# Kepler orbits and gravitational scaling

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

Course: [Orbits, missions and exoplanets](https://studybrix.com/en/courses/orbits-missions-and-exoplanets/index.md)
Level: university
Prerequisites: Newtonian gravity, algebra and energy.

## 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.

## Practice questions

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

Answer: 8 years

Explanation: P=sqrt(4³)=8 years.

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

Answer: 5

Explanation: sqrt(100/4)=5.

### 3. What is the semi-major axis a?

A. The orbital inclination
B. Always the current distance
C. Always the nearest distance
D. Half the longest diameter of an ellipse

Answer: Half the longest diameter of an ellipse

Explanation: The semi-major axis is a geometric orbital parameter.

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

Answer: 3

Explanation: sqrt(2×18/4)=3.

### 5. Why can an orbiting astronaut appear weightless?

A. The Moon removes all gravity
B. Gravity is exactly absent
C. Mass becomes zero
D. Astronaut and spacecraft are in free fall together

Answer: Astronaut and spacecraft are in free fall together

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

Explanation: Credit units and mass dependence.

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

Answer: 0.5

Explanation: Speed scales as r^(−1/2).

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

Sample answer / marking guidance: Additional gravitating bodies and atmospheric drag can matter; radiation pressure, nonspherical gravity or relativistic effects can also matter depending on context.

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

## Further reading

- https://science.nasa.gov/solar-system/orbits-and-keplers-laws/

Original AI-assisted educational draft; not independently teacher-reviewed. References support further study, not copied exercises, affiliation or curriculum accreditation.