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Astronomy & space · University

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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Prerequisites

Newtonian gravity, algebra and energy.

Study notes

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

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

P=sqrt(4³)=8 years.

8 years

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

sqrt(100/4)=5.

5

Check your understanding

  1. What is the semi-major axis a?

    1. The orbital inclination
    2. Always the current distance
    3. Always the nearest distance
    4. Half the longest diameter of an ellipse

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

  3. Why can an orbiting astronaut appear weightless?

    1. The Moon removes all gravity
    2. Gravity is exactly absent
    3. Mass becomes zero
    4. Astronaut and spacecraft are in free fall together

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

Answers and marking guidance

Answers and marking notes
  1. Half the longest diameter of an ellipse

    The semi-major axis is a geometric orbital parameter.

  2. 3

    sqrt(2×18/4)=3.

  3. Astronaut and spacecraft are in free fall together

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

  4. 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.

Further reading and review references

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