Questions tagged [mathematical-astronomy]
For questions related to the mathematical operations and analysis of astronomical and astrophysical observations, processes and dynamics.
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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 ...
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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 $\delta$:
$$\delta \approx -23.45° \cdot \cos\left(\...
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Solve smallest circle problem in polar coordinates - that is, smallest cone
I'm working on a problem relevant to astronomy. I'm looking at a few dozen stars at the same time, and I want to identify a centroid direction. Not the average, such that most stars are as close as ...
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What function would model Mercury's orbital velocity around the Sun?
I am working on a mathematical investigation for my school work and in my investigation, I am trying to model Mercury's velocity around the sun. I picked up data for the velocity from the NASA ...
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is it safe to use Trigonometric of large number in pocket scientific calculator
In this book (from 90s), there is a suggestion to reduce angle interval down to between 0 and 360 degrees, when the angle is larger than $360^\text{o}$ or when the angle is negative.
So here is my ...
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Estimate the coverage shift of a satellite given its distance to the original satellite
I am doing some satellite-related research and would like to verify whether an assumption I am making about the satellite geometry is reasonable or not.
Essentially, a LEO (low-earth-orbit) satellite ...
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39
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Orbit determination and State Transition Matrix for asteroid
I'm trying to determine the orbit of an asteroid based on the least squares solution.
$$ \delta x=(H^t H)^{-1}\times(H^t \delta z) $$
$$H= \overline{H}{\phi}$$
The observations are made in five years, ...
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27
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Understanding Robin Green's derivation of the equation of the centre
I'm struggling to understand the derivation of the equation of the centre given in Robin Green's Spherical Astronomy (p147-148).
To order $e^{2}$, the equation of the centre is $$\nu-M=2e\sin M+\frac{...
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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?
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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 ...
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Graph theory interpretations of constellations
Bit of an informal question - but are there any notable connections between constellations and their respective graphs? i.e. do certain constellations have a hamiltonian cycle, are certain ...
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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 ...
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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 ...
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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 ...
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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 ...
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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
$$\frac{1}{\xi^2} \frac{\mathrm{d}}{\mathrm{d}\xi} \left(\xi^...
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Is the covariant derivative of the inverse metric zero?
I know that covariant derivative of metric is zero [source, and my lecture notes confirm this]: $$\nabla _{\mu}g_{\alpha \beta} = 0,$$
How can I know if $$\nabla _{\mu}g^{\alpha \beta} = 0?$$
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Hamilton's function according to the Delaunay variables
Consider the following system:
\begin{equation}
\ddot{x}_1=-\frac{\mu_{\oplus} x_1}{r^3}-\frac{\mu_\oplus R_{\oplus}^2J_2}{r^5}\bigg(\frac{3}{2}x_1-\frac{15}{2}\frac{x_1x_3^2}{r^2}\bigg)\\
\ddot{x}_2=-...
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Prove that potential is non-decreasing for any spherically symmetric system
I am trying to prove that the potential of a spherically symmetric stellar system is non-decreasing (source of problem 2016 Paper 4 Question 7, Astrophysics Tripos).
What I did is the following.
We ...
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How to get velocity dispersion from density profile of a stellar system
I know that one of the Jeans equations can be written in this way [source]:
$$\frac{\partial}{\partial t}\left(\nu \overline{v_{j}}\right)+\frac{\partial\left(\nu \overline{v_{i} v_{j}}\right)}{\...
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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 ...
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Do all spherically symmetric stellar systems have zero anisotropy?
My notes are telling me that anisotropy of a stellar system is given by:
$$\beta \equiv 1-\frac{\sigma_{\theta}^{2}+\sigma_{\phi}^{2}}{2 \sigma_{r}^{2}}=1-\frac{\overline{v_{\theta}^{2}}+\overline{v_{\...
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Poisson's equation-like formula in axis-symmetric Galactic potential in terms of Oort constants
I am tackling this problem, from a 2016 Cambridge Astrophysics Tripos past paper:
Let $\Phi(R, z)$ be the axi-symmetric Galactic potential. At the Solar location, $(R, z) = (R_0, 0)$, prove that
$$\...
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142
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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 ...
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2
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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 ...
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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 ...
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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:
$$
r=A(1-\cos \theta), \quad t=B(\theta-\sin \theta), \quad ...
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I need to transform an equation into another using taylor series
I have an astrophysics class and I am trying to finalize some exercices. My astrophysics teacher didn't teach us any of this saying that we should have learned it from other classes by other teachers (...
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Deriving closed form of function
I am working on a project regarding gravitational decay. I have a function $$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_{...
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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 ...
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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 ...
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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 ...
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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}{\...
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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(...
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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 ...
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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 ...
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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^*(...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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". ...
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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 ...
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Finding the radiation pressure at a distance from an isotropic black body star.
Question
A star, radius $R$, emits isotropically with a black body spectrum with temperature T.
I want to show that the radiation pressure at a distance $d>R$ from the centre of the star is $P_{...
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Showing that an equation for radiation pressure from a star obeys the inverse square law and why this law breaks down near the surface.
If the radiation pressure a distance $d>R$ from the center of an isotropic black body star is found to be $P_{rad}=\large{\frac{4\sigma T^4}{3c}}\left[1-\left(1-\frac{R^2}{d^2}\right)^{\frac{3}{2}}\...
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Constant circular motion: understanding $\underline{e_{\theta}}=\frac{d(\underline{e_r})}{d\theta}$
Context: 1st year BSc Mathematics, Vectors and Mechanics module, constant circular motion.
This may be trivial, but can someone tell me what's wrong with the following reasoning?
$$\underline{e_r}=\...
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Calculate times of full moon and new moon (Parallel vectors?)
I am going to find an expression for the times when the earth is on the straight line between the moon and the sun, and for the times the moon is on the straight line between the earth and the sun.
...
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