Questions tagged [mathematical-astronomy]
For questions related to the mathematical operations and analysis of astronomical and astrophysical observations, processes and dynamics.
119
questions
1
vote
0
answers
24
views
How can I identify which planet I'm in ? Using only sensors
I'm working on a personal project where I want to identify on which planet I'm in using the following sensors:
Accelerometer, Gyroscope, Pedometer, Magnetometer and an Altitude Sensor.
My first idea ...
2
votes
0
answers
27
views
Kepler's Equation of Elliptical Motion in Terms of Cosine
Virtually all material discussing Kepler's Equation of Elliptical Motion resolves its solution to solving the following equation numerically:
$$
E - \textit{e}\;\sin(E) = M
$$
where E is the Eccentric ...
0
votes
0
answers
31
views
Which function resembles this graph?
I am plotting the apparent magnitude of all visible stars sorted from brightest (-1.46) to dimmest (8ish) and this graph is the result. I feel like I've seen this graph's shape before, but I can't ...
0
votes
0
answers
49
views
What does it mean for the dots in the concentric circles when their wave functions intersect in a Fourier representation?
I saw a piece of code on github which transforms the planetary movement into the Fourier wave function.
These circles are given by the $x$ and $y$ ordinates: $x=\cos (\omega t)$, $y=\sin (\omega t)$, ...
3
votes
0
answers
47
views
Mathematical formulation of a gravitational law for galaxies beyond newtonian
The author is not familiared with the current theories and the specific model for dark matter, nor the PDE aspect of mathematical theories related to the dark matter problem, so I apologize in advance ...
1
vote
1
answer
198
views
What is the equation describing earth's orbit around the sun in 3 dimensional space?
I'm trying to draw a 2d ellipse in 3d space, which describes earth's orbit around the sun. such as image of a 2d ellipse in 3d space or the same image but different perspective. I'd like to be able to ...
1
vote
1
answer
41
views
Formula to calculate number of sunsets per year based on lattitude
How can I create a formula to turn latitude into number of sunsets per year?
Let's keep it relatively simple. Assuming the Earth is a smooth ball, assuming the sun is a single point, not getting ...
3
votes
1
answer
80
views
Declination angle
Currently I'm studying spherical geometry. I have come across the declination angle $\delta$ of the sun and there is an approximation to calculate $\delta$:
$$\delta \approx -23.45° \cdot \cos\left(\...
0
votes
2
answers
49
views
Solve smallest circle problem in polar coordinates - that is, smallest cone
I'm working on a problem relevant to astronomy. I'm looking at a few dozen stars at the same time, and I want to identify a centroid direction. Not the average, such that most stars are as close as ...
1
vote
0
answers
29
views
What function would model Mercury's orbital velocity around the Sun?
I am working on a mathematical investigation for my school work and in my investigation, I am trying to model Mercury's velocity around the sun. I picked up data for the velocity from the NASA ...
0
votes
0
answers
59
views
is it safe to use Trigonometric of large number in pocket scientific calculator
In this book (from 90s), there is a suggestion to reduce angle interval down to between 0 and 360 degrees, when the angle is larger than $360^\text{o}$ or when the angle is negative.
So here is my ...
0
votes
1
answer
25
views
Estimate the coverage shift of a satellite given its distance to the original satellite
I am doing some satellite-related research and would like to verify whether an assumption I am making about the satellite geometry is reasonable or not.
Essentially, a LEO (low-earth-orbit) satellite ...
0
votes
0
answers
32
views
Understanding Robin Green's derivation of the equation of the centre
I'm struggling to understand the derivation of the equation of the centre given in Robin Green's Spherical Astronomy (p147-148).
To order $e^{2}$, the equation of the centre is $$\nu-M=2e\sin M+\frac{...
0
votes
0
answers
42
views
Time and date from shadow
Given longitude, latitude and the height of a vertical stick at sea level, is it possible to determine date and time from the direction and length of its shadow? If yes, what are the formulas?
2
votes
1
answer
64
views
Deriving the equation of the centre
I'm failing to understand the derivation of the equation of the centre given on page 11 of Determining planetary positions in the sky for ±50 years to an accuracy of $\lesssim1$ degree with a ...
0
votes
0
answers
50
views
Calculating specific degrees when the sun is below the horizon given 3 other known values
Reference: Types of Twilight
Hi all, I have a question that I'm unsure how to approach and I was hoping someone could help me.
**
I'm trying to calculate the time when the Sun is 16.1 degrees below ...
1
vote
0
answers
132
views
How do I convert units from per radian seconds to arcseconds per year?
I am working on calculating axial precession of a planet orbiting a star, starting by attempting to follow along with Wikipedia's article on the topic. I have gotten through most of it without much ...
