One wrong unit: how we navigate to Mars, and how we once lost a spacecraft

At 09:00:46 on 23 September 1999, in Coordinated Universal Time (UTC), controllers saw NASA’s Mars Climate Orbiter fire its main engine to brake into orbit around Mars. (That is the time the signal reached Earth; out at Mars it had happened about 11 minutes earlier [Reference 1].) Four minutes later the spacecraft slipped behind the planet, as planned, and its radio signal disappeared. That happened 49 seconds earlier than predicted. The signal was due back about 21 minutes later. It never came. Controllers kept searching until 25 September [Reference 2].

Six days after the loss, engineers found the reason. A ground computer file describing small thruster firings gave its numbers in pound-force seconds, a unit from the US customary (pound-based) system. The navigation software read them as newton-seconds, the metric unit. Both measure the size of a push: a force multiplied by how long it acts. Each push was booked at less than a quarter of its true size, and the spacecraft reached Mars about 170 km lower than planned [Reference 2].

“One wrong unit” is true, but not the whole story. To see why a few tiny pushes could doom a spacecraft, you need to know how anyone hits Mars, and how navigators know where a spacecraft is. The central lesson of NASA’s mishap investigation board was not about units. It was about the checks that should have caught one.

Aiming at a moving target

Planets, asteroids and spacecraft are all falling around the Sun. An orbit is that fall seen from outside: an object pulled toward the Sun, but moving sideways so fast that it keeps missing. Earth does this at about 29.78 km/s [Reference 3]. A spacecraft on the launch pad is already in orbit around the Sun, because it moves with Earth [Reference 4].

So you do not “fly” to Mars like a plane; you change the orbit you are already on. The rule that makes this possible is the vis-viva equation: a single line of algebra that tells you how fast something on an orbit is moving at any distance from the Sun, given only the size of its orbit [Reference 5]. Its message is simple. Go faster at the same distance, and your orbit gets bigger: its far end swings out further from the Sun.

The cheapest way to reach Mars uses exactly that. Speed up in the direction Earth is already moving, so that your new orbit is an ellipse whose closest point to the Sun is on Earth’s orbit and whose farthest point just touches Mars’s orbit. This half-ellipse is called a Hohmann transfer [Reference 4]. For two circular orbits lying in the same flat plane, like two rings drawn on one sheet of paper, it is the route that needs the least fuel using two engine firings, or “burns” [Reference 6].

Pretend the planets move on perfect circles in that one plane, and the numbers are easy to compute. Earth circles the Sun at 29.78 km/s. To leave on the transfer ellipse the spacecraft needs 32.73 km/s around the Sun, so it must gain 2.95 km/s relative to Earth. That change of speed is its delta-v, the basic currency of any space mission. (It must also climb out of Earth’s gravity. From a low “parking” orbit around Earth, where it already moves at 7.7 km/s, escaping alone would take about 3.2 km/s, so you might expect 3.2 + 2.95 ≈ 6.1 km/s in all. In fact one burn of about 3.57 km/s does both, because what adds up is energy, which grows with the square of speed: a push given while already moving fast buys more energy than the same push given later, once the craft has slowed while climbing away from Earth.) Then it coasts. As it climbs away from the Sun it slows down, like a ball thrown upward, and reaches Mars’s distance at only 21.48 km/s. In this circular picture Mars moves at 24.13 km/s (computed from its fact-sheet distance), so it catches up with the spacecraft from behind at 2.65 km/s [Reference 6,Reference 7].

That sounds as if the spacecraft should speed up to match Mars, yet orbiters arrive by braking. Both are true. Mars is catching up at 2.65 km/s, so relative to Mars the spacecraft is rushing in and must get rid of the energy of that approach to stay: speeding up to match Mars, seen from the Sun, is braking, seen from Mars. Mars’s gravity also speeds the spacecraft up as it falls in, and, as at Earth, a burn made there, where it moves fastest, removes that energy cheaply. Escape speed, the speed at which an object breaks free of Mars’s pull, is 4.9 km/s at the planned closest point, 226 km up, and Mars Climate Orbiter was expected to be doing 5.5 km/s as its braking burn began. The burn was planned to change its speed by 1.37 km/s [Reference 1], capturing it into a loop of 13–14 hours [Reference 2]. In the ideal case about 1.0 km/s would do; by our calculation the real orbiter met Mars faster, at about 3.35 km/s rather than 2.65 km/s, which gives 1.36–1.38 km/s. Without the burn it would swing past and fly off [Reference 6].

