward. Just as atmospheric drag acts to circularize the orbits of low Earth-orbit satellites, the initial protoplanetary disk would have acted to circularize the orbits of everything in that disk. The "too many impact events" is also a bit unlikely, at least so far as primitive life forming is concerned. Life formed in the Hadean, when the Earth was still undergoing heavy meteor bombardment. No tectonics: This is a good point. I would place this as part of what distinguishes a planet as having "an environment suitable for life"; category ne. The same goes for mineralogy. As I mentioned in my first post, I was intentionally overgenerous with my assessment of 0.25 such planets per solar system. I suspect this number is much lower. Life evolves slowly, particularly so primitive life. Our own planet is the only one we can draw upon. The initial formation of life was quick, but subsequent developments were not. I see the key events (note well: I am not a biologist) to be the development of photosynthesis, the development of a nucleus, the development of sex, the development of semi-intelligence, the development of near-intelligence (tool use), and the development of true intelligence (complex speech and writing). There is a long gap between each development. These developments, unlike life itself, are not inevitable. That a lot of semi-intelligent species has some sort of hand equivalent but only one has developed true intelligence indicates to me that the probability is far, far smaller than 1%. For most species there is no need to progress further than they have on the intelligence scale. Our complex brain is an incredible energy hog. Excess intelligence may well be a maladjustment in most species.

What makes you think planets migrate toward their sun?

The version On Wikipedia is the same as that seti.org and replicated on many other sites: [math]N=R^*\times f_p\times n_e\times f_l\times f_i\times f_c\times L[/math] where (from http://www.seti.org/seti/seti-science/) N = The number of civilizations in The Milky Way Galaxy whose electromagnetic emissions are detectable. R* = The rate of formation of stars suitable for the development of intelligent life. fp = The fraction of those stars with planetary systems. ne = The number of planets, per solar system, with an environment suitable for life. fl = The fraction of suitable planets on which life actually appears. fi = The fraction of life bearing planets on which intelligent life emerges. fc = The fraction of civilizations that develop a technology that releases detectable signs of their existence into space. L = The length of time such civilizations release detectable signals into space. Here's my take: R* = One per year, a fairly standard estimate. Blue giants and brown dwarfs don't count as "suitable" star, nor due most double stars. fp = 0.9. I'll be generous. Most young stars appear to be surrounded by a dust nebula. ne = 0.25. This might well be generous given the number of star systems we have observed with jovian planets orbiting very close to their star. fl = 0.9. Life started so soon after the formation of the Earth that I will take this as a near-given. fi = 1/1,000,000. I'll defend this later. fc = 1/100. I'll defend this later as well. L = 100,000 years. I'm being optimistic and assuming we will make it past global warming, resource depletion, global thermonuclear war, plague, name your disaster. Put this together: 1/year*0.9*0.25*0.9*1e-6*1e-3*1e5 years = 1/5,000. We are almost certainly alone in our galaxy, very likely alone in the local group, possibly alone in the Virgo supercluster, but we far from alone in the universe. About my numbers: Only one out of a million planets that do develop life of any sort develop "intelligent" life. I'll define intelligent life as life capable of something akin to written communication. The Drake equation is missing a very big factor that militates against developing intelligent life. That big factor is the fraction of stars that develop complex life. Life on Earth was stuck in the form of single-celled organisms for 3.5 billion years or so, and then stuck in the form of non-intelligent life for the next 550 million years. The latter number indicates that intelligent life is a fluke. The first number indicates that complex life is an incredible fluke. Most planets will succumb to runaway global warming (Venus) or runaway global cooling (Mars). Even Earth nearly succumbed to the latter. Of the few planets that do survive that, there are external threats such as meteors, massive solar flares, passing stars that render the solar system chaotic, gamma ray bursts, etc., and internal threats such as excessive tectonic activity and frozen tectonic activity. Of the few planets that do survive life-destroying disasters, simple life is most likely to remain simple. Of the very, very few planets that do develop complex life, life is like to remain unintelligent. Only one out of a hundred planets that do develop "intelligent" life develop the ability to communicate electronically. Developing the ability to communicate electronically requires plenty of resources. Without our abundant metals and combustibles we would still be hunter-gatherers. Planets that form too far out from the center of the galaxy will not have the abundant metals the propelled humanity to progress beyond the stone age. (Planets that form too close to the center of the galaxy are much more prone to life-destroying disasters.) Planets where life struggles rather than thrives will not develop the hydrocarbons that propelled the Industrial revolution. On a few planets life will develop the ability to communicate electronically at just about the same time they develop the ability to obliterate themselves. Planets on which the ability to communicate electronically for but a decade or so don't count for much.