3
votes
1
answer
121
views
How do I account for the tilting of Earth's axis in right ascension and declination in a model solar system?
I am building a simplified model solar system in GeoGebra. Celestial objects are placed in a heliocentric coordinate system with the sun at the origin, the x-y-plane as the ecliptic, and the x-axis ...
2
votes
0
answers
88
views
Orbital resonances - expansion of disturbing function
I want to study the orbital resonance type $3:1$ between an asteroid and Jupiter. For this purpose, I found the expansion of the disturbing function in Celletti A., Stability and Chaos in Celestial ...
3
votes
1
answer
169
views
Could there be exact solutions to the Lane-Emden equation for real n≥0 other than 0, 1, or 5?
This Astronomy SE answer says
With a constant $k$ and the polytrop index $n$. This is a result of the solutions of the Lane-Emden equation
$$\frac{1}{\xi^2} \frac{\mathrm{d}}{\mathrm{d}\xi} \left(\xi^...
1
vote
0
answers
414
views
Is the covariant derivative of the inverse metric zero?
I know that covariant derivative of metric is zero [source, and my lecture notes confirm this]: $$\nabla _{\mu}g_{\alpha \beta} = 0,$$
How can I know if $$\nabla _{\mu}g^{\alpha \beta} = 0?$$
1
vote
0
answers
46
views
Hamilton's function according to the Delaunay variables
Consider the following system:
\begin{equation}
\ddot{x}_1=-\frac{\mu_{\oplus} x_1}{r^3}-\frac{\mu_\oplus R_{\oplus}^2J_2}{r^5}\bigg(\frac{3}{2}x_1-\frac{15}{2}\frac{x_1x_3^2}{r^2}\bigg)\\
\ddot{x}_2=-...
0
votes
1
answer
27
views
Prove that potential is non-decreasing for any spherically symmetric system
I am trying to prove that the potential of a spherically symmetric stellar system is non-decreasing (source of problem 2016 Paper 4 Question 7, Astrophysics Tripos).
What I did is the following.
We ...
1
vote
1
answer
46
views
How to get velocity dispersion from density profile of a stellar system
I know that one of the Jeans equations can be written in this way [source]:
$$\frac{\partial}{\partial t}\left(\nu \overline{v_{j}}\right)+\frac{\partial\left(\nu \overline{v_{i} v_{j}}\right)}{\...
0
votes
1
answer
107
views
Algebric troubles while trying to get energy of orbit in isochrone (or plummer) potential
I am facing this problem, from a 2016 Astrophysics Tripos past paper:
The gravitational potential
$$\Phi=-\frac{G M}{b+\sqrt{b^{2}+r^{2}}}$$
where $G$, $M$ and $b$ are constants and $r$ is the ...
1
vote
0
answers
18
views
Do all spherically symmetric stellar systems have zero anisotropy?
My notes are telling me that anisotropy of a stellar system is given by:
$$\beta \equiv 1-\frac{\sigma_{\theta}^{2}+\sigma_{\phi}^{2}}{2 \sigma_{r}^{2}}=1-\frac{\overline{v_{\theta}^{2}}+\overline{v_{\...
2
votes
1
answer
38
views
Poisson's equation-like formula in axis-symmetric Galactic potential in terms of Oort constants
I am tackling this problem, from a 2016 Cambridge Astrophysics Tripos past paper:
Let $\Phi(R, z)$ be the axi-symmetric Galactic potential. At the Solar location, $(R, z) = (R_0, 0)$, prove that
$$\...
1
vote
2
answers
219
views
Equation of Rotated Ellipse - Semi Major Axis is Changing
I am looking at astronomical observations of gas. The gas is orbiting a black hole with circular radius, $R$. However, from Earth it appears that the gas is an inclined ellipse. This is because ...
2
votes
2
answers
151
views
Is it special for a Lagrangian to equal its first integral?
Inspiration for this question: a this thread made me wonder if there is more to a story of a Lagrangian being equal to one of its first integral than "coincidence".
Background
I know that ...
27
votes
2
answers
637
views
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 ...
1
vote
0
answers
26
views
Parametrized solution for motion of radially infalling object
I have this equation:
$$\ddot{r}=-\frac{G M}{r^{2}}$$
I have to show that the solution of $r(t)$ can be parametrized by $\theta$ like this:
$$
r=A(1-\cos \theta), \quad t=B(\theta-\sin \theta), \quad ...
0
votes
2
answers
51
views
I need to transform an equation into another using taylor series
I have an astrophysics class and I am trying to finalize some exercices. My astrophysics teacher didn't teach us any of this saying that we should have learned it from other classes by other teachers (...
