Revision summary

Choose centre distance, distinguish a field quantity from a body or system quantity, and state the direction and energy reference before substituting.

Force and field strength

G is universal; M is the source mass and m the test mass. F and g in the first two expressions are magnitudes, with attraction towards the source. Equal and opposite interaction forces act on different bodies.

For the exterior of a spherical body, use its centre: r = R + h. Do not use altitude alone or extend the external point-source model into the body's interior. Doubling r quarters the field magnitude; doubling the small test mass changes its force but not the field strength.

Over height changes small compared with R and a small surface region, g and its direction change little, giving an approximately uniform field. When gravity alone acts, free-fall acceleration equals the local g; N/kg and m/s² are equivalent units. Support or drag changes the resultant acceleration, not the source's field.

Potential, interaction energy and work

Potential is external work per unit mass to bring a small test mass from infinity to the point without changing kinetic energy. It is measured in J/kg. U_G is the two-mass gravitational potential energy, in J, with zero at infinite separation. E_p = U_G for the gravitational energy account here.

Moving outward makes negative potential energy less negative: ΔU_G is positive and gravity does negative work. External work equals ΔU_G only when kinetic energy is unchanged and no other transfer matters. Include a final kinetic-energy change and any losses when needed.

Use the negative local gradient

For outward r positive, the isolated-source potential graph rises from a negative value towards zero. Its positive local slope gives negative g_r, an inward field. Use a tangent at the required radius, not the graph height or a long chord. Distinguish tangent guide points from actual curve values.

Reflecting the negative curves across zero gives |φ| proportional to 1/r and |g_r| proportional to 1/r² over the exterior domain. Neither reaches zero at a finite radius. With more than one source, field vectors can cancel at a point while their negative scalar potentials still add.

Set the escape boundary condition

Minimum escape has speed and potential energy tending to zero at indefinitely large separation. The model assumes a small body, an effectively fixed source, no relevant atmosphere, losses, other sources or propulsion after release, and a path avoiding collision.

The minimum speed is independent of test mass, but the needed kinetic energy is not. Above the threshold, residual kinetic energy remains at large distance; below it, the unpowered body cannot reach infinity in this model. Constant surface-g mgh is unsuitable for an infinite-height calculation.

Specify the complete orbital state

Gravity supplies the actual inward force of a circular orbit. Velocity is tangent; acceleration is inward. No extra centripetal interaction or outward balancing force is needed in the inertial-frame account. Larger circular radius gives lower speed and longer period for the same source.

Compute both E_k = ½mv² and U_G = -GMm/r for an orbital energy account. Raising a body to the final radius without giving it the required speed is a different final state. From rest on the supplied nonrotating model surface, the 400 kg example needs a mechanical-energy increase of 1.50 × 10¹⁰ J, not merely its 5.00 × 10⁹ J potential-energy increase.

A geostationary orbit is circular and equatorial, travels west to east like Earth, and has Earth's rotation period. Equal period alone is insufficient. Its fixed apparent direction supports communication antennas and continuing weather views, while its equatorial geometry limits where it is overhead and how well it views polar regions.

For radius-period data, use centre radii in metres and periods in seconds. Plot T² against r³; the ideal gradient is 4π²/(GM), in s²/m³. A freely fitted intercept can differ slightly from zero when supplied periods are rounded. Actual observational differences also require checking uncertainties and model conditions.

GM has units m³/s². Potential and potential energy differ by mass; field magnitude and its signed component differ by the direction convention, not by their units.