The coast takes half the time of one full trip around the ellipse: about 259 days, or 8.5 months. That raises the real difficulty. You are not aiming at Mars; you are aiming at the place Mars will be in 259 days. The spacecraft goes halfway round the Sun, 180°. Mars takes 687 days to go once round, so in 259 days it moves 136°. That means Mars needs a 180° − 136° = 44° head start on launch day. Mars starts ahead, yet catches up from behind: the faster spacecraft, on its inner track, draws level with Mars after about four months and edges ahead, then slows, and Mars closes in as they meet. NASA’s introduction to spaceflight compares it to throwing a dart at a moving target: you have to lead the aim point by just the right amount [Reference 4].

Earth laps Mars, and the same arrangement repeats every 780 days, about 26 months. This repeat time is the synodic period [Reference 7,Reference 8]. The stretch of days around each good alignment, when the trip is affordable, is a launch window. Miss it and you wait about two years.

Real trajectories are not quite Hohmann

The tidy picture assumes circular orbits in one plane. Mars’s orbit is off-center: its distance from the Sun swings between 207 and 249 million km. It is also tilted by about 1.85° to Earth’s [Reference 7]. The tilt matters more than it sounds. Every orbit around the Sun lies in a flat sheet through the Sun. Picture a book lying open on a table: the table is the plane of Earth’s orbit, each page a possible orbit, and the spine the line from the launch point through the Sun to the point exactly opposite. A trip that goes exactly halfway round must end at the far end of the spine. Mars sits slightly above or below the table there, so the only page that reaches it stands upright: an orbit at right angles to the plane of Earth’s orbit. Paths that go nearly halfway round must still tilt steeply. Tilting the orbit costs a lot of fuel [Reference 8,Reference 6].

So real missions go a little less or a little more than halfway round. Designers call these Type I (less than 180° around the Sun: shorter trips) and Type II (between 180° and 360°: longer trips, often cheaper than Type I) [Reference 8,Reference 4]. To find them they solve a classic puzzle called Lambert’s problem: given where Earth is on launch day and where Mars will be on arrival day, which orbit joins the two in exactly that time? They solve it for every pair of dates and draw the cost of each as contour lines, like a hiking map, over a chart of launch date against arrival date. The cost is measured by the launch energy, called C3: the square of the speed the spacecraft still has once it has climbed clear of Earth’s gravity (squaring makes it a measure of energy). The chart is known as a porkchop plot, for its shape [Reference 8].

The 2026 window falls in October–November. NASA’s Mars design handbook lists the trip needing the least launch energy in this window as a Type II path leaving Earth on 31 October 2026 and taking 292 days [Reference 8]. On that day Mars is about 63° ahead of Earth, not the textbook 44°. The path sweeps about 196° instead of 180°, and although the trip is 33 days longer, Mars, near the slow far point of its orbit, covers slightly less ground than in the ideal case (133° instead of 136°), so it needs a 196° − 133° = 63° head start. Some windows need almost twice the launch energy of others, because the orbits are off-center and tilted [Reference 8]. A window spans weeks of usable dates: Japan’s space agency, JAXA, scheduled its Martian Moons eXploration (MMX) spacecraft for launch on 20 October 2026 (Japan time) [Reference 10].

Mars Climate Orbiter flew this kind of path too. It launched on 11 December 1998 on a Type II trajectory and reached Mars 286 days later [Reference 2,Reference 1].

How do you find a spacecraft 196 million kilometers away?

When Mars Climate Orbiter reached Mars, it was 196.2 million km from Earth in a straight line, and a radio signal took about 11 minutes each way [Reference 1]. The navigators, at JPL in California, which managed the mission [Reference 2], worked out where it was from radio signals exchanged with NASA’s Deep Space Network: dish antennas 34 and 70 meters across [Reference 1], at complexes in Goldstone (California), Madrid and Canberra [Reference 12].

The workhorse measurement is Doppler tracking, the radio version of a siren dropping in pitch as an ambulance drives away. A station sends a radio tone, the spacecraft sends it straight back, and if the spacecraft is moving away the returning waves are stretched. In practice the station counts waves: every time the returning signal slips one full cycle behind the outgoing one, the round-trip path has grown by one wavelength, just 3.6 cm at the frequencies used, about 8.4 billion waves per second [Reference 13,Reference 12]. That makes Doppler superbly precise along the line of sight from Earth.