1
vote
0
answers
54
views
Deriving closed form of function
I am working on a project regarding gravitational decay. I have a function $$f(t)=-\dfrac{64G^{3}(M_{1}M_{2})(M_{1}+M_{2})}{5c^{5}\Big(r-(f(1)+f(2)+...f(t-2)+f(t-1))\Big)^{3}}=-\dfrac{64G^{3}(M_{1}M_{...
0
votes
0
answers
25
views
Equivalence of minor epicycle and eccentric
In epicycle-deferent astronomy, adding a second minor epicycle to account for observational discrepancies is mathematically equivalent to shifting the deferent into a so-called eccentric, or a circle ...
-1
votes
1
answer
72
views
What is the variable(s) or such written before the integration symbol?
From 'Distance measures (cosmology)' on Wikipedia:
Cosmologists commonly use the following measures for distances from the observer to an object at redshift $z$ along the line of sight:
Comoving ...
0
votes
1
answer
52
views
Probability question regarding mars, venus, and pluto
Background:
Venus cycle is $584$ days
Mars cycle is $780$ days
Pluto cycle, in this case, is 245.5 YEARS
Question:
What are the odds of all three converging at the same degree point on the same ...
2
votes
1
answer
61
views
Deriving the first moment of Collisionless Boltzmann Equation in Spherical Polar Coordinates
I am following these notes: Dynamics and Astrophysics of Galaxies.
After equation 6.37, we have:
\begin{equation*}
p_r\,\frac{\partial f}{\partial r} + \frac{p_\theta}{r^2}\,\frac{\partial f}{\...
1
vote
0
answers
78
views
Lane-Emden equation. Quasi-linearization method.
How to prove this theorem? I have a doubt.
Theroem: Suppose that $w(x,\alpha)$ solved
$\ddot{w}+\frac{2}{x}\dot{w}+\alpha^{2}w=0$ with $w(0)=1$,
$\dot{w}(0)=0$, $w(1)=0$.
Then $v(x,\alpha) := \omega w(...
0
votes
1
answer
38
views
Spherical Geometry Distance Between 2 Points
I try to calculate distance between 2 points on Earth. I have an a car and this car goes with speed 100 km/h and I know the start point latirude and longitude and car goes 5 hours. I want to calculate ...
1
vote
1
answer
101
views
Fixing an orbit in space using r and v (Keplerian orbits)
I'm wondering what would be a good geometric method to compute orbital elements that fix the orbit in space, given that one is given the position vector $\vec{r}$ and the velocity vector $\vec{v}$ for ...
0
votes
0
answers
31
views
How to solve an integral in two dummy variables if both variables are forced to be the same?
What happens to a general integral of this form in two dummy variables, $x$ and $y$
\begin{equation}
\langle \int\int f(x)g^*(y) dxdy \rangle
\end{equation}
if the expectation product $\langle f(x)g^*(...
1
vote
1
answer
202
views
Help With Euler Angles
I apologize in advance for the lack of meaningful formulas or calculations, I am not a mathematician and am using excel to try to compute everything. I think there is likely a simpler way to approach ...
0
votes
0
answers
217
views
direct conversion from az/el to ecliptic coordinates
Background:
I'm trying to build a lightweight antenna tracker with two servos. For mechanical reasons, I'm first mounting servo 1 on a base so that it tilts forward/backwards, then mount servo 2 on it ...
12
votes
2
answers
164
views
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 ...
0
votes
1
answer
386
views
How do I compute a vector given Right Ascension, Declenation?
So, I want to calculate the vector between two points on Earth. I know the Right Ascension/Declination of a telescope’s beam-pointing-center when posting at a distant star for both points on earth. I ...
0
votes
2
answers
225
views
If the orbital period of a smaller body is longer, why does the moon not fall behind the earth in their orbit around the sun.
If you calculate the orbital period of an earth sized object around the sun at 1 au, it is 31554651 in seconds (1 year). (Why does Google say a year is 31536000 seconds?!)
If you calculate the ...
1
vote
0
answers
87
views
Mathematical astrophysics subjects for math graduate students
I'm looking for a cool and interesting subject in mathematical astrophysics to study for my master's project. What I really aim for is certain processes in the cosmos (multiple body-problem, black ...
1
vote
2
answers
73
views
Is the vector product of a vector with its derivative equal to the linear product of their magnitudes?
I’m reading “Fundamentals of Astrodynamics” by Roger Bate and in the first chapter he states “in general
$ \vec{a }\cdot\dot{\vec{a}}= a\dot{a}. $
My intuition tells me that the dot product of the ...
19
votes
1
answer
612
views
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". ...
1
vote
1
answer
110
views
Circle about a point on the Celestial Sphere
Given a star at some celestial (spherical) coordinates (RA,Dec) or (Az,Alt) and an angle that represents the radius of the circle surrounding the star- what is the general form of the equation of this ...