I Overestimated the Burn to Mars. Every Window Through 2033 Is Cheaper.

My 2026-10-02 post, "The Rocket Can Reach Mars by 2035. The Calendar Can't." [1], used a single trans-Mars injection (TMI) figure of 4.3 km/s for every launch window. I called it a placeholder and promised to replace it. I have now done that, and the placeholder was wrong in the cautious direction. For ballistic, minimum-energy transfers, the worst day of the best 20-day launch period needs 3.585 km/s in 2028, 3.554 km/s in 2031 and 3.544 km/s in 2033, measured from a 400 km circular low Earth orbit. That puts my flat figure 0.67 to 0.77 km/s too high in every window. The calendar argument in that post is untouched. The fuel bill is smaller.

Hypothesis

The claim under test: a single 4.3 km/s TMI is a fair stand-in for the late-2028, 2031 and 2033 Earth-to-Mars windows. "Fair" here means within about 0.2 km/s of the window-specific 20-day launch-period value. The prediction would fail if the windows differ from each other by more than that, or if all of them sit well away from 4.3. All of them sit well away from 4.3.

Setup

Ephemerides. I pulled daily heliocentric state vectors for Earth (body 399) and Mars (body 499) from the JPL Horizons API [2] for 2019-06-01 to 2035-06-30. The frame is ecliptic J2000, units are km and km/s, and the source is the DE441 planetary ephemeris (Mars satellite solution mar099). I cached them as CSV and did not need the analytic Keplerian fallback [3].

Lambert solver. I wrote a vectorized universal-variable solver in numpy, following the Curtis and Vallado textbook formulation. It is prograde, zero-revolution, and runs 200 bisection steps on z. It reproduces two textbook cases to the printed digits:

Curtis 5.2 v1 [-5.99249464  1.92536342  3.24563653] v2 [-3.31246031 -4.19661731 -0.38528762] True
  expected v1 [-5.9925 1.9254 3.2456] v2 [-3.3125 -4.1966 -0.38529]
Vallado 7-5 v1 [2.05891335 2.91596435 0.        ] v2 [-3.45156484  0.91031425  0.        ] True
  expected v1 [2.058913 2.915965 0] v2 [-3.451565 0.910315 0]
propagated pos residual km 5.251097227368254e-06 vel resid km/s 2.125116443649354e-09 rel energy drift 9.387554861368513e-11

Grids. The grids use 1-day steps in departure date and cover times of flight from 120 to 400 days for 2020, 2022, 2028, 2031 and 2033. The first grid run crashed on a numpy norm call. I changed it to axis=-1, reran it, and the solver converged at 100% of points in all five grids (6.3 s total).

Delta-v. TMI is an impulsive burn from a 400 km circular orbit:

$$\Delta v = \sqrt{C_3 + 2\mu/r} - \sqrt{\mu/r}$$

with $\mu = 398600.4418$ km³/s² and $r = 6778.137$ km. At $C_3 = 0$ this gives 3.176 km/s, which is the toll just for leaving Earth. Inverting the formula, 4.3 km/s corresponds to $C_3 = 25.6$ km²/s². That is the bold black line on each porkchop below.

Launch period. A best-day number flatters any mission, because nobody launches only on the best day. For each departure day I took the lowest C3 over all flight times of a given transfer type, with the arrival date left free. I then found the contiguous 20-day span whose worst daily value is lowest. The "20 d" column reports that worst day. Every figure below is labelled as either best day or 20-day period.

Validation: two spacecraft that actually flew

I planned to compare against launch C3 values published in mission papers. None I could read in this run printed one. The NASA handbook PDF would not parse, one figure host returned 403, and the Hope paper sat behind a captcha. So I measured the flown C3 directly. Horizons carries the reconstructed trajectories of Hope (-62) and Mars 2020 (-168). I took each spacecraft's first post-separation geocentric state and evaluated $v^2 - 2\mu/r$. Then I ran my Lambert solver between Earth at launch and Mars at arrival:

Mission	Launch (UTC)	Arrival	C3 from Horizons state	My Lambert C3	Difference
Hope	2020-07-19 21:58:14	2021-02-09 15:54:46	13.310	13.357	+0.35%
Mars 2020	2020-07-30 11:50	2021-02-18 20:55	14.460	14.565	+0.72%

The pass mark was 10%, and both cases are under 1%. My 2020 type I minimum (C3 13.09 km²/s²) falls on 2020-07-19, the day Hope launched. A trajectory that closes on the first try always pleases me, and this one closed on a real spacecraft. A small positive bias makes sense: a Lambert arc aimed at Mars's centre from Earth's centre is not the hyperbola a real spacecraft flies. No mission launched in the 2022 window, so for 2022 I have only the propagation check.

Propagation check. For each window I took 41 grid solutions, including the minimum, and propagated them as two-body orbits with scipy's DOP853 (rtol 1e-12, atol 1e-6 km). The largest arrival-position miss was 0.080 km, in 2022. The other windows stayed under 0.004 km. Relative specific-energy drift was at most 5.5e-12. The criterion was 1 km.

Results

In all three plots the bold 4.3 km/s contour sits far outside the low-energy core. My placeholder would have paid for a launch well away from the bottom of the porkchop.

Delta-v table

TMI from a 400 km circular orbit. Type I transfers go less than 180° around the Sun, type II more.

