Questions tagged [mathematics]

Questions explicitly involving numerical analysis.

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1answer
51 views

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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0answers
93 views

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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84 views

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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1answer
134 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 ...
4
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1answer
111 views

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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2answers
137 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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2answers
192 views

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 ...
4
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2answers
338 views

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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4answers
215 views

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 ...
2
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1answer
76 views

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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1answer
843 views

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 ...
3
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2answers
222 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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2answers
354 views

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 ...
3
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1answer
239 views

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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0answers
61 views

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 ...
3
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1answer
97 views

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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3answers
1k views

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 ...
5
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1answer
116 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 $\...
3
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1answer
77 views

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 ...
3
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1answer
250 views

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 ...
9
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1answer
384 views

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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4answers
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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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1answer
105 views

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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1answer
116 views

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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1answer
673 views

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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2answers
181 views

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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0answers
58 views

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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1answer
93 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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0answers
135 views

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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1answer
225 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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1answer
119 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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1answer
226 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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1answer
103 views

Are solar sail spirals logarithmic? Can this be shown analytically or by dimensional analysis alone?

There are several questions and answers here about spiraling towards or away from the Sun using solar sails tilted by roughly 45 degrees to convert the sunlight's incoming radial momentum into ...
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1answer
179 views

Why did my vis-viva math solution get so close despite being wrong? Under what conditions would it have been a good approximation?

I tried to answer Equation for Velocity and distance from Sun of a solar sail powered spacecraft but I'm missing something. I set up a math solution and got about 0.4 year to reach zero velocity based ...
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5answers
895 views

Why is specific impulse equivalent to effective exhaust velocity?

I am very comfortable with specific impulse in seconds. I know how to calculate it, I know what it is, I know what happens to a rocket when you double in or cut it in half. But it feels like the ...
3
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1answer
327 views

How to solve the two-body problem in the ECI frame through numerical integration?

I need to know how to solve two-body problem by solving a system of first order equation derived from the equation below. $$\ddot{\mathbf{r}} = -\frac{\mu}{r^3}\mathbf{r}$$ How do I go about this, and ...
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1answer
195 views

What is the highest number of impulses required for an optimal orbital transfer?

Given two arbitrary orbits around a point mass, there exists some optimal transfer between them in terms of delta-v. What's the highest number of impulses such a transfer could require? (That is, I'm ...
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2answers
433 views

When and where on Planet Mars are the Sun's rays the most blueshifted?

Let's say for a moment that the spin of a planet causes the Sun's rays to be blue-shifted in the morning. As far as I know, redshift occurs in steps of like 2.6 km/s or something like that. We can't ...
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1answer
87 views

Can I "tack" towards a laser beam from which I am powering my 100% efficient & massless ion engine?

As far as I know, for light and particles taking special relativity into account: \begin{align} E^2 &= (T + m_0c^2)^2\\ &= p^2 c^2 + m_0^2 c^4 &\text{ (particles)}\\[1.5em] E &= p ...
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124 views

How should mass specific impulse calculated for relativistic exhaust?

This answer says: Optimising for Isp only is problematic, as it's simply: $$I_{sp} = \frac{v_e}{g}$$ Which is the same as optimising for exhaust velocity. With no constraints on thrust, particle ...
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0answers
53 views

Why does Gravity Simulator's rendering of TESS' orbit look like a pretzel? How can I unpretzefy it?

Below is a screen shot from the amazing Gravity Simulator driven by our own @HappyKoala. If you haven't seen it recently go check it out now, then come back in 30 minutes and read the rest of my ...
2
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1answer
93 views

Finding the mass flow rate of a blowdown system?

This is a rather amateur question I will admit but how does one find the mass flow rate of an engine with a given tank pressure? Is it really as simple as calculating the cross-section of all plumbing,...
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1answer
553 views

How much delta-v have I used here? What's the "official" equation for delta-v from parametric thrust?

I took a break from Stack Exchange, jumped in my spacecar and flew the following squiggle: $$a_x = \cos(10 \ t)$$ $$a_y = \sin(5 \ t)$$ $$a_z = \cos(2 \ t)$$ starting at ...
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1answer
141 views

Why is "delta-v + $\sqrt{{v_E}^2 + C_3}$ where ${v_E}^2 = $11.19 km/s" the correct way to calculate total propulsive delta-v? Please show all work

There is one answer to What spacecraft has had the greatest total propulsive delta-v? and I can not understand how its numbers have been calculated. Response to comments does not seem forthcoming so I'...
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1answer
272 views

How (the heck) was coherent synthetic aperture radar (SAR) implemented using photographic emulsion aboard Apollo 17?

This answer to Is there really microfilm on the Moon? includes a block quote which includes the following subquote: The equipment was installed in the service module and consisted of a coherent ...
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2answers
263 views

Does the Newton-Rhapson solution to Kepler's equation diverge?

I have the following simple solution to the Kepler's equation which uses newton-rhapson iterative technique. I have read various papers about the initial guess and divergence near an eccentricty of 1, ...
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1answer
173 views

Find the jump condition of the mass costate of a Launch Vehicle ascent problem solved via an indirect method (Pontryagin Minimum Principle)

I'm trying to solve a trajectory optimization problem for a class of problems like the old Altas-Centaur SLV3 Centaur launch vehicles. Those are a stage and a half design where the 2 LR-87 engines ...
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1answer
135 views

Keplers equation: written in change in eccentric anomaly

I am reviewing the "Fortran Astrodynamics Toolkit (can be found here) and even more specifically at the two body propagation method implemented in this Kepler module for elliptical and circular ...
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85 views

Is a 2:1 "figure-eight" Lissajous orbit possible in the Circular Restricted Three Body Problem?

Was Queqiao in a halo or Lissajous orbit? Why do sources disagree? says Proper halo orbits have the same period for their in-plane oscillations and out of plane oscillations, so they are closed ...
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1answer
178 views

Implementation of NASA Breakup model Area to Mass distribution

In the research "NASA's new breakup model of evolve 4.0" the implementation for the area to mass distribution of fragments produced by collisions and explosions is provided. The distribution ...