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Regarding the Use of Mean Anomaly ($M$) or Phase ($\Phi$) In the answer by Stan Liou he uses a Taylor Series approximation of the Mean Anomaly to derive a nice formula which determines the CPTS (Cumulative Perihelion Time Shift) value as a function of $t^2$. This formula produces results very close to those graphed by Weisberg & Taylor. As it took me ...

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Finding $\Delta t_N$ using a Binomial Series approximation and assuming $\dot P$ is constant over 30 years. This method does not make any assumptions about Mean Anomaly. Orbit is defined as the cycle from one periapsis to the next periapsis. Summary The time from the $0$th to the $N$th perihelion can be obtained from the following sum $$T_N = ... 0 Regarding the differentiation of the Mean Anomaly function M. Currently under revision Based on Wikipedia entry:- In celestial mechanics, Mean Anomaly is a parameter relating position and time for a body moving in a Kepler orbit. It is based on the fact that equal areas are swept in equal intervals of time by a line joining the focus and the orbiting ... 2 Write the mean anomaly as a Taylor series (n\equiv 2\pi/P):$$\begin{eqnarray*} M(t) \equiv \int_0^tn\,\mathrm{d}t &=& M_0 + \dot{M}_0t + \frac{1}{2}\ddot{M}_0t^2 + \mathcal{O}(t^3)\\ &=&n_0t + \frac{1}{2}\dot{n}_0t^2 + \mathcal{O}(t^3)\\ &=&\frac{2\pi}{P_0}t - \pi\frac{\dot{P}_0}{P_0^2}t^2+\mathcal{O}(t^3)\text{.} \end{eqnarray*}$$... 2 As for solving the equation, dP/dt=K/P^{5/3} is a separable differential equation. See the first section here. Google also brings up several pages describing how to solve such equations. In particular, we rewrite the equation in differential form as:$$P^{5/3}\, dP = K\, dt.$$Loosely speaking, I've cleared the denominators, with a goal to get the P ... 2 For reference, the relevant bit from the paper is: The observable pulsar is a weak radio source with a flux density of about 1\,\mathrm{mJy} at 1400\,\mathrm{MHz}. ... Our most recent data have been gathered with the Wideband Arecibo Pulsar Processors (“WAPPs”), which for PSR B1913+16 achieve 13\,\mathrm{\mu s} time-of-arrival measurements in ... 2 I think there's several points that need addressing. Firstly how expansion operates on scales below the intergalactic is very much an open question. The approximation of spatial homogeneity and isotropy fails before you reach these scales. Whilst it may be tempting to think that cosmic expansion could be modeled as something akin to a small repulsive force ... 0 I do not know where that formula was first published in full, but Oppenheim at least does something very close to it. First, keep in mind some of the relevant symbols in Oppenheim, though they're rather standard:$$\def\anode{☊}\begin{eqnarray*} k &=& \small\sqrt{G} = \small\text{Gaussian gravitational constant}\\ \anode &=& ...

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Excepting a Big Rip scenario, there is no eventual 'clash'. Consider a Friedmann–Lemaître–Robertson–Walker universe: $$\mathrm{d}s^2 = -\mathrm{d}t^2 + a^2(t)\left[\frac{\mathrm{d}r^2}{1-kr^2} + r^2\left(\mathrm{d}\theta^2 + \sin^2\theta\,\mathrm{d}\phi^2\right)\right]\text{,}$$ where $a(t)$ is the scale factor and $k\in\{-1,0,+1\}$ corresponds to a ...

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Definitely too small to measure. The expansion isn't even enough to measurably impact galaxy superclusters. What you are otherwise thinking of is known as the Big Rip. This is not guaranteed to happen, though. It depends on a constant describing the density of dark energy over time, usually denoted $w$. When $w<-1$ the Big Rip occurs in finite time, ...

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Since I don't have Walter's book, I'm uncertain as the context of the derivation of the equation you quote. Therefore, I've simply re-derived it here; apologies if there's some repetition of things you already know, but perhaps it'll be useful for anyone else reading this regardless. Constants of Motion The Schwarzschild solution is the unique nontrivial ...

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Gravity plus dark energy reportedly can combine to do awful things to the amount of energy in the universe: If dark energy does exist, then it ultimately causes the expansion of the Universe to accelerate. On their journey from the CMB to the telescopes like WMAP, photons (the basic particles of electromagnetic radiation including light and radio waves) ...

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Gravity travels at light speed (or less, possibly), so even in an infinite non-expanding universe of finite age you'd only be gravitationally interacting with a finite mass in a finite volume. Our universe is observed to be expanding, further inhibiting us from coming into contact with new objects. Furthermore, a common assumption of cosmological models is ...

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Quantum gravity could be the reason protons overcome the coulomb force and bond at the nucleus of an atom (strong force). If you extrapolate the mass energy of the plank units that fill up the volume of the proton nucleus it could satisfy the Schwarzschild condition, which is indeed, relatively quite a bit of gravity.

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