Questions tagged [mathematics]
Questions explicitly involving numerical analysis.
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Objects move clockwise instead of counter-clockwise
I'm trying to write an orbital simulation with patched conics. For my sources I'm using https://orbital-mechanics.space/ and Fundamentals of Astrodynamics (Second Edition).
Both sources suggest to use ...
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Object slowest at periapsis - despite correct position calculation
I'm coding up a simulation of celestial bodies using patched conics. At the moment I'm stuck with calculating the velocity of the object in the perifocal frame.
For the position I'm simply using the ...
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Person falling from space
A person at rest 500 km above the Earth falls straight downwards. She has a snug magical force field around her that is totally rigid and completely protects her from outside heat. The force field ...
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Euler Angle Notation
Fundamental Question
Correct me if I am wrong here. For Euler Rotations, we use the right hand rule. So a positive rotation in X-Axis will represent a clockwise rotation; whereas a positive rotation ...
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Which notes and/or books on mathematics, physics, 3D geometry preparatory to the study of orbital dynamics?
I have a master's degree in aerospace engineering and I'm preparing for job interviews for orbital mechanics/space flight dynamics engineer positions and I need material about maths, physics, 3D ...
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What is the velocity of the ISS relative to the Earth's surface?
This was asked once before, but I believe there was no clear answer given and accepted.
The ISS has an orbital period of 92.9 minutes, or 5574 seconds. If it were in a completely equatorial orbit, ...
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Rewriting the equation of motion as a system of first order differential equation
I am required to rewrite the equation
$$\ddot{\mathbf{r}} = - \frac{\mu}{r^3} \mathbf{r}$$
as a first order differential equation. How do I go about doing this?
I understand how this equation was ...
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How can I determine a rotation matrix from unit vectors?
I want to determine the rotation matrix from which compute the Euler Angles to rotate my spacecraft in order to point in a specific direction. For example, i want the z body axis of the spacecraft to ...
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Optimal numerical method to solve BVP for optimal low thrust control
I need to solve a boundary value problem (BVP) for an optimal control in case of low thrust trajectories optimization. I want to include the J2 perturbations to the Edelbaum solution (multiple ...
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Calculating period/SMA delta and other parameters from Clohessy-Wiltshire
I'm working on the camera system for a game about relative motion. The orbital motion is completely described by Clohessy-Wiltshire.
I would like to be able to calculate some things like eccentricity ...
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How can I show the ideal path from Earth to Mars in a vacuum using calculus?
I would like to use calculus to try and model the optimal path to mars in space, assuming the rocket starts with an initial velocity in the vacuum of space, and only accelerates due to the Sun's ...
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Is it possible to determine the classical orbital elements for an Elliptic Repeating SSO?
As in the title, it is possible to have a repeatable Sun-synchronous orbit but with an elliptic eccentricity?
If yes, which equations I have to consider to determine its classical orbital elements?
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Relationship between osculating eccentricity and Brouwer-Lyddane short eccentricity
I was under the impression that the mean elements are given by taking the average over an orbital period of the satellite, i.e.:
$e_{mean}(t) = \int_{t}^{t+T(t)}e_{osc}(t')dt'[0.367in]$
where $T(t)$ ...
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GMAT: What causes the error "ArrayTemplate error : dimension error"?
I have made various scripts that work with GMAT, but once in a while I get an "ArrayTemplate error : dimension error" error. I do not know what causes it, as in this case my script appears ...
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How does one calculate the look angle for non-geo satellites (i.e. LEO, HEO, etc.)
I'm trying to find a way to calculate look angles to not just GEO satellites, but LEO, MEO, and HEO sats. The only calculator I've found myself is here, which was referenced for a similar problem on ...
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What is the maximum distance between two satellites orbiting at 400 km that would still allow them to communicate with each other using radio waves?
All satellites have the same inclination.
At an altitude of 400 km, the circumference of their orbit is $2\pi \cdot (6371+400) = 42\ 543$ km.
If, for example, you have only 2 satellites separated by ...
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How does one generate a formula(s) for an artificial satellite's location at time t given a set of known locations at discrete times?
