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Questions tagged [mathematics]

Questions explicitly involving numerical analysis.

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77 views

How were the geodetic and geocentric latitudes of the Space Shuttle defined and calculated?

The image below is cropped from https://i.stack.imgur.com/4XFaJ.jpg as described in this answer. It shows a geographic state of a space shuttle at a given instant in time. There is only one value for ...
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1answer
48 views

What do the axis labels mean in this plot (apparently from Sutton 8th edition)?

In the Analytical Graphics Inc. video Low-Thrust Space Trajectory Design and Optimization the speaker Pradipto Ghosh of AGI shows a slide at around 07:00 shown ...
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0answers
30 views

What Methods are Used for Satellite to Site Visibility Calculation?

In the paper, Rapid Satellite-to-Site Visibility Determination Based on Self-Adaptive Interpolation Technique it is claimed that "Satellite-to-site visibility periods are typically determined by the ...
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2answers
60 views

How to find instantaneous positions of a central body of specified mass that results in an osculating orbit?

If I understand correctly, osculating orbital elements such as those posted in JPL's Horizons represent the mathematical osculating (tangential or "kissing") Keplerian orbit about a specified location ...
4
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1answer
50 views

Mechanics of Sun-Synchronous Orbit & North Korea's KMS-4

QUICK QUESTION: Is North Korea's KMS-4 in a proper Sun-synchronous orbit? https://www.n2yo.com/?s=41332 2016-009A FOLLOW-UP QUESTION: How do I tell? I want to understand the mechanics. RESEARCH:...
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1answer
163 views

Launching east from a mountain on the equator at midnight during a new moon; ranking of each contribution?

This answer is intriguing and I'm curious if it's correct, and especially which effects are most important. If you want to avoid gravity assists, the most fuel-efficient way out of the Solar System ...
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1answer
94 views

Help with my tensor tension; how to derive and calculate this rigid body gravity gradient torque?

Tensors make me tense. Imagine a long thin rod in a circular orbit. The gravity gradient will produce a net torque on the rod whenever it is not oriented parallel, or perpendicular to the radius ...
3
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1answer
113 views

Help with Scott Manley's rigid body dynamics narration

I've just watched Scott Manley's video SpaceX's Water Landing Reveals Rocket "Secrets" (or, What We Learned from CRS-16) (again) and there's some rigid body dynamics in there where I'm not sure I am ...
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82 views

What is wrong with my orbit sim equations? How can I fix them?

Edit: Partial question: The object now moves in an ellipse like it's supposed to, but it orbits around a different point than I intend. How can I make it orbit around a particular object at a point? ...
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1answer
53 views

How can I convert HGI coordinates to J2000 coordinates?

I am looking at old mission data such at this one from Vega 1. The coordinate system provided is HGI. Most of the data I have is in ICRF/J2000.0, as JPL Horizons provides. How can I convert between ...
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2answers
317 views

Explanation of weird values in New Horizons data?

In this answer I computed (methods given later in post) the estimated time, distance, range, and resolution for the science capture for New Horizon's closest approach to Ultima Thule (2014 MU69). ...
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101 views

Ideal shape for a long, skinny reaction mass for LEO to cis-lunar and beyond? (a “space rail gun”)

@Ingolifs' answer got me procrastinating thinking further. What you're asking about is a mass driver space station in low earth orbit that a payload or ship launched from earth can dock with, and ...
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63 views

How to make satellite ground track on map?

I want to make satellite ground track using matlab. I already had a position of satellite(ECI) by solving Kepler equation. And I know convert ECI to ECEF using rotation determinant. Then I convert ...
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2answers
519 views

From the General Thrust Equation towards Tsiolkovsky, how to explain dropping these terms along the way?

The NASA Glen Research Center tutorial page Rocket Thrust Equation links to the General Thrust Equation page which starts with: Which transcribes into MathJax as: $ \ \ \ F = \dot{m}_\mathrm e ...
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209 views

Analytical approximations for the shapes of these plots of ISP vs nozzle diameter? (Raptor engine)

A comment on the currently unanswered question Is there an authoritative catalog for rocket launch identifiers? led me to this Github page which links to r/spacexlounge on which currently appears a ...
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0answers
34 views

What is this perturbation effect called, and what would be an analytical expression for the resulting eccentricity oscillations?

In this answer I've generated some oscillations of eccentricity by being somewhat naughty and propagating a TLE for +/-250 days from epoch. I believe it's okay for the purposes of demonstrating gross ...
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2answers
237 views

What exactly means universal variable x and z?