There have been only 20 or so generations since the introduction of tobacco to the old world and even fewer generations since the onset of widespread use of tobacco products. That is hardly enough time for a genetic mutation to have appeared and spread. The obvious explanation is that the genetic propensity for nicotine addiction was already present before the introduction of tobacco. Probably something very primitive, because even rats can become addicted to nicotine.

That factor of 2*pi is wrong. Ignoring the eccentricity and inclination of the Moon's orbit, Lagrange's planetary equations dictate that [math]\frac {d\mathcal X}{dt} = -\,\frac 2{\omega_Mr_M}\,\frac{\partial \mathcal R}{d r_M}[/math] where [math]\mathcal X[/math] is the difference between the argument of latitude and that expected from a pure 2-body perspective and [math]\mathcal R[/math] is the perturbative potential energy. The Sun is the driving factor here, so [math]\mathcal R = -GM_S\left(\frac 1{||\mathbf r_E + \mathbf r_M||} - \frac 1{r_E}\right)[/math] Expanding this out and averaging over one lunar orbit yields [math]\frac {d\mathcal X}{dt} \approx \frac {{\omega_E}^2}{\omega_M}[/math] and, yep, that factor of 2*pi doesn't belong. Your 5,000 km is right on the mark. Finally, I reiterate that comparing results against reality is always a good idea.

I did a back-of-the-envelope caclulation using Lagrange's planetary equations (I ignored the eccentricity and inclination of the Moon's orbit) and got an advance of 30,000 km via [math]2\pi \,\frac {{\omega_E}^2}{\omega_M}\, t\, r_M[/math]. I think that factor of 2*pi might not be right, in which case that 30,000 km becomes 5,000 km. Either way, 5000 km is not out of line because the simplifications. Comparing against reality is always a good idea.

I am not suggesting you use Lagrange's Planetary equations for propagation. I am merely suggesting you use them to understand why the Moon's position is advanced in the n-body model compared to the simple 2-body solution. Qualitatively, the Sun acts as a perturbing force on the 2-body problem. If you just use a first order approximation, this perturbation force more-or-less averages out to zero over one Moon orbit. However, the gravity gradient is a bit higher sunward than anti-sunward. Averaging a second-order expansion over a period of one orbit yields a small sunward acceleration.

Phate, do you know anything about Langrange's planetary equations?

There is no snapping here. This is simple Newtonian mechanics. All it takes here is a quick google for the journal name. For example, http://en.wikipedia.org/wiki/Journal_of_Scientific_Exploration, This article is a bit better. The journal is at least indexed by GeoRef. However, it is not one of their priority journals. A couple of points. First, this article only concerns itself with the size of the Earth, not the Earth's mass. Secondly, this article is essentially making the same kind of "god of the gaps" argument that creationists use to argue against evolution. The problem with such arguments is that science fills in those gaps. Take for instance one of the articles you cited in your first post: If you take that article at face value, it says the Earth is shrinking, not expanding. What is really happening is that our ability to measurement the size and shape of the Earth is improving. I mentioned earlier that one of the fundamental laws of physics is the law of conservation of angular momentum. Another such law is conservation of energy. Mass is energy, as demonstrated quite well at the end of World War II. Colliders do this all of the time. They do not violate conservation of energy. Also note that the this article does not discuss conversion of a positron into a proton. Positrons are leptons while protons are baryons. One will not and cannot turn into the other. No. The Earth does not have a significantly varying gravitational field. The Burgess shales are about 540 million years old and are one of the most remarkable fossil finds ever. These fossils mark what is called the Cambrian Explosion, when sea life rather suddenly transitioned from very primitive forms into all of the basic forms of life present on Earth today plus a whole lot of oddball life forms. There was no life on land 540 million years ago. Every single Burgess shale fossil is some form of sea life. So where are these fossils? At about 8,500 feet above sea level in the Canadian Rockies. The plate tectonics model explains quite well why the oldest part of the ocean floor is a mere 200 million years old. The ocean plates are primarily basaltic rock while the continental plates are primarily less dense granitic rock. The lighter continents literally float on top of the denser oceanic crust. As the plates move around, it is inevitably the oceanic crust that subducts into the Earth because of this large difference in density. The movement of the plates is a bit random over long periods of time. Continents can merge, covering up all signs of the ancient ocean that used to separate them. Continents can also split along rift boundaries, creating new oceans between formerly conjoined plates. Since it is the ocean floor that inevitable subducts into the Earth, this random jostling means that oceanic floor has a rather short life span. You're looking at things wrong. Our ability to measure practically everything is improving over time. No. Lack of evidence is lack of evidence. Pinning a goofy model on lack of evidence is, in general, a bad idea in science because the evidence will appear in time as our measurements get better and better. Intelligent advocates of religious belief quite some time ago saw the same kinds of arguments made with regard to religion as problematic. Such arguments are called "god of the gaps" arguments, and the problem is that science has a tendency to fill such gaps. You still have not addressed the two biggest problems with this goofy model. (1) Where does the mass come from? Any proposed mechanism flies in the face of physics, which is the most thoroughly vetted of all sciences. (2) If scientists had indeed found that the Earth were expanding, the results would be trumpeted in the most important journals rather than skulking about in obscure and/or psychoceramic journals. A couple of examples of this are the Michelson-Morley experiment and the discovery that the ocean floor is quite young. The Michelson-Morley was supposed to measure the speed of the Earth with respect to the luminiferous ether. The experimenters instead obtained a completely unexpected result: The speed of light was apparently constant. The negative result garnered a lot more publicity and interest than would have a positive result. The same thing happened with measurements of the ocean floor in the mid 1900s. Geologists had every reason to suspect that the ocean floor was the oldest of all rock. This discovery was one of the major drivers in the development of the plate tectonics model. Scientists love bizarre, unexpected results.

