# Questions tagged [mathematics]

Questions explicitly involving numerical analysis.

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### 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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### 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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### 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 ...
110 views

### Kepler prediction problem, fundamentals of astrodynamics

I needed help with problem 4.3 of Fundamentals of Astrodynamics by Roger R. Bate, Donald D. Mueller, Jerry E. White (dover, archive.org) Given that $r_0 = (I+j)$ and $v_0 = 2k$ find $r$ and $v$ for ...
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### Calculate flight path angle given semi-major axis, eccentricity and distance from the focal point

How can I, given the semi-major axis (α), the eccentricity (ε), and the distance from the focal point (r), calculate the flight path angle (φ)? (As shown above)
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### Are there some three-body orbits that can't be escaped? Can we know without propagating forever?

update: Searching for "choreographies" I found this Physics SE question which is related but different because it asks only if periodic solutions can be proven to be periodic numerically and my ...
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### Who developed the mathematics used to correct the trajectory of Apollo 13?

The 1967 NASA technical note D-4246, Midcourse Guidance Procedure with Single Position Fix Obtained From Onboard Optical Measurements describes correcting the trajectory of a spacecraft traveling ...
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### Analytical expression for the ground track of the International Space Station

For a maths task in school, I am investigating the orbit of the International Space Station around the Earth. I understand that when the 3D movement in space is represented on a 2D surface, the ...
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### How do you calculate the percentage of light-speed a ship with a certain delta-v can achieve, accounting for relativistic effects?

I was doing some back-of-a-postage-stamp calculations on high-isp interstellar spacecraft earlier and realised something. If I design a spaceship which according to the rocket equation has a very high ...
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### Do Lagrange points still exist if there is significant radiation pressure on the third body from the first?

From this answer: To obtain the distance to L1, find the smallest value of $r$ such that $$\frac{M_2}{r_1^2} + \frac{M_1}{R^2} - \frac{r_1(M_1+M_2)}{R^3} - \frac{M_1}{(R-r_1)^2} = 0.$$ To obtain the ...
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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 ...
111 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 ...
96 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
295 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 ...
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### 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 ...
299 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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?
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### 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/...
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### 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 ...
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### 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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### 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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### 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 ...
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### 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 ...
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### 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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### 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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### 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 ...