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Astronomy & space · lesson 1 of 4

Kepler orbits and gravitational scaling

Keplerian orbits approximate isolated Newtonian two-body motion. Real trajectories can include perturbations, drag or relativistic corrections.

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Learn the ideas

Prerequisites: Newtonian gravity, algebra and energy.

Two-body model

Keplerian orbits approximate isolated Newtonian two-body motion. Real trajectories can include perturbations, drag or relativistic corrections.

Period law

P²=4π²a³/[G(M+m)]. For a negligible planet mass orbiting one solar mass, years and AU give the familiar approximate P²=a³.

Circular motion

Circular speed is sqrt(μ/r), with μ=GM for a dominant central mass. Orbiting bodies are in continuous free fall rather than outside gravity.

Escape

Escape speed from radius r in the isolated Newtonian model is sqrt(2μ/r). This is not a propulsion budget including drag or powered trajectory details.

Worked examples

Example 1

In the approximate solar-mass AU-year model, find P for a=4 AU.

P=sqrt(4³)=8 years.

8 years

Example 2

For μ=100 in compatible units and circular radius 4, find circular speed.

sqrt(100/4)=5.

5

Check your understanding

8 questions

Decimals or fractions are accepted for numeric answers. Proofs and written responses use sample answers and require human marking. This is practice, not a protected examination.

1 In the approximate solar-mass AU-year model, find P for a=4 AU.years

Show answer and explanation

8 years

P=sqrt(4³)=8 years.

2 For μ=100 in compatible units and circular radius 4, find circular speed.

Show answer and explanation

5

sqrt(100/4)=5.

3 What is the semi-major axis a?

Show answer and explanation

Half the longest diameter of an ellipse

The semi-major axis is a geometric orbital parameter.

4 For μ=18 and radius 4 in compatible units, find escape speed.

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3

sqrt(2×18/4)=3.

5 Why can an orbiting astronaut appear weightless?

Show answer and explanation

Astronaut and spacecraft are in free fall together

Apparent weightlessness is free fall, not generally zero gravitational field.

6 Why does the simplified P²=a³ relation need units and central mass specified?

Sample answer and marking guidance

Its numerical constants incorporate AU, years and a solar-mass approximation. Other units or central masses change the numerical relation.

Credit units and mass dependence.

7 At fixed μ, circular-orbit radius quadruples. Find new speed divided by old speed.

Show answer and explanation

0.5

Speed scales as r^(−1/2).

8 Name two limitations of an isolated two-body orbit model.

Sample answer and marking guidance

Additional gravitating bodies and atmospheric drag can matter; radiation pressure, nonspherical gravity or relativistic effects can also matter depending on context.

Credit two physical effects, not a claim that Keplerian modelling is useless.

Further reading and review references

These are reference links, not copied exercises, institutional endorsement or proof of independent review.