More that it is a minor publication, and as such the quality of the papers and the quality of the peer-review tend to be suspect. That is true for the Earth's orbital rate about the Sun, but that is because the Earth is so much less massive than the Sun. In general, the period of two bodies orbiting about their center of mass due to gravity is [math]P=2\pi\sqrt{\frac{a^3}{G(m_1+m_2)}}[/math] where [math]P[/math] is the period, [math]a[/math] is the semi-major axis (this is the radius of a circular orbit) of the orbit, and [math]m_1[/math] and [math]m_2[/math] are the masses of the bodies. They are extremely accurate. We know the distance to the Moon to millimeter level accuracy, thanks to retroreflectors left on the Moon by the Apollo astronauts. One of the consequences of this so-called expanding Earth theory is that to conserve angular momentum, the Moon would have to be moving closer to the Earth. Conservation of angular momentum is one of the fundamental laws of physics. The angular momentum resulting from the Earth and Moon orbiting each other is [math]l_{\text{orbit}} = m_Em_M\sqrt{\frac{Gr_M}{m_E+m_M}}[/math] The Earth and Moon also have angular momentum due to their rotation about their axes: [math]l_{\text{rot}} = I_E\omega_E + I_M\omega_M[/math] Increasing the Earth's mass (and size) would increase both the Earth-Moon orbital angular momentum the Earth's rotational angular momentum. Something has to give, and the only thing left is the Earth's rotation rate. The Earth's would have to slow down considerably to account for a significant increase in the Earth's mass and an increase in the Earth-Moon separation. We know the Earth's and the Moon's rotation rates and the Earth-Moon distance to extremely high levels of accuracy. The evidence flies in the face of this theory. That the Moon is moving away from the Earth means the Earth's rotation rate has to be slowing down, and it is. It's conservation of angular momentum again. The Earth spun considerably faster hundreds of billions of years ago. So, what causes this? It's not a change in the Earth's mass. Its the tides. If the ocean bed were frictionless, the tidal bulge would be perfectly aligned with the Moon. The ocean bed is not frictionless. This does two things. The tides moving against the Earth's surface slows the Earth's rotation rate. At the same time, the tidal bulge on the side of the Earth closer to the Moon is a bit in front of the Earth-Moon line, and this continuously gives the Moon a little boost of energy. The end result is a transfer of angular momentum from the Earth's rotation about its axis to the Moon's orbit about the Earth. ======================================================= There are many, many problems with this expanding earth hypothesis. One is the mass itself. Where does it come from? Physicists have been pondering the deep nature of matter for some time now, and no one has seen a positron turn into a proton. It does not and cannot happen. Another problem is the angular momentum problem I just described. Yet another problem is that the tectonic plate model of geology fits very well with observed behaviors and with the fossil record. How in the world does this expanding earth model explain the Burgess shales? One final problem, and this is the biggest one of all. Suppose we truly had found that the Earth was indeed increasing in mass to the extent expounded in this hypothesis. Such a result would not be published in some obscure journal. It would be published in one (or both) of Nature and Science, the two scientific journals at the top of the scientific journal pecking order.

