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Lambert's problem

Lambert's problem is a astronomy 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 Lambert's problem rather than just read about it. In short: In celestial mechanics, Lambert's problem is concerned with the determination of an orbit from two position vectors and the time of flight, posed in the 18th century by Johann Heinrich Lambert and formally solved with mathematical proof by Joseph-Louis Lagrange. It has important applications in the areas of rendezvous, targeting, guidance, and preliminary orbit determination.

Lambert's problem — main illustration
Lambert's problem — illustration

Key takeaways

  • Lambert's problem belongs to astronomy; place it in that map before memorising details.
  • Learn the definition first, then one example that makes the definition concrete.
  • Connect Lambert's problem to a quantity you can measure, compute or draw — that is where exam questions come from.
  • Reproduce the core statement of Lambert's problem from memory before moving on to harder problems.

Reference excerpt

In celestial mechanics, Lambert's problem is concerned with the determination of an orbit from two position vectors and the time of flight, posed in the 18th century by Johann Heinrich Lambert and formally solved with mathematical proof by Joseph-Louis Lagrange. It has important applications in the areas of rendezvous, targeting, guidance, and preliminary orbit determination. Suppose a body under the influence of a central gravitational force is observed to travel from point P1 on its conic trajectory, to a point P2 in a time T. The time of flight is related to other variables by Lambert's theorem, which states:

The transfer time of a body moving between two points on a conic trajectory is a function only of the sum of the distances of the two points from the origin of the force, the linear distance between the points, and the semimajor axis of the conic. Stated another way, Lambert's problem is the boundary value problem for the differential equation

r ¨ = − μ r ^ r 2 {\displaystyle {\ddot {\mathbf {r} }}=-\mu {\frac {\hat {\mathbf {r} }}{r^{2}}}}

of the two-body problem when the mass of one body is infinitesimal; this subset of the two-body problem is known as the Kepler orbit. The precise formulation of Lambert's problem is as follows: Two different times t 1 , t 2 {\displaystyle t_{1},\,t_{2}} and two position vectors r 1 = r 1 r ^ 1 , r 2 = r 2 r ^ 2 {\displaystyle \mathbf {r} _{1}=r_{1}{\hat {\mathbf {r} }}_{1},\,\mathbf {r} _{2}=r_{2}{\hat {\mathbf {r} }}_{2}} are given. Find the solution r ( t ) {\displaystyle \mathbf {r} (t)} satisfying the differential equation above for which

r ( t 1 ) = r 1 r ( t 2 ) = r 2 {\displaystyle {\begin{aligned}\mathbf {r} (t_{1})=\mathbf {r} _{1}\\\mathbf {r} (t_{2})=\mathbf {r} _{2}\end{aligned}}}

Initial geometrical analysis

The three points

F 1 {\displaystyle F_{1}} , the centre of attraction,

P 1 {\displaystyle P_{1}} , the point corresponding to vector r ¯ 1 {\displaystyle {\bar {r}}_{1}} ,

… excerpt ends here. Continue reading the full article.

Illustrations

Lambert's problem: Figure 2: Hyperbola with the points P 1 {\displaystyle P_{1}} and P 2 {\displaystyle P_{2}} as foci passing through F 1 {\displaystyle F_{1}}
Figure 2: Hyperbola with the points P 1 {\displaystyle P_{1}} and P 2 {\displaystyle P_{2}} as foci passing through F 1 {\displaystyle F_{1}}
Lambert's problem: Figure 3: Ellipse with the points F 1 {\displaystyle F_{1}} and F 2 {\displaystyle F_{2}} as foci passing through P 1 {\displaystyle P_{1}} and P 2 {\displaystyle P_{2}}
Figure 3: Ellipse with the points F 1 {\displaystyle F_{1}} and F 2 {\displaystyle F_{2}} as foci passing through P 1 {\displaystyle P_{1}} and P 2 {\displaystyle P_{2}}
Lambert's problem: Figure 4: The transfer time with * r1 = 10000 km * r2 = 16000 km * α = 120° as a function of y when y varies from −20000 km to 50000 km. The transfer time decreases from 20741 seconds with y = −20000 km to 2856 seconds with y = 50000 km. For any value between 2856 seconds and 20741 seconds the Lambert's problem can be solved using any y-value between −20000 km and 50000 km
Figure 4: The transfer time with * r1 = 10000 km * r2 = 16000 km * α = 120° as a function of y when y varies from −20000 km to 50000 km. The transfer time decreases from 20741 seconds with y = −20000 km to 2856 seconds with y = 50000 km. For any value between 2856 seconds and 20741 seconds the Lambert's problem can be solved using any y-value between −20000 km and 50000 km

Worked examples

Example 1 — a first encounter with Lambert's problem

Start with the simplest possible case. Write down what Lambert's problem claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In astronomy, 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 Lambert's problem 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 Lambert's problem 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 Lambert's problem

In research
Lambert's problem appears in astronomy 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 Lambert's problem 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
Lambert's problem is common in secondary-school and first-year university syllabi. It links to neighbouring topics Astrodynamics, Conic sections, Orbits, so understanding it makes those chapters shorter.
In everyday life
Look for Lambert's problem 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 Lambert's problem in 20 minutes

  1. Read the reference excerpt below once, without taking notes.
  2. Close the page and write down what Lambert's problem means in your own words.
  3. Compare your version with the excerpt and mark what you missed.
  4. Work through the three examples above with pen and paper.
  5. Explain Lambert's problem out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Lambert's problem in simple terms?

In celestial mechanics, Lambert's problem is concerned with the determination of an orbit from two position vectors and the time of flight, posed in the 18th century by Johann Heinrich Lambert and formally solved with mathematical proof by Joseph-Louis Lagrange. It has important applications in the…

Why does Lambert's problem matter?

Because it connects several astronomy 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 Lambert's problem?

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 Lambert's problem.

Tags

  • Astrodynamics
  • Conic sections
  • Orbits

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