The Oberth effect means that the same Δv adds more energy when it is spent
deep in a gravity well, at high speed. Mission design often treats that burn as instantaneous
at periapsis. This project measures the correction. η is the share of the impulsive
Oberth bonus — the gain in excess speed beyond Δv itself — retained by a real burn, and
Π compares the burn duration with the time required to travel one periapsis distance
at periapsis speed.
The view above is a simulator, not a recording of a result. Moving the
slider integrates the full flyby again with the same Δv distributed across a longer or shorter
burn. The bright arc marks the burn and the blue line marks coast; dashed lines show the
unpowered arrival and textbook impulsive departure. η and loss come directly from the
simulator output.
Held fixed: arrival speed equal to the escape speed at periapsis,
Δv = 0.1 vp, Δv/c = 0.3, prograde thrust centred on periapsis, a point-mass
planet. An illustrative case, not a mission. Π is the burn's duration in units of
rp/vp.
What the sweep shows.
01 — Short burns
Loss scales quadratically with Π.
For a prograde burn centred on periapsis, the loss is at most about
Π²/96 of Δv, for Π up to about 1 and modest Δv/vp and mass ratio; larger
values scale it by a closed-form factor. A burn shorter than rp/vp
(about 16% of a circular orbit's period at rp) therefore loses roughly 1% of
its Δv or less. Across the 20,860 prograde sweep runs with Π ≤ 1, none exceeds 1.010
times the closed-form bound for its own Δv/vp and Δv/c.
It is a leading-order bound, not a strict inequality, and it is
probably close to a rule of thumb practitioners already use. It is reported as a clean
statement for flybys, not as a discovery.
02 — Long burns
Half the bonus is gone at Π of about 8–40.
The energy-based efficiency falls to one half at Π between 8.4
and 40 across all 1,143 engine and arrival families in the sweep, with a median of
26. The sweep is dimensionless, so this holds at every body: a burn longer than 10–40
periapsis timescales forfeits about half the Oberth bonus.
Beyond the quadratic regime, slow arrivals (v∞/vesc
≲ 0.5) follow a Π−1/3 law until a crossover set by the arrival speed, then
ln Π / Π; faster arrivals have no such middle regime. That is why the Sun,
Jupiter and Saturn, with low arrival speeds, keep far more at a given Π.
03 — One number, nearly
Π alone does not collapse the data.
At small Π, log(1−η) scatters by 0.2–0.3 dex against Π. Against
Π√C, with C the closed-form prefactor from the theory, it scatters by about 0.001
dex, some 200 times tighter. The scatter that remains after collapsing on Π alone is
mostly the arrival speed: v∞/vesc explains 73–92% of it for
prograde burns, and the mass ratio is irrelevant to the collapse.
04 — Real engines
Retiming the burn rarely changes much.
Optimising where the burn sits relative to periapsis recovers a median
of 7×10−6 percentage points of η for chemical engines, 8×10−5
for nuclear thermal and 0.0068 for electric, over 5,895 valid mission samples. The
largest single case is 4.1 points (nuclear thermal), and every gain above 0.1 points comes
from starting the burn later than centred. A chemical burn centred on periapsis is within about 0.1
points of the optimum of the prograde family.
One solar burn, three engine classes.
Hibberd, Eubanks and Hein's solar Oberth burn to catch 3I/ATLAS spends
8.36 km/s at 3.2 solar radii. Flown with their motors, the impulsive model holds. Flown
with other engines at the same geometry, it does not.
Same 8.36 km/s prograde burn, centred in time on perihelion at 3.2 R☉, bound arrival from 5.2 au · loss is the equivalent-Δv loss
| Engine |
Burn time |
Π |
Δv lost |
Share of Δv |
| Solid stack, CASTOR 30B + STAR 48B | 210.8 s | 0.033 | 0.099 m/s | 0.0012% |
| Nuclear thermal, Isp 900 s, a₀ 0.5 m/s² | 10,808 s (3.0 h) | 1.67 | 233 m/s | 2.8% |
| SEP-class, Isp 6000 s, a₀ 3.3×10⁻³ m/s² | 2.38×10⁶ s (27.5 days) | 368 | 6,344 m/s | 75.9% |
The solid stack loses 0.099 m/s when the burn is centred in time and
0.090 m/s at the best timing, and stays under 0.2 m/s across every sensitivity tried
(staging coast, arrival conic, thrust profile). The SEP row is hypothetical: that thruster
could not operate at 3.2 R☉, and it is there to show the propulsion physics at fixed
geometry. It keeps 24.1% of its Δv, against 23.0% from the large-Π law.
The loss reaches 1% of Δv at Π of about 1. For the solid stack that means
cutting the thrust by about 30; a single nuclear-thermal stage crosses it at an initial
acceleration of roughly 0.83–0.88 m/s².
Built so it can be checked.
Every number and figure comes from code in the repository, and figures
carry the script and the git commit that made them in their metadata. The in-browser
simulator above is the repository's own JavaScript port, checked against the Python one on
ten cases to within 10−8 in η.
Verification
- Tests passing
- 509
- Fresh-clone rebuild
- 56 / 56 outputs identical
- Sweep
- 135,720 runs · 0 errors
- Sweep η error, 99th percentile
- 3.3×10⁻⁸
- Rebuild, one command
- 43–79 min
Every run carries its own numerical error estimate, from an
energy-balance residual integrated alongside the orbit.
The rebuild audit found problems before it passed. Committed outputs had
gone stale after later code changes (Phase 1 files, one numbers file and the Phase 3
optimisation campaign), and an animation's first frame kept a marker from the still drawn before it. Each was regenerated or fixed, with a regression test for the bug, and none of the
numbers quoted in the paper outline changed.
What is not yet established.
Tracked as carefully as the code, because the distinction is the point.
- The novelty is provisional. The literature check was a web search, not a database
search, and the 1966 paper by Robbins that the small-Π result is compared with could not
be accessed, so claims that go beyond his scaling are provisional. The repository's
claim register lists nine claims as provisional or assumption-dependent, each with what
would settle it.
- Point-mass gravity. Planet oblateness is neglected. A check on ten Jupiter and
Saturn cases at 1.1 radii changes η by at most 0.0026, but absolute outgoing speeds shift
by 21–28 m/s, so absolute results stay point-mass.
- The planet-centred frame has limits. It stops being valid for a burn that starts
outside the planet's sphere of influence. Those mission samples are masked, not given a
number.
- Representative presets, limited steering. The mission atlas uses representative
presets, not hardware data, and only smooth pitch laws and one piecewise law were tested,
so the optima hold within those families.
- One earlier result is only partly reproduced. The model reproduces a large overestimate
reported in a thesis by Confraria (2020) at a thrust-to-weight of 0.1, but not its exact
size at higher thrust. The raw data behind that figure were not pursued.
- The solar case is a simplified model. It is planar and point-mass, with constant
thrust per motor, bracketed by two-level profiles. The unknown coast between stages is
bracketed at 10 s and 60 s, and thermal limits are not modelled.
Not yet a paper. The repository holds a working outline, the figures and
the numbers behind it, and has been in feature freeze since 6 October. The only planned
follow-up is the check against Robbins (1966).
Repository · reproducible from a clean clone, with a self-contained explorer
github.com/vyom-aggarwal/oberth-efficiency-atlas