science

Pseudostate trajectory model

Pseudostate trajectory model is a science topic covered in the lgStudy science library. This page brings together a partial reference excerpt, illustrations, worked examples, real-world applications and a short study plan, so you can understand Pseudostate trajectory model rather than just read about it. In short: The pseudostate trajectory model is an approximation method to calculate spacecraft trajectories in the presence of more than one planetary-sized bodies. This method was developed as an astodynamical calculations model by J.S.

Key takeaways

Pseudostate trajectory model belongs to science; place it in that map before memorising details.
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Reproduce the core statement of Pseudostate trajectory model from memory before moving on to harder problems.

Reference excerpt

The pseudostate trajectory model is an approximation method to calculate spacecraft trajectories in the presence of more than one planetary-sized bodies. This method was developed as an astodynamical calculations model by J.S. Wilson in order to improve upon the method of patched conic approximation.

Method The pseudostate model was first proposed by J. S. Wilson in 1969 to study Earth-Moon spacecraft transfer trajectories. In the context of an Earth-Moon-spacecraft three-body system, the pseudostate method usually is more computationally efficient than the traditional patched-conic method of trajectory design. The patched-conic method essentially seeks to "patch" together two (Keplerian) two-body ellipses (the conics) at a point of intersection defined by the Moon's gravitational sphere of influence, while taking into account the various physical constraints. This method can result in large errors that may be controlled by a possibly unstable and time-consuming iterative computational process. The pseudostate model modifies the conic-patching method by defining a pseudostate transformation sphere (PTS), a region in which the spacecraft trajectory is calculated as an approximate solution to the restricted three-body problem. The method starts by calculating an initial simple two-body Earth-spacecraft ellipse and using it to propagate the spacecraft's position to a point within the Moon's PTS (the spacecraft's pseudostate), next the approximate restricted three-body solution is applied and the pseudostate is backward propagated to a point on the surface of the Laplace sphere, which defines the beginning of the Moon's gravitational sphere of influence, and finally a two-body Moon-spacecraft conic is calculated and the spacecraft location is forward propagated from the surface of the Laplace sphere to an arbitrary perilune point. The Laplace sphere and the gravitational sphere of influence concepts used in the two models, when applied to Earth-Moon system with an approximately circular orbit, is given by

R Laplace = D ( m M ) 2 / 5 {\displaystyle R_{\text{Laplace}}\;=\;D\;\left({\frac {m}{M}}\right)^{2/5}}

where R Laplace {\displaystyle R_{\text{Laplace}}} is the radius of Laplace sphere, D {\displaystyle D} is the average Earth-Moon distance, m {\displaystyle m} is the mass of the Moon and M {\displaystyle M} is the Earth's mass. Strictly speaking, the Laplace sphere is not a sphere but a changing hypersurface defined at each point of the path of a gravitational mass. The criterion for calculating the Moon's Laplace sphere is to analyze the Moon's gravity as the primary force acting in the region under consideration while the Earth's gravity is treated as a perturbing force. The Laplace sphere differs from the Hill sphere because the calculation of the latter sphere requires the presence of stable orbits while the former does not.

Application An example where this approximation method was used to calculate spacecraft trajectories was published in the September 2023 issue of the Journal of Astronautics by Ding et al.

See also Patched conic approximation Two-body problem N-body problem Sphere of influence

References

Worked examples

Example 1 — a first encounter with Pseudostate trajectory model

Start with the simplest possible case. Write down what Pseudostate trajectory model claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In science, the smallest case is usually a single object, a single equation or a single measurement. Check that every symbol or term in your sentence has a meaning in that case.

Example 2 — changing one variable

Take the situation from Example 1 and change exactly one quantity: double it, halve it, or set it to zero. Predict what should happen to Pseudostate trajectory model before you calculate. Comparing your prediction with the result is the fastest way to find out whether you understand the idea or only the words.

Example 3 — an exam-style question

Typical questions about Pseudostate trajectory model ask you to (a) state it precisely, (b) apply it to given data, and (c) explain a limitation. Practise writing all three answers in under five minutes; the third part is what separates a full-mark answer from an average one.

Applications of Pseudostate trajectory model

In research: Pseudostate trajectory model appears in science research whenever the underlying quantities have to be modelled precisely. Papers usually cite it as a starting assumption and then explore where it breaks down.
In technology and industry: Engineering practice reuses Pseudostate trajectory model in design rules, simulations and safety margins. Knowing the idea lets you read a specification sheet and understand why the numbers look the way they do.
In the classroom: Pseudostate trajectory model is common in secondary-school and first-year university syllabi. It links to neighbouring topics Astrodynamics, so understanding it makes those chapters shorter.
In everyday life: Look for Pseudostate trajectory model outside the textbook — in sport, cooking, traffic, electronics or the sky above you. An example you found yourself is remembered far longer than one you were given.

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How to study Pseudostate trajectory model in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Pseudostate trajectory model means in your own words.
Compare your version with the excerpt and mark what you missed.
Work through the three examples above with pen and paper.
Explain Pseudostate trajectory model out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Pseudostate trajectory model in simple terms?

Why does Pseudostate trajectory model matter?

Because it connects several science ideas at once: it gives you a definition you can apply, a quantity you can calculate, and a way to check whether a result is plausible.

How should I study Pseudostate trajectory model?

Read the excerpt, restate it from memory, then work through the examples and applications listed on this page. The five-step study plan above takes about twenty minutes.

What does this page cover?

It gives you a compact reference excerpt plus original lgStudy explanations, examples, applications and study material on Pseudostate trajectory model.