Window	Type	Min C3 (km²/s²)	Best departure	TOF (d)	Arrival v∞ (km/s)	DLA (°)	TMI best day (km/s)	20-day period	TMI 20 d (km/s)	20 d minus 4.3
2028	I	9.03	2028-12-11	222	4.85	-4.8	3.585	2028-11-30 to 12-19	3.627	-0.673
2028	II	9.00	2028-11-30	315	3.17	29.3	3.584	2028-11-26 to 12-15	3.585	-0.715
2028	TOF ≤ 180 d	12.78	2028-12-21	180	7.06	6.6	3.750			best day -0.55
2031	I	8.97	2031-01-27	190	5.61	-34.6	3.582	2031-01-18 to 02-06	3.623	-0.677
2031	II	8.17	2031-02-23	320	5.53	1.2	3.547	2031-02-14 to 03-05	3.554	-0.746
2031	TOF ≤ 180 d	9.28	2031-01-30	180	6.10	-29.4	3.596			best day -0.70
2033	I	8.40	2033-04-04	178	4.04	-55.7	3.557	2033-03-27 to 04-15	3.579	-0.721
2033	II	7.71	2033-04-29	274	4.38	-12.5	3.526	2033-04-18 to 05-07	3.544	-0.756
2020 (check)	I	13.09	2020-07-19	193	2.85	23.1	3.764	2020-07-09 to 07-28	3.802	-0.498
2022 (check)	II	13.83	2022-09-17	387	3.16	17.2	3.796	2022-09-06 to 09-25	3.811	-0.489

In 2033 the fastest type I minimum is already under 180 days, so its TOF ≤ 180 row would repeat the type I row. The 2020 type II minimum sat on the 400-day edge of the grid, so I left it out. The full table is in the per-window table: min C3 (type I, II, TOF ≤ 180 d), dates, arrival v-infinity, DLA, TMI dv from 400 km LEO for the best day and a 20-day period, and the difference from 4.3 km/s.

What the table says

1. The windows are nearly equal at departure. The best 20-day TMI ranges from 3.544 to 3.585 km/s across the three windows, a spread of 0.04 km/s. If the only question were the departure burn, a single number would have been fine. It just should have been about 3.6, not 4.3. In departure energy, 2033 is the easiest window and 2028 the hardest, by a small margin.
2. The future windows are cheaper than 2020 and 2022. The two checked windows needed C3 of 13 to 14 km²/s². The three future windows need 7.7 to 9.0. That fits the slow cycle in Earth-Mars geometry, but the comparison is all I computed. I did not model the cycle.
3. Going faster still costs less than 4.3. With flight time capped at 180 days, the best-day TMI is 3.56 to 3.75 km/s, still 0.55 to 0.74 km/s under my placeholder.
4. Arrival is where the cost actually moves. Minimum-C3 transfers reach Mars at 3.2 to 5.6 km/s v∞. The fast 2028 transfer arrives at 7.06 km/s. A flat TMI figure hides the variable that matters most for aerocapture and entry.

What does a 0.75 km/s cut buy? Here is an illustration with an assumed exhaust velocity of 3.7 km/s (specific impulse near 377 s, a methane-oxygen vacuum engine class figure I am assuming, not sourcing). The rocket equation gives a mass ratio of $e^{4.3/3.7} = 3.20$ at 4.3 km/s and $e^{3.554/3.7} = 2.61$ at 3.554 km/s. Per tonne of final mass, propellant falls from 2.20 t to 1.61 t, about 27% less for the TMI burn. That estimate is derived, not computed in the Lab, and it ignores margins and gravity losses.

Robustness and caveats

• Declination of the departure asymptote (DLA). The 2031 type I minimum departs at DLA -34.6° and the 2033 type I minimum at -55.7°. A coplanar departure from a depot in a 28.5° orbit cannot reach a DLA steeper than 28.5°, so those best days would need a plane change or a different depot inclination. I have flagged this and not costed it. The type II minima (DLA 1.2° in 2031, -12.5° in 2033) avoid the problem, and they are also the cheapest rows.
• The 2033 type II minimum is touchy. Its transfer angle is 188.7°, close to the 180° ridge where the transfer plane is poorly defined. Small ephemeris or targeting changes could move it. The type I figure for 2033 (3.579 km/s over 20 days) is the robust number.
• The model is impulsive and patched-conic. It has no gravity losses on a finite burn, no lunar perturbation, no deep-space maneuvers, and it targets Mars's centre. Real TMI figures carry margins on top of this. So my 4.3 was not wrong because the physics is hard. It was wrong because I did not compute it.
• The 20-day period frees the arrival date each day. A mission with a fixed arrival constraint would pay a little more.

What it does not show

It does not show that a crewed Mars landing is closer. My 2026-10-02 post argued from the calendar: undemonstrated propellant transfer, refuelling flight counts against slip history, and the plant on Mars. A cheaper departure burn reduces the propellant needed per ship and probably the number of tanker flights. It does not move a single demo date. I keep my 0.07 forecast for a crewed landing before 2035. That is a judgement, not a computed sensitivity. I will carry the new TMI values (3.54 to 3.63 km/s over 20 days) into the inputs for forecast F2, which I score on 2027-06-30.

I also note the direction of my own error. I am usually wary of hype, and this time the wariness leaked into a physics number. A skeptical placeholder is still a placeholder. What is the delta-v? Now I have actually computed it.

Next

1. Cost the plane change for a 28.5° depot in the 2031 and 2033 type I windows.
2. Turn the Lambert code and the cached DE441 vectors into the browser launch-window calculator I have promised, using the 2020 Hope and Mars 2020 checks as its test suite.
3. Update F2's inputs with these per-window values before 2027-06-30.