Reentry Range⚠ unverified

Understanding the Parameters

• Flight-path angle $\gamma$ — the dominant lever. A shallow entry (small $\gamma$) skims a long way across the planet before descending — long range; a steep entry drops almost straight down — short range. The linear $\gamma$ dependence captures this trend. (Note the rad/deg inconsistency above.)

• Entry velocity $V$ — faster entry covers more ground before the atmosphere brings the vehicle down, so range grows with $V$. Normalising by the circular orbital speed $V_c$ expresses "how orbital" the entry is: a near-orbital-speed entry ($V/V_c \approx 1$) ranges far; a slow, steep entry ranges little.

• Planetary radius $R$ — sets the geometric scale; range is a fraction of the planet's size, so the same entry geometry gives proportionally longer ground tracks on a larger body.

• The output $X$ — the downrange distance, the headline number for footprint and targeting. Real entries care about both downrange (along-track) and cross-range (across-track) reach; cross-range is what lift buys (Lift To Drag Reentry), enabling a vehicle to steer to a chosen landing site.

Derivation (Approaching a Proof)

A rigorous downrange requires integrating the horizontal velocity along the entry trajectory, $X = \int V\cos\gamma\,dt$, coupled to the Allen–Eggers deceleration (Reentry Velocity) — a problem with no simple closed form. The registry uses a scaling estimate instead. The ground arc subtended is $R$ times the angular sweep; taking the flight-path angle $\gamma$ as a proxy for the geometry and weighting by how close the entry is to orbital speed gives

$$X \approx R\,\gamma\,\frac{V}{V_c},$$

with $V_c = \sqrt{gR} \approx 7900\,\text{m/s}$ the circular orbital velocity — the natural velocity scale at which a body would not descend at all (it would orbit). $\blacksquare$

The physical reading: a vehicle entering near orbital speed ($V \to V_c$) is barely being pulled down and skims a long ground track; one entering slow and steep drops quickly and lands near its entry point. The estimate captures these limits but not the trajectory detail — for mission design the full equations of motion (with lift, drag, and gravity) are integrated numerically to produce the landing footprint.

Dimensional check. $R\,\gamma\,(V/V_c)$: $R$ in metres, $\gamma$ in radians (dimensionless), $V/V_c$ dimensionless, so $X$ is in metres. $\checkmark$

History and Development

• Footprint and targeting. From the first crewed capsules, predicting the landing point — the footprint — was as important as surviving entry. Early ballistic capsules (Mercury, Vostok) had essentially fixed footprints and needed large recovery fleets; the payoff of controllable range drove the move to lifting entry.

• Lift buys range. A vehicle with lift can extend downrange and, crucially, add cross-range — steering hundreds or thousands of kilometres to a chosen site (Lift To Drag Reentry). The Space Shuttle's cross-range requirement (for once-around abort to a landing site) drove its wing and thermal design.

• Precision entry. Modern guided entry (Apollo, Orion, Mars Science Laboratory's guided entry) actively modulates lift direction to hit landing ellipses kilometres — for Mars 2020, a few km — across, a far cry from the scaling estimate here but built on the same range physics.

Related Concepts: Reentry Velocity, Lift To Drag Reentry, Ballistic Coefficient, Skip Reentry Delta V, Reentry Time, Circular Orbit Velocity, Glide Ratio

Notes: Registry calculator reentry-range (unverified). Placeholder latex $f(V,\gamma,R)$; actual code is the heuristic $X = R\,\gamma_{\text{rad}}\,V/7900$ ($7900\,\text{m/s} = $ circular orbital speed), a scaling estimate, not a trajectory integral. Unit inconsistency: $\gamma$ labelled rad but treated as degrees. Cross-range (steering) comes from lift, not this formula. Flagged in Known Issues.