Stop with the videos already! How about a paper in a peer-reviewed journal? If you can't find any (you won't), how about any written material? The Moon is indeed moving away from the earth. What implications does this have? If the earth is indeed growing how can you explain the moon moving away from the earth? That is the point. The Earth and Moon are moving away from one another by a very well-explained mechanism. If the Earth and Moon were getting more massive they should be moving toward one another. They aren't. How about an article from NASA itself, rather than a crackpot saying that NASA said the Earth's radius is growing at 18 mm per year? The only references you provided in support of this concept are videos and their ilk. None of the articles that you posted mention this concept whatsoever.

Moved to pseudoscience. There is no plausible physics behind these speculations. Just to name a couple of non-plausible ideas, positrons do not turn into protons and there are no magnetic monopoles. There is no plausible geology behind these speculations. The plate tectonics model is falsifiable but has not been falsified. To the contrary; the plate tectonics model is very well-documented and very well-observed. Neal Adams' stuff falsifies itself. For example, if the Earth and Moon are increasing in mass the Earth and Moon would be moving toward each other rather than away from each other. Videos are a terrible mechanism for communication scientific ideas. Videos by a comic book artist, doubly so.

No. You need not only a tin foil hat to believe in the hollow earth garbage, you need full body suit tin foil armor.

The Sun is losing 4 million metric tonnes per second due to fusion, not 400 million. You are off by a factor of 100. 4 million tonnes per second sounds like a lot of mass loss. It isn't when compared to the mass of the Sun itself. The solar mass is about 2×1027 metric tonnes. At the current rate, it will take the Sun another 15 billion years to lose a mere 1/1000th of its mass. The Sun will turn into a red giant and then a a white dwarf long before then.

Dude, you missed a factor of 1/2: [math]d=\frac 1 2 at^2[/math] (I prefer 'd' for displacement; using 's' is confusing as it is also used for speed.) Combining with [math]v=at[/math] and solving for the final velocity in terms of distance and time, [math]v=at = 2\frac d t[/math]

Funny you should mention Star Wars ... From Ask an Astrophysicist No. There is no attraction toward the center of mass, only to the two stars individually. One scenario for planetary formation in a binary system is a wide binary system -- a pair of stars orbiting each other at with tens of AUs between the stars (e.g., the Centauri system). Planets can form close to one of the stars and remain in relatively stable orbits for quite some time. As an analogy, think ofJupiter being a star and its moons being planets. The other scenario for planetary formation in a binary system is a close binary system -- a pair of stars orbiting each other at fractions of an AU. Planets can still form, but only at some distance from both stars. The stars will distort the orbits of close-in planets significantly; Kepler's laws will not hold in such a system. A paper: http://www.astroscu.unam.mx/rmaa/RMxAC..22/PDF/RMxAC..22_lissauer.pdf

"Fairness" is not a good metric for at least two reasons: What does the term "fair" mean? Any arguments about the fairness of the tax code revolve around unstated assumptions regarding the term "fairness". What liberals think is fair is anything but fair in a conservative's mind, and vice-versa. Flat tax proponents claim "fairness" as one of the virtues of the flat tax. Some extremists even go so far as to argue that the "fairest" tax is a flat fee. Devils advocacy: The rich don't drive any more than do the middle class. Why should they pay any more in taxes? There is no such thing as a "fair" tax because there is nothing "fair" about taxation. Taxation is legalized armed robbery. Note well: I am not advocating doing away with taxation. Saying that requires a lot more than a mere tin foil hat. Only those who wear full-body tin foil armor can make such a claim. I am merely advocating throwing out the idea of 'fairness'. I suggest a metric of minimizing unfairness in its stead, and in particular spread the pain-of-suffering as evenly as possible. For example, consider the case of a hypothetical 25% across-the-board flat tax. A poor person who makes $10,000 per year has to pay $2,500 in taxes, while the person who makes one million per year has to pay $250,000. The pain felt by the typical $10,000 per year wage earner as a result of paying $2,500 in taxes is a whole lot more than the pain felt by the typical million dollar per year wage earner as a result of paying $250,000 in taxes. The across-the-board flat tax does a terrible job of spreading the pain-of-suffering. Some level of progressivity is needed to spread the pain-of-suffering as evenly as possible. Another advantage of this metric is that it dissociates the issue of who pays how much from the issue of how the tax monies are spent. This is not the case when the "fairness" metric is used. Collecting the lion's share of the federal tax receipts from one group (the wealthy) and giving that money to others (the poor) is anything but fair.

