For questions related to the mathematical operations and analysis of astronomical and astrophysical observations, processes and dynamics.
Questions tagged [mathematical-astronomy]
131 questions
27
votes
2 answers
Why is this a first integral? - particle near Schwarzschild black hole
Background
I know that the Schwarzschild metric is:
$$d s^{2}=c^{2}\left(1-\frac{2 \mu}{r}\right) d t^{2}-\left(1-\frac{2 \mu}{r}\right)^{-1} d r^{2}-r^{2} d \Omega^{2}$$
I know that if I divide by $d \lambda^2$, I obtain the…
zabop
- 1,031
20
votes
1 answer
Can there be an energetically unbounded three-body orbit where escape is impossible due to conservation of angular momentum?
This question evolved from a discussion below this answer which explains (among other things) that the total energy of a system offers insight as to the possibility of one (or all) members "escaping".
The total energy would be the sum of the…
uhoh
- 1,967
12
votes
2 answers
Why is my value for the length of daylight wrong?
I was watching a YouTube video where it showed how length of daylight changes depending on the time of year, and I was curious and wanted to try calculating the value of how long the daylight is in the Tropic of Cancer (23.5 degrees latitude) during…
user525966
- 5,957
9
votes
2 answers
How to calculate sunrise and sunset times?
I need to create a function (in C++) to calculate the sunrise and sunset times, but I am not a mathematician and I cannot find a correct (and easy) way to do that.
I need to get the same results as can be found…
KelvinS
- 255
8
votes
2 answers
How to prove the shape of the "Flower of Venus"
This is the gif that inspired this question. So, considering that Venus and Earth's orbits are coplanar concentric circles and that their orbits follow a 13:8 (difference of five) ratio, how can I prove this five-fold symmetry?
I've been playing…
Carlos Gruss
- 153
7
votes
2 answers
Seasonal changes in hours of daylight
I will post my own answer to this question unless someone else posts the same answer first, but I am curious to know what other points of view might lead to different ways of answering it.
Temporarily (for the duration of this question) assume for…
7
votes
0 answers
Keplerian orbits and closest approaches to Earth.
This question arose out of a discussion on Space.SE, but I think it will appeal to mathematicians more than astronomers:
Let's consider a small astronomical object following an ideal elliptic Keplerian orbit around the sun. For concreteness I'm…
hmakholm left over Monica
- 291,611
7
votes
5 answers
What fraction of a sphere can an external observer see?
Here is a geometry problem.
Let there be a ball of radius R and let's call it the Moon.
Let there be an external observer: A.
A is at a distance d to (the surface of) the Moon.
[Edit] A is a Cyclope, he has only one eye.
Question:
What…
6
votes
1 answer
Is $\frac{\arccos\left((\sqrt{r}+1)/(r+1/\sqrt{r})\right)}{\pi \left|1-r^{-3/2}\right|}$ analytic at $r=1$?
Is the function $f:(0,\infty)\rightarrow(0, 1)$, defined below, analytic at $r=1$?
$$f(r) := \frac{\arccos\left(\frac{\sqrt{r}+1}{r+\frac{1}{\sqrt{r}}}\right)}{\pi
\left|1-\frac{1}{r^{3/2}}\right|}\quad
\mathrm{if\ } r>0\mathrm{\ and\ }…
irchans
- 1,995
5
votes
3 answers
Are Position and Velocity (or Velocity and Acceleration) Vectors Always Parallel?
While reading Chapter 1 of an astrodynamics textbook, I came across the statement:
$$\mathbf{v}\cdot \mathbf{{\dot{v}}}=v{\dot{v}}$$
In other words, the dot product of velocity and the time-rate-of-change of velocity is simply equal to the product…
Bryson
- 53
5
votes
1 answer
Solving Kepler's second law
Kepler's second law, about equal areas in equal times, is a differential equation: it gives velocity as a function of location.
Where are the best expository accounts of the process of solving this equation, giving position as a function of time?
4
votes
1 answer
κ₀ for Mercury—Formula
I refer here to Ptolemy’s epicycle-and-deferent model of the Solar System, specifically that of Mercury (see drawing).
In this model, Mercury (not shown) revolves on an epicycle of center C, which itself turns on an eccentric circle (later called…
Pierre Paquette
- 235
- 2
- 14
4
votes
2 answers
Is there a way to solve $x\left(\frac{e^x+1}{e^x-1}\right)=4$ for x besides just plugging numbers in?
This comes into play in the equation for the shift in Cosmic Microwave Background (CMB) photon frequency due to inverse Compton scattering:
$\frac{\Delta T}{T_{CMB}} = y \left( x\left(\frac{e^x+1}{e^x-1}\right)-4 \right)$
Where $y$ is essentially…
SneezyTheDwarf
- 41
- 2
4
votes
1 answer
I would like to know if there is an easy expression to define the path of a point on the Earth (city of Los Angeles is one example) around the Sun?
We are a few hours away from spring equinox. I was thinking if there is a mathematical expression that would define the path of a given point on the Earth, given by longitude and latitude say 34.05, 118.25 Los Angeles, around the Sun. Considering…
kamran
- 151
- 5
3
votes
1 answer
Prove Kepler's second law of planetary motion
An object moves in $\mathbb R^3$ it's position $r(t)$ satisfies $$r''(t) = s(t)r(t)$$ for some scalar function $s$ (a central force field, in which all acceleration is directly towards or opposite the origin). Show that
(i) it's motion is confined…
SRobertJames
- 6,117
- 1
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