A second measurement, ranging, times a coded signal’s round trip to measure distance. Doppler measures speed toward Earth; ranging measures distance. If the computer’s model is missing a push, it can find a path that fits one set of measurements, but none that fits both perfectly, so paths fitted to each drift apart: a known warning sign of an unaccounted push [Reference 13].

Both share a blind spot: they measure along the line of sight. Motion across that line shows up only indirectly. As Earth turns, each dish is carried toward and then away from the spacecraft, which adds a daily wobble to the Doppler signal; the timing and size of that wobble reveal the spacecraft’s direction in the sky. One pass, a station’s tracking session while the spacecraft is above its horizon, gives its speed toward Earth and that direction. Its speed across the line of sight shows up only as a very slow drift of its position across the sky, which takes several days of data to see [Reference 13,Reference 12].

To fix that, the network developed delta-DOR. Two dishes on different continents, about 10,000 km apart, record the spacecraft’s signal, and the tiny difference in its arrival time gives its direction on the sky. The dishes then swing to a quasar, the bright core of a galaxy so distant that it serves as a fixed marker, close to the spacecraft on the sky, and the comparison cancels most errors [Reference 12]. In 1992–93, for Mars Observer, an earlier NASA mission lost three days before reaching Mars in 1993, the system’s errors were about 22.5 billionths of a radian (a radian is about 57°); by 2013 it reached one to two billionths [Reference 12]. At Mars Climate Orbiter’s arrival distance those correspond to about 4.4 km and 0.2–0.4 km sideways, far finer than the roughly 200-km error in its aim described below.

Mars Climate Orbiter did not use it. According to a JPL history of the technique, the mission had originally planned to use delta-DOR, but the requirement was dropped to save money, and it relied on two-way range and Doppler alone [Reference 12].

The small forces

To point its antenna and instruments, Mars Climate Orbiter used three reaction wheels: spinning wheels inside the body that turn the craft by speeding up or slowing down [Reference 1]. Small, steady twists from outside, such as the pressure of sunlight (light carries a tiny push, which adds up on a big solar wing), are absorbed by spinning the wheels ever faster, like pushing a spinning office chair again and again in the same direction. The wheels must stay below their top speed (engineers say “unsaturated”), so every so often thrusters fire to slow them down again. These firings were called angular momentum desaturation events (angular momentum is the physics word for amount of spin) [Reference 2].

A thruster firing that unloads the wheels also gives the whole spacecraft a tiny shove. The board called modeling these “small forces” critical for navigation [Reference 2]. For scale, the small thrusters used only to turn the orbiter produced about 0.9 newtons each; the larger ones, used for course corrections and turning, produced 22 newtons [Reference 1]. Our sources do not say which set did the desaturation firings.

Data about each firing came down to Earth. On one side, the spacecraft operations team in Colorado [Reference 14] ran a program on Earth, SM_FORCES (“small forces”), that wrote it into a file, the angular momentum desaturation (AMD) file. On the other side, the navigation team at JPL used that file to model the shoves. A project rulebook, the Software Interface Specification, said the file had to use newton-seconds. SM_FORCES wrote pound-force seconds instead, and the navigators, trusting the rulebook, assumed the numbers were metric [Reference 2].

One pound-force is the weight of a one-pound mass under standard Earth gravity, about 4.45 newtons (exactly 4.448 221 615 260 5, by definition) [Reference 15]. So a number written in pound-force seconds is about 4.45 times smaller than the same push written in newton-seconds. Read as newton-seconds, each firing was booked at 1 ÷ 4.45 = 22.5 % of its real size, and about 77.5 % of every shove was missing. In the board’s words, the navigation software “underestimated the effect on the spacecraft trajectory by a factor of 4.45” [Reference 2].

The spacecraft’s own software used metric units correctly; the mistake was only on the ground [Reference 2].

Three more things turned a small error into a fatal one.

First, the firings happened 10 to 14 times more often than the navigators expected. Mars Climate Orbiter had a single, lopsided solar wing, unlike the symmetrical arrays of Mars Global Surveyor, an earlier NASA Mars orbiter, so sunlight twisted it much harder. A daily 180° “barbecue” flip planned to cancel the twisting was dropped from the operations plan, and nobody told the navigators. The board suggested that their assumption that the new craft was much like Mars Global Surveyor had left them knowing too little about it [Reference 2]. More firings meant more missing pushes.

