Questions tagged [mathematics]

Questions explicitly involving numerical analysis.

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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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5 votes
2 answers
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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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2 votes
1 answer
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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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2 votes
1 answer
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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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7 votes
5 answers
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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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2 votes
0 answers
69 views

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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2 votes
2 answers
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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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2 votes
0 answers
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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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1 vote
1 answer
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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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2 votes
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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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1 vote
1 answer
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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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3 votes
1 answer
494 views

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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0 answers
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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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2 answers
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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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1 answer
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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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1 answer
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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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5 votes
2 answers
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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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2 votes
1 answer
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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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1 vote
0 answers
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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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4 votes
1 answer
170 views

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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3 votes
2 answers
149 views

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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2 votes
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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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4 votes
2 answers
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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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5 votes
4 answers
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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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2 votes
1 answer
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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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9 votes
1 answer
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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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4 votes
2 answers
277 views

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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4 votes
2 answers
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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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3 votes
1 answer
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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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3 votes
1 answer
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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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14 votes
3 answers
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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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5 votes
1 answer
166 views

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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3 votes
1 answer
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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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3 votes
1 answer
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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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9 votes
1 answer
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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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18 votes
4 answers
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What is the "pendulum rocket fallacy" as it relates to analogizing a pencil balanced on a finger to maintaining attitude of a hovering rocket?

I've received comments that explain to me that the analogy between rocket attitude control during a hovering maneuver and the act of balancing pencil on the end of a finger is a helpful and good one; ...
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1 vote
1 answer
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What are the limitations to using this "trick" to quantitatively relate altitude changes (via their speed) with applied forces that cause them?

In this comment I used a "trick" to double check a calculation in the post above it. Using the vis-viva equation I first determined that if the ISS lost 10 meters of altitude in 86400 ...
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0 votes
1 answer
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Solution to 4 equations and 3 unknowns?

I have these 4 equations: and I am looking for the values of Px, Py and Pz, so these are equations to find the X, Y and Z coordinates (kind of see what a GPS is doing. I wanted to immitate a GPS and ...
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6 votes
1 answer
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Understanding the rocket equation - calculating Starship delta v

I am trying to calculate delta v of Starship - both as separate stages (Starship [SS], Super Heavy [SH]) and as a fully stacked rocket including 100t payload. I know that I have to calculate 2 ...
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1 vote
2 answers
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Are bicomplex numbers including tessarines ever used in spaceflight (as an alternative to quaternions)?

The answer to What are quaternions and how are they used to represent spacecraft dynamics? gives us a thorough overview of the topic and is worth a good read. I saw the HNQ Why are quaternions more ...
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2 votes
0 answers
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Understanding the paper "Determination of Satellite Orbital Elements by Means of On-Board GPS Receiver and Its Verification" by Yoshiwo Okamoto et. al

I have been assigned with implementing the algorithm proposed in the paper "Determination of Satellite Orbital Elements by Means of On-Board GPS Receiver and Its Verification" by Yoshiwo ...
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-2 votes
1 answer
107 views

Coord. system:collection of charts (local homeomorphisms to ℝ𝑛) for differentiable manifold 𝑀, frame:section of tangent vector fiber bundle over 𝑀

This answer to Are ECI and ECEF both frames and/or coordinate systems? Is there a difference? (which asks about the difference between coordinate systems and frames) says: In the language of pure ...
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1 vote
0 answers
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How to find/extract the semi-major axis from a TLE with Skyfield?

I have a TLE: 1 33591U 09005A 21024.46740879 .00000049 00000-0 52095-4 0 9997 2 33591 99.1927 41.5349 0014084 172.6670 187.4709 14.12449679616592 And I would like ...
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6 votes
1 answer
260 views

Displaying spacecraft attitude from rotation matrix or quaternion?

Navigation and guidance systems generally use rotation matrices or quaternions to uniquely specify 3D rotations. But these are not the least bit intuitive, so we like to convert them to Euler angle ...
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2 votes
1 answer
131 views

What is the purpose of eccentric, parabolic and hyperbolic anomaly?

I am currently reading David Vallado's 'Fundamental of Astrodynamics and Applications' and I have this doubt on the 2nd chapter named 'Kepler's Equation and Kepler's Problems'. Although I understand ...
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3 votes
1 answer
290 views

Does apsidal precession rate have any dependence on the argument of periapsis? Perhaps higher order?

Writing this comment got me thinking, which invariably leads to confusion in my case. @DavidHammen's answer to show that for Juno's near 90° polar orbit the major precession is apsidal; periapsis is ...
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