First off, a correction: Oops! That is the rotating-to-inertial transformation matrix. I'll start very generically here. Suppose you have at hand the representations in some n-dimensional Cartesian reference frame [math]A[/math] of the n unit vectors of some other Cartesian reference frame [math]B[/math]. Denoting these unit vectors as [math]\lbrace \hat{\mathbf u}_{B_i:A} \rbrace[/math], where the subscript [math]B_i[/math] indicates the ith unit vector of reference frame [math]B[/math] and the subscript [math]:A[/math] indicates the unit vector is expressed in terms of reference frame [math]A[/math]. The matrix that transforms column vectors from frame A to frame B in terms of these unit vectors is [math]\mathbf T_{A\to B} = \bmatrix \hat{\mathbf u}_{B_1:A}^{\;T} \\ \hat{\mathbf u}_{B_2:A}^{\;T} \\ \cdots \\ \hat{\mathbf u}_{B_n:A}^{\;T} \endbmatrix [/math] As a sanity check, [math]\mathbf T_{A\to B}\,\hat{\mathbf u}_{B_i:A} = \hat {\mathbf e}_i\quad(\hat {\mathbf e}_{i_j} \equiv \delta_{ij})[/math], as desired. Back to the problem at hand: In [math]\mathcal R^3[/math], the time derivative of some vector [math]\mathbf q[/math] as observed by an observer fixed in reference frame A is related to the derivative as observed by an observer fixed in reference frame B via [math]\frac{d\mathbf q}{dt}\Bigl|_A = \frac{d\mathbf q}{dt}\Bigl|_B\, +\,\boldsymbol{\omega}_{A\to B}\times \mathbf q[/math] where [math]\boldsymbol{\omega}_{A\to B}[/math] is the rotation rate of frame B with respect to frame A. The subscript [math]A\to B[/math] on [math]\boldsymbol{\omega}[/math] is going to make the math get ugly, so I will simply use [math]\boldsymbol{\omega}[/math] from this point on. The above equation can be expressed in matrix form by means of the skew symmetric cross product matrix. Denoting [math]\boldsymbol{\omega}^{\mathbf X}[/math] as this matrix, [math]\frac{d\mathbf q}{dt}\Bigl|_A = \frac{d\mathbf q}{dt}\Bigl|_B\, +\,\boldsymbol{\omega}^{\mathbf X}\, \mathbf q[/math] The natural way to express the derivative of some vector as observed by an observer fixed in some reference frame is to express the vector in terms of that reference frame and differentiate the vector element-by-element: [math]\frac{d\mathbf q}{dt}\Bigl|_{A:A} = \frac{d\mathbf q_{:A}}{dt}[/math] With the vectors expressed in the reference frame of the observer, the above relation between the two time derivatives becomes [math]\frac{d\mathbf q_{:A}}{dt} = \mathbf T_{B\to A} \left( \frac{d\mathbf q_{:B}}{dt} +\,\boldsymbol{\omega}_{:B}^{\;\mathbf X}\, \mathbf q_{:B}\right)[/math] Another way to arrive at the relation is to differentiate the transformed vector [math]\mathbf q_{:A} = \mathbf T_{B\to A}\,\mathbf q_{:B}[/math] [math]\frac{d\mathbf q_{:A}}{dt} = \mathbf T_{B\to A}\,\frac{d\mathbf q_{:B}}{dt} + \dot{\mathbf T}_{B\to A}\,\mathbf q_{:B}[/math] Equating the two results, [math] \mathbf T_{B\to A} \left( \frac{d\mathbf q_{:B}}{dt} +\,\boldsymbol{\omega}_{:B}^{\;\mathbf X}\, \mathbf q_{:B}\right) = \mathbf T_{B\to A}\,\frac{d\mathbf q_{:B}}{dt} + \dot{\mathbf T}_{B\to A}\,\mathbf q_{:B}[/math] Canceling the common term yields [math] \mathbf T_{B\to A}\, \boldsymbol{\omega}_{:B}^{\;\mathbf X}\, \mathbf q_{:B} = \dot{\mathbf T}_{A\to B}^{\;T}\,\mathbf q_{:B}[/math] For the above to be true for any vector quantity [math]\mathbf q[/math], [math] \mathbf T_{B\to A}\, \boldsymbol{\omega}_{:B}^{\;\mathbf X} = \dot{\mathbf T}_{B\to A}[/math] The above can be