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# Questions tagged [mathematics]

Questions explicitly involving numerical analysis.

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### How to find instantaneous mass of a central body that results in a given osculating orbit?

This question was underspecified, so I'm going to try again. Osculating orbital elements are Keplerian elements of a hypothetical orbit around a specified center which would be tangent to a specified ...
75 views

### Analytical treatment of relative positions between satellites?

I have been searching the internet for estimation of relative position between two satellites. However, all sources seem to be pointing to methods used for proximity operations, where the math ...
92 views

### Why does SpaceIL's Beresheet landing graphics show m\s instead of m/s?

I just noticed that the software displayed in the video of the landing had the velocity and acceleration units with a backslash (m\s) instead of the usual convention of a forward slash (m/s). Out of ...
78 views

### Given r/a, what are the limits on the direction that an orbiting body could be moving (e.g. solid angle vs r/a)?

An orbiting object at distance $r$ and semimajor axis $a$ will move at $\sqrt{2 - \displaystyle \frac{r}{a}}$ times the speed of a circular orbit at $r$, no matter what the eccentricity or which ...
91 views

### When is it okay or at least helpful to use energy arguments in orbital mechanics?

When faced with energy-based questions about spaceflight we often comment something like orbits are about momentum, not energy and use delta-v math in answers. But in this answer to How big a nuke ...
82 views

### How can Earth-Centered Inertial (ECI) coordinates be properly called “inertial” if the Earth's orbit is always accelerating towards the Sun?

I've just realized (again) that I don't understand anything, something that often happens to me after reading a @DavidHammen answer. How can Earth-Centered Inertial (ECI) coordinates be inertial if ...
165 views

### At a height of 100 km, what speed do you need to be going to escape Earth's gravity?

I understand that the escape velocity of Earth is 11 km/s. However, Earth's gravitational sphere of influence is not infinite, so it is possible to go slower than that and still escape the sphere of ...
248 views

### What information can I get if an elliptical orbit intersecting a circular orbit of radius 𝑟 and velocity of ellipse is equal to velocity of circular?

How can I approach solving the question below? Consider an elliptical orbit intersecting a circular orbit of radius $r$ as shown below. If $v_{ellipse} = v_{circular} = v$, find: True ...
130 views

### How much less delta-v would it take to reach the Sun using Venus and Earth flyby's compared to direct?

@TomSpilker showed that the minimum delta-v to go from Earth orbit into the Sun is about 20.89 km/s. That's going from 200 km LEO to "perihelion at the sun's photosphere (guaranteed complete entry!)" ...
649 views

### What's the eccentricity of an orbit (trajectory) falling straight down towards the center?

Discussion below this answer has got me thinking. A circular orbit has eccentricity of $\epsilon = 0$, ellipses $0 < \epsilon < 1$ (between but not including 0 and 1), and parabolic and ...
74 views

### Understanding line of sight between two satellite with oblate Earth using “scaling k-component”

I have been doing some reading on accuracy of determining line-of-sight between two satellites give a WGS84 oblate earth. The paper Rapid Determination of Satellite Visibility Periods (Alfano, Negron ...
71 views

### How does one impulsively null spacecraft nutation?

The Scott Manley video The Only Pilot to Shoot Down A Spacecraft - A Space Ace talks about the ASM-135 ASAT and its successful test launched from an F-15. He shows images from the page The F-15 ASAT ...
265 views

### What is the optimal inclination change strategy?

Consider a transfer between two circular orbits of similar radius, the only difference being the inclination difference, $\alpha$. What's the minimal $\Delta v$ required to perform this transfer? ...
82 views

### How do you calculate the ∆v required to increase apoapsis to a certain height?

To calculate the ∆v needed to increase our apoapsis from a low orbit of 100km to 400km, we find the velocity at 100km using the semi major axis (which is 250km, in this case), and then find the ...
113 views

### How do you calculate gravity loss (simplified)?

Assume a rocket with constant mass and thrust. To simplify things, we can ignore aerodynamic drag, and we'll also not perform a gravity turn, hence our flight plan is as follows: -Burn radially from ...
23 views

### How can I “debounce” these TDRS satellite inclinations? (reconstruct the zero-crossings)

The plot below is from the question Why do the geosynchronous TDRS satellites have this distribution of inclinations?, and the explanation says Inclination and mean motion (rev/day) for TDRS ...
793 views

### Why does the eccentricity vector equation always equal -1?

This is the eccentricity vector equation, $$e=\frac{1}{\mu}[(v^2-{\mu \over r})r-(r \cdot v)v]$$ $$e=|e|$$ Now this equation is written differently from many different sources but they essentially ...
44 views

### Mean to Osculating conversion for non-J2 averaged elements

What is the appropriate technique to convert mean elements (true anomaly averaged) into osculating elements when the model has J2, SRP, drag and additional forces modeled. Brouwer's transformation is ...
51 views

### How accurate can a point in space be determined by the likes of the ICRF?

How accurately could a 3D point in open space (i.e. not on an object), say at an elevation of 30,000m above sea level, be located or modelled using the likes of the International Celestial Reference ...
7k views

### Time at 1 g acceleration to travel 100 000 light years

How long would it take to go 100,000 light years at a constant 1 g acceleration?
64 views

### Can we calculate elliptical properties in a spherical coordinate system?

Can we calculate the elliptical properties like the semi-major axis and eccentricity using the state vectors (position and velocity) in the spherical coordinate system? http://www.braeunig.us/space/...
130 views

### What is the formula for Legendre Polynomials from EGM96 calculated in F447.f program?

The F447.f program is designed to calculate the geoid undulation for EGM96. I'm trying to understand how it accomplish this calculation. I stuck in ...
129 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 ...
52 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 ...
59 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 ...
82 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 ...
69 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:...
197 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 ...
141 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 ...
124 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 ...
89 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? ...
66 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 ...
341 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). ...
164 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 ...