Given a set of spatio-temporal points in space {(Lat_1, Lon_1, Alt_1, Time_1), ... (Lat_n, Lon_n, Alt_n, Time_n)}, I need a function (i.e. a formula) that takes these points (or subset of points) as ...
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Optimizing trajectories with GMAT: how to understand the "Vary" parameters, and how to know what values to use for them?
I've been using GMAT in my free time to optimize trajectories, and have varied burn component values and spacecraft states, usually with success. The vary command in GMAT, with the Yukon optimizer ...
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GMAT: How do I prevent infinite propagation into the future, which causes the "requested epoch 95008.513478505 is not on the DE file" error?
I have a spacecraft I am trying to get from Earth to Mars. I use an Earth propagator until the SC is out of the Earth's sphere of influence, then a deep space propagator until Mars' sphere of ...
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Formula for Determining Time to Perform Delta-V Burn Given Propulsion System Parameters
Question is as stated. Determining how much delta-V one needs to change orbits is pretty straight forward with the rocket equation. However, I’d like to see if there’s a way to calculate delta-V from ...
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How to calculate laser link spot size
In the reseach paper " A laser link from lunar surface..." , chapter 2.1.3.:
When employing, for example, 20 cm diameter apertures at the
transmitter telescopes and considering a carrier ...
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What is the equation to find how much fuel you need to reach a velocity? [closed]
Theoretically, I need to launch a rocket, and I need to find how much fuel I need to reach a velocity however I do not know the equation.
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What is the equation for fuel equals distance for a rocket? [closed]
Theoretically, I am trying to launch a rocket from space and trying to figure out how much fuel I would need to get to a certain distance.
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how I can find the radius vector and the velocity vector satellite in elliptic orbits? [closed]
In order to evaluate the following expression at each point in an elliptical orbit, I first need the state vector $\mathbf{r}, \mathbf{v}$.
$$
\omega_j=\frac{1}{r^2||\textbf{v} \times \textbf{r}||} \ \...
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Satellite Revisit Time Analysis
I am currently making up my mind about revisit analysis. I am searching for clever algorithms to implement this in Matlab. I want to define targets on Earth surface and get the revisit times for each ...
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Different/better ways to calculate the shadow region passage of the satellite and the effect of orbital elements, other than shadow cone technique?
The NASA Technical Publication Method for the calculation of spacecraft umbra and penumbra shadow terminator points; TP-3547 describes the following:
A method for calculating orbital shadow ...
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What are "transcritical, pitchfork, period-doubling, torus, and subharmonic bifurcations" in the context of three-body orbits? (e.g. butterflies)
MattB's excellent and well-sourced answer to Where do the butterflies land on this bifurcation plot? (Earth-Moon three-body butterfly orbits) begins:
I'm fairly certain that the butterfly family is ...
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Why is the eccentricity vector used to describe near-circular orbits?
When dealing with near-circular orbits I've seen the usage of the eccentricity vector defined as
$$(e_x,e_y) = (e\cos\omega, e\sin\omega)$$
I've seen this in Schaubs' "Analytical mechanics of ...
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Difficulties with higher order Runge-Kutta implementations
I've been trying to implement a higher order Runge-Kutta numerical integrator in MATLAB for a bit now with ambitions of n-body trajectory fun. Unfortunately, I have not had success in reaping the ...
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Questions about orbital ring math from Birch 1982 paper
I have been reading the original Paul Birch paper published in JBIS 1982. I have been using this copy found here.
In particular I am wondering about generalizing the ideas, like for a different planet ...
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How to make an algorithm that finds time to a true anomaly at epoch?
I am having trouble creating a function that finds the time for an inputed true anomaly when I know my current eccentric and mean anomaly (my equations are in vector form).
Note that I only intend to ...
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James Webb Space Telescope Velocity Loss Formula or how fast will JWST be going on L+26
I was never great at math but now I wish I would have taken it more seriously.
We have an office bet going and I need to come up with a formula to figure out the future velocity the JWST. I was able ...
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Calculating time for several transfer orbits
I was watching a video that displayed how the time to get to Mars varied with the production of varying delta-v and was looking for the mathematics behind such a calculation. Here is the link to the ...