I have been studying by myself Orbital Mechanics, and when solving the Lambert Problem, it's common to use the universal variable approach. I understand the algorithm, but I haven't found any book ...
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69 views

Simplified WGS84 or EGM2008 model for rapid computations?

I am searching for a simplified model of EGM2008 that can be used for rapid, repetitive computations. It needs to be valid only up to GPS altitudes. I'm thinking such a model already exists but cannot ...
129
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2answers
18k views

How did the Apollo computers evaluate transcendental functions like sine, arctangent, log?

Navigation with a sextant or maneuvers using gimbal angles might be two examples of cases where an Apollo computer might need to do trigonometry. Trigonometric functions like sine, arctangent, etc. ...
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1answer
341 views

What is the terminal velocity on Mars?

How would I calculate the terminal velocity of Mars? What is the terminal velocity of a balloon entering Mars' atmosphere? Would a balloon pop if dropped from space?
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1answer
134 views

How can articulated spacecraft components increase the risk of gimbal lock?

This answer has got me baffled. In theory, yes. In practice, no. The problem is gimbal lock... There seems to be some risk of inducing gimbal lock if a spacecraft component of sufficient mass is ...
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112 views

How to get from the perturbative acceleration formula to the acceleration on the 3 axis in the case of the oblate Earth, considering j2?

If we start from the pertrubative acceleration formula, how do we get the expression for the acceleration for the 3 axis written under that form. I've used the partial derivative for the gradient ...
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147 views

What is the optimal angle for a solar-sail deorbit towards the Sun when radial thrust is included?

In this answer there's a geometrical derivation of the optimal angle of a solar sail to deorbit a spacecraft into the Sun. The naive answer is 45° which would direct the reflected light directly ...
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0answers
78 views

Lighting an area of the Earth Surface from LEO

What does the math look like to calculate the size of an object in LEO that when angeled off the sun, would illuminate a part of the Earth's surface? For example, if I wanted to 'lightup' a square ...
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2answers
146 views

What is the functional form for r(t) for a solar-sail deorbit into the Sun?

This answer describes a spacecraft in a heliocentric orbit using a solar sail to "deorbit" into the Sun by reflecting sunlight into roughly the prograde direction. In comments it was proposed that ...
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0answers
60 views

Aerobraking on Mars

I am looking for a calculator or formulas that can be used to calculate the aerobraking on Mars mission. I have found some data on previous Mars flights where aerobraking was used and all my current ...
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64 views

Can Lissajous orbits have stable/unstable manifolds?

The question Did DSCOVR travel “along the stable manifold of it's future SE L1 Halo orbit” to get there? is specific to DSCOVR's trajectory from Earth to its primarily heliocentric orbit near Sun-...
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1answer
138 views

Calculating the Earth Geopotential effect

I'm calculating the acceleration due to the harmonics in ECEF frame. Gravity potential in spherical harmonics is shown here (I just removed "$1+$", as considering only harmonics effect). $U_{har}=\...
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2answers
147 views

How to prepare for an interview for a Pure Mathematics position at ISRO's Space Application Center?

I have completed a Master's in Pure Mathematics. I have read topics like Abstract Algebra, Linear Algebra, Functional Analysis, Topology, Cryptography, Course in C++, etc. Recently I found an ...
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0answers
123 views

Software (besides Ltool) to simulate CR3BP (three body) problems?

Are there any other softwares for CR3BP besides Ltool? As I've learnt here Ltool is not generally available to the public. I would like to simulate three body orbits within the standard circular ...
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1answer
353 views

Using of the coordinate systems in orbit propagation

For the orbit propagation in ECI coordinate system, considering the effects of zonal harmonics and Sun/Moon perturbations the following procedure was formed: Convert the state vector to ECEF ...
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0answers
53 views

variables affect the trajectory in a stochastic way

What are the different variables affecting the trajectory of a rocket in a stochastic way for getting potential three-sigma trajectory?
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1answer
161 views

Low-thrust spiraling to escape, is the flight path angle (gamma) at C3=0 always 39 degrees?

@MarkAdler's comment led me to ask Why would a slow spiral from a C3 of zero take about 2.4 times as much ΔV as an impulsive maneuver? which resulted in this tidy and efficient @MarkAdler answer which ...
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2answers
2k views

How to calculate the flight path angle, γ, from a state vector?

