clair002

Let's say we have two people viewing the moon who are 10300km apart along the same line of latitude at the same time. How much of a shift against the more distant background stars should the moon appear to shift? This approximation assumes that the sublunar point is roughly between the two observers. Well, first we need to find their actual linear distance (rather than the distance over the curvature of the Earth, which is what you get from Google Earth). Let's define our variables: \[ R = 6371393 m \;\; \text{Earth Average Radius} \] \[ d = 10300000 m \;\; \text{Earth distance along curvature} \] We can find the angle in radians from the arc length using: \[ \theta = d/R \] and we can find the length of a chord using the half-sine rule: \[ \text{crd} \, \theta = 2 \sin \frac{\theta}{2} \] Plugging that in we find that the straight-line distance is slightly shorter than around the curvature: \[ 6371393 \times 2 \sin(\frac{1}{2} \frac{10300000}{6371393}) \approx 9214000m \] Nex...

Nothing new here, just wanted to capture these proofs into a single location for easy reference. I've tried to arrange them into a form that is easy to understand and follow. Kepler's First Law: Ellipses \(\require{cancel}\)Proof that a Newtonian force between two masses will produce an elliptical orbit - this is a very textbook approach using polar coordinates, I'm just capturing it here for reference. There are many other approaches , and this has likely been done millions of times now. Remember that Newton's Law is: \[ \begin{align} F &= G \frac{Mm}{r^2} \\ &\therefore \nonumber \\ F &= m a \end{align} \] We define a unit vector in the radial direction \( \hat{r} \) along the angle \( \theta \): \[ \hat{r} = \hat{x} \cos \theta + \hat{y} \sin \theta \] Therefore \( \hat{r} \) changes as per angle \(\theta\) perpendicular to \( \hat{r} \), giving us the tangential unit vector \( \hat{\theta} \) \[ \frac{\mathrm{d}\hat{r}}{\mathrm{d}\theta} = \hat{x} ( ...

This is something I've been working on for a while and I hope you will find something new in it and you will almost certainly find some place where I've made an error (contact me on twitter @ColdDimSum to report errors). I don't think I've made any grievous errors but possibly have some things out of order, misattributed, or wrong in the fine details or due to my clumsy wording (I'm not a professional scientist nor writer nor science writer, I'm a professional software developer and I have been writing programs for over 35 years). For any errors I apologize -- but please consider these my notes on the subject and always find a good Primary source to substantiate anything specific. Thankfully, scientific theories are valid based solely on the authority of the evidence and not on who thought of them or when. But I do think I have some insights I can share based on my studies and I hope my errors do not detract from the overall story. In all likelihood, nothing...