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Computationally solving the two-body problem for patched conics approximation
I have been trying to create an orbital simulation using the patched conics approximation like Kerbal Space Program. I initially tried solving Kepler's equation using Newton's method, however this ...
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Can "prolateness" be characterized by a $J_2$ coefficient like "oblateness"?
When an imperfect sphere can be obtained by rotating an ellipse around an axis, it is called a spheroid. There are two types of spheroids, oblate ones and prolate ones.
Most solar bodies can be ...
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Any exact analytical solutions for non-Keplerian orbits; those around non-radially symmetric mass distributions (e.g. J₂≠0)?
Keplerian orbits, those around spherically symmetric mass distributions (Newton's Shell theorem collapses them to a point) have analytical solutions in that you can write $t(\theta)$ as a simple ...
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Understanding g₀ to calculate delta-v
I am trying to calculate the exact delta-v from various existing upper stages with a given payload in a vaccum and in orbit, say LEO. I understand that according to the Tsiolkovsky rocket equation:
$$\...
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What is the numerical procedure to find the next closest approach between two bodies on keplerian orbits?
Given two bodies on Keplerian orbits, what is the procedure to find the next closest approach of the two bodies? The bodies would be effectively massless objects like two spaceships orbiting around a ...
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How to calculate the height above the horizon of a geostationary satellite on a given latitude?
Wikipedia's Geostationary orbit; Communications says:
Geostationary communication satellites are useful because they are visible from a large area of the earth's surface, extending 81° away in both ...
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Is there a neat formula for hyperbolic Kentucky windage?
When doing some back of a napkin calculations for a recent question Where would the Pascal B manhole cover be now?, I realised a quick way of solving these class of orbital mechanics question is to ...
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Can mascons have Lagrange-like points? In principle? At the Moon?
Background
Lagrange points are a mathematical consequence of the The Circular Restricted Three Body Problem (CR3BP or CRTBP); two massive bodies orbiting around their center of mass and a third ...
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Modeling a rocket using Tsiolkovsky's equation and ordinary differential equations
So I have this uni assignment to make a model out of ODEs, and my idea was to use rockets. After some research I found about Tsiolkovsky's equation and tried using it on a Falcon 9 (disconsidering ...
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Estimating thrust and kg fuel for a generation ship's journey to Alpha Centauri
Premise: A generation spaceship leaves Earth around the year 2060 on a journey to colonize Alpha Centauri A (ACA). In this fiction, fusion power is achieved in 2040, improved over 20 years, and used ...
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Could this three-thruster spherical spacecraft de-tumble itself with zero final velocity?
@Joshua's answer to What is the minimum number of RCS thrusters capable of stabilizing a satellite against an arbitrary rotation? begins:
I know this question is old, but I randomly discovered a ...
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Do geodesics close on an ellipsoid?
I want to calculate the shortest distance between two points (Point A and Point B) on the ellipsoid surface. For this I need to use geodetic passing through these two points. Well, if I continue this ...
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Is there an intuitive reason for why the shape of the orbit at perigee is the mirror image of that at the apogee?
Is there an intuitive reason for why the shape of the orbit at perigee is the mirror image of that at the apogee? It seems strange to me, because the situation is so different at the perigee compared ...
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Difference between rotated frame and rotating frame
Suppose we have a spacecraft orbiting around Earth and we neglect all effects of Earth's $z$-axis motion, i.e., no wobble, precession, or nutation.
For simplicity, we only assume a rotation rate $\...
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Attitude determination with phototransistors and magnetometer
This was originally in Engineering Stack Exchange but someone suggested posting it here.
This is for a University Engineering project. The project is the design of small satellite which would be ...
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Pendulum rocket fallacy - so why *do* skyrockets have sticks?
This question is inspired by two, linked questions.
Skyrockets (of the pyrotechnic/firework kind) have guide sticks, which are (according to internet sources) to enhance stability: "The guide ...
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Star-shaped artifacts in SAR images of the "Suez Canal traffic jam seen from space"
Wikipedia's 2021 Suez Canal obstruction links to the Sentinel-1 synthetic aperture radar (SAR) image of the
Traffic jam in the Gulf of Suez caused by the obstruction as seen by the Sentinel-1 ...
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