@Julio's excellent answer describes a flight path angle, and explains that it is the angle between the tangential direction (perpendicular to the radial vector to the central body) and the current ...
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1answer
296 views

Spiraling out from circular orbit to escape via low thrust, what is γ (gamma)?

@MarkAdler's comment led me to ask Why would a slow spiral from a C3 of zero take about 2.4 times as much ΔV as an impulsive maneuver? which resulted in this tidy and efficient @MarkAdler answer which ...
2
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1answer
171 views

Why would a slow spiral from a C3 of zero take about 2.4 times as much ΔV as an impulsive maneuver?

I just read a fascinating comment that I don't understand. In part it says ...the Oberth effect does not depend on the mass of the object. A slow spiral in to a low orbit from a C3 of zero will ...
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1answer
537 views

How does the poliastro python package “Going to Mars with Python” example work? What's it really doing?

@astrojuanlu's answer mentions the existence of the Python package poliastro, and in the documentation there I found an example titled Going to Mars with Python using poliastro among others. Input ...
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1answer
201 views

What does this paper say is wrong (quantitatively and procedurally) with WISE & NeoWISE asteroid data?

I don't have daily access to the journal Icarus so when I read the New York Times article Asteroids and Adversaries: Challenging What NASA Knows About Space Rocks; Two years ago, NASA dismissed and ...
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1answer
234 views

I've almost learned to spell Chebyshev, why has JPL switched to Hermite interpolation for DE438?

According to @ChrisR's thorough answer One caveat however is that the interpolation in DE438 is a Hermite interpolation, and no longer a Chebychev interpolation, so you may need to update your code....
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1answer
257 views

How can I verify my reconstructed gravity field of Ceres from spherical harmonics?

Based on @DavidHammen's very helpful answer I've made progress reconstructing the gravity field of Ceres from the Dawn radiometric data. The question there contains further information, but suffice it ...
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183 views

How can I calculate the delta-v correctly, this way does not seem to be correct?

I'm learning about Astrodynamics, and I would like to ask how can I calculate the $\Delta V$ for an interplanetary mission. I used a website for the porkchop plots (http://sdg.aero.upm.es/index.php/...
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66 views

How many kinds of closed or periodic orbits are there in the circular restricted three-body problem?

Question: How many kinds of closed or periodic orbits are there in the circular restricted three-body problem? Has anyone made an attempt to enumerate and classify/categorize them? Notes: If there is ...
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0answers
86 views

Will these equations give me the approximate apparent RA and Dec using positions from Horizons?

In this answer @DavidHammen showed me how to get coordinates from JPL's Horizons using the Earth's equatorial plane as a reference plane by selecting ...
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2answers
220 views

How to intuitively explain that tri-elliptic transfers are not better than bi-elliptic transfers?

The question How to intuitively explain that reaching geostationary orbits consume more Delta V than escape orbits? includes the graphic below, which illustrates that when transferring between ...
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1answer
103 views

What are the symbol definitions in the equations for finding orbital elements?

I've been looking at the answer to this question on how to convert position and velocity to orbital elements, but some of the symbols are not defined and there are no reference links for those symbols....
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1answer
92 views

What does the phrase “topological point of view” mean when applied to two-body orbits in this context?

A recent question includes the passage: From a topological point of view, it seems likely that any rigid body would have to admit at least one stable low orbit, regardless of mass concentrations. I ...
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2answers
157 views

Do all rigid bodies of radius r have at least one stable orbit with perihelion p such that 2r > p > r?

In reading about the TESS mission, I was ultimately led to this NASA commentary from 2006, discussing how lumpy Earth's moon is. One of the points made is that there exist only 4 orbital inclinations ...
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3answers
424 views

How to calculate cone angle between two satellites given their look angles?

I have the azimuth, elevation and distance of 2 satellites relative to a ground station, defined by latitude and longitude, as I've discussed in my previous question. I'm using the satellites' TLEs ...
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1answer
222 views

Equation for orbital period around oblate bodies, based on J2?

In this answer I point out that the period of items (ring particles, moons, spacecraft, etc.) around an oblate body will not scale exactly as $a^{3/2}$ because the closer you are to the planet, the ...
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3answers
155 views

Interpolate the ephemerides of the Earth

I have a table of state vectors generated from ephemerides of the Earth with time steps of 1 minute. Now I would like to interpolate this to a finer time increments in order to use it as part of an ...