used to determine the rotation rate if the transformation matrix and its time derivatives are known: [math] \boldsymbol{\omega}_{:B}^{\;\mathbf X} = \mathbf T_{B\to A}^{\;T}\,\dot{\mathbf T}_{B\to A}[/math] Denoting [math]\hat u, \hat v, \hat w[/math] as the unit vectors for frame B expressed in terms of frame A, [math]\boldsymbol{\omega}_{:B}^{\;\mathbf X} = \bmatrix \hat u\cdot\dot{\hat u}&\hat u\cdot\dot{\hat v}&\hat u\cdot\dot{\hat w}\\ \hat v\cdot\dot{\hat u}&\hat v\cdot\dot{\hat v}&\hat v\cdot\dot{\hat w}\\ \hat w\cdot\dot{\hat u}&\hat w\cdot\dot{\hat v}&\hat w\cdot\dot{\hat w}\endbmatrix [/math] This matrix is indeed skew symmetric since [math]\hat u_i\cdot \hat u_j = \delta_{ij}[/math] Differentiating wrt time, [math]\frac{d}{dt}(\hat u_i\cdot \hat u_j = hat u_i \cdot \dot{\hat u}_j + \dot{hat u}_i \cdot \hat u_j = 0[/math] This reduces to zero for [math]j=i[/math]. From the above, [math] \begin{aligned} \omega_u &= \hat w\cdot\dot{\hat v} = -\hat v\cdot\dot{\hat w} \\ \omega_v &= \hat u\cdot\dot{\hat w} = -\hat w\cdot\dot{\hat u} \\ \omega_w &= \hat v\cdot\dot{\hat u} = -\hat u\cdot\dot{\hat v} \end{aligned} [/math] The frame rotating with the Moon's orbit about the Earth is defined in terms of the displacement vector [math]\mathbf r_{E\to M:I}[/math] from the center of the Earth to the center of the Moon expressed in some inertial system; [math]\dot{\mathbf r}_{E\to M:I}[/math], the inertial time derivative of this vector; and [math]\mathbf h_{E\to M:I} = \mathbf r_{E\to M:I} \times \dot{\mathbf r}_{E\to M:I}[/math], the specific angular momentum of the Moon with respect to the Earth. With these, the rotating frame unit vectors expressed in inertial coordinates are [math] \begin{aligned} \hat {\mathbf r} &= \frac{\mathbf r_{E\to M}}{||\mathbf r_{E\to M}||} \\ \hat {\boldsymbol \theta} &= \hat {\mathbf r} \times \hat {\mathbf h} \\ \hat {\mathbf r} &= \frac{\mathbf h_{E\to M}}{||\mathbf h_{E\to M}||} \end{aligned} [/math] Applying the general expression for the components of the frame angular velocity vector to this situation, [math] \begin{aligned} \omega_r &= \hat h\cdot\dot{\hat \theta} = -\hat {\theta}\cdot\dot{\hat h} \\ \omega_{\theta} &= \hat r\cdot\dot{\hat h} = -\hat h\cdot\dot{\hat r} \\ \omega_h &= \hat {\theta}\cdot\dot{\hat r} = -\hat r\cdot\dot{\hat {\theta}} \end{aligned} [/math] Since the Moon's angular momentum vector is normal to the velocity vector, the [math]\hat \theta[/math] component of the frame angular velocity vector is zero. With a little manipulating, the [math]\hat h[/math] component of the frame angular velocity vector is simple [math]h_{E\to M}/r^2_{E\to M}[/math]. The radial component involves the out-of-plane acceleration of the Earth and Moon toward the Sun, Jupiter, ...

The top 1% of the population already pays 38.8% of all federal income tax receipts. It certainly is feasible to make them take on an even greater burden. Whether it is fair to do so is a different question, and whether the top 1% of the households would willingly fork over several several hundred thousand dollars each is yet another question. One huge problem with asking any one person to fork over an extra a hundred grand or ten to the government is that it becomes very worthwhile to that person to pay a bevy of creative tax attorneys to ferret out as many loopholes as possible. The rich will focus their energies on escaping taxation (and on bribing Democratic congresscritters) rather than expanding jobs.