Second, for the first four months the AMD files had format errors and could not be used at all; navigators modeled the firings themselves from e-mail notices. The correctly formatted, mis-scaled files came into use only in April 1999, and within about a week they showed signs that the firings’ effects were being underestimated [Reference 2]. The signs were noticed but not formally reported, a thread we come back to below.

Third, the shoves pointed the wrong way to be noticed. Two directions matter. Their main push was across the line of sight from Earth, exactly where Doppler tracking is weakest. It was also mainly at right angles to the spacecraft’s course [Reference 2], so each shove moved the path a little sideways, aiming it slightly closer to the middle of Mars. At Mars, how far off-center the incoming path is aimed decides how deeply it swings past: aim closer to the center, and the closest approach is lower. The board found exactly that: the real path at closest approach was lower than the navigators’ estimate [Reference 2].

Real navigators do not predict a path once and wait. They keep adjusting their computed path until it matches the latest measurements as closely as possible, and re-aim with course corrections. But because tracking can hardly see motion across the line of sight, a path that was off in that direction could still match the measurements. The unbooked pushes made the matched path consistently wrong, an error “present but undetected in the trajectory estimation” [Reference 2], and the final correction was aimed using it.

The last week

The final planned course correction, trajectory correction maneuver 4 (TCM-4), was worked out on 8 September 1999 and carried out on 15 September. It aimed for a first periapsis, the lowest point of the path around Mars, of 226 km [Reference 2]. Then the estimates slid:

Source: the board’s first report [Reference 2].

Throughout the approach, path estimates based on Doppler alone consistently put the spacecraft closer to Mars than estimates combining Doppler and range: the warning sign of an unaccounted push. Worries were raised at the working level, but only informally, and the difference was never resolved [Reference 2].

There was a way out. An emergency fifth correction, TCM-5, could have raised the pass. It was discussed verbally shortly before the arrival sequence began but never carried out. The team had not prepared it, the onboard arrival sequence left no time to upload one, and its urgency was not fully understood [Reference 2].

The engine started at 09:00:46 UTC (the time the signal reached Earth). The signal vanished at 09:04:52, 49 seconds earlier than predicted [Reference 2]. In hindsight, disappearing early is consistent with a lower path, which can duck behind the planet sooner, though an error in arrival time could also contribute; the board’s report records the timing without drawing a conclusion.

On 29 September, after the navigators went over the discrepancies with the spacecraft engineers, it was discovered that the small-force velocity changes were too low by a factor of 4.45, because the AMD file used pound-force seconds. With corrected values, the reconstructed first pass was about 57 km above the surface. The board concluded that the orbiter “either was destroyed in the atmosphere or re-entered heliocentric space after leaving Mars’ atmosphere”, that is, skipped off the air and flew on into orbit around the Sun [Reference 2]. Nobody knows for certain, though NASA’s spacecraft catalog says that at such a low altitude atmospheric stresses and friction would have destroyed it [Reference 16].

How big an aiming error was this? Mars’s gravity bends incoming paths inward, like a funnel, and that squeezes them together near the planet: paths aimed 200 km apart pass only about 155–170 km apart in height. So the 169-km drop from 226 km to 57 km corresponds, by our calculation, to aiming about 200 km closer to Mars’s center, after a curving journey of 669 million km [Reference 1]. In our illustrative model, about 50 nudges, each changing its speed by about 1 mm/s, over the last 60 % of the cruise are enough: with three-quarters of each unbooked, 3–4 cm/s goes missing in total. Small, but it lasts: in a straight line, 1 cm/s kept up for 100 days adds up to about 86 km. Real navigation fitted and corrected the path along the way, so this shows only the scale of the problem.

It was never just a unit

The board, chaired by Arthur Stephenson of NASA’s Marshall Space Flight Center, said it plainly: “mistakes occur on spacecraft projects. However, sufficient processes are usually in place on projects to catch these mistakes before they become critical to mission success” [Reference 2]. On Mars Climate Orbiter, they were not. Alongside the root cause, the board listed eight contributing causes [Reference 2]:

1. Errors in modeling the small velocity changes went unnoticed.
2. The navigators did not know the spacecraft well enough.
3. The emergency maneuver TCM-5 was not performed.
4. Systems engineering mishandled the handover from the builders to the flight team.
5. Communication within the project was inadequate.
6. The navigation team was understaffed.
7. Training was inadequate.
8. Testing did not properly cover the ground software.

Near arrival, navigation was handled by two people, the team lead and the mission’s navigator, while the operations project ran three Mars missions at once. When numbers did not agree, people relied on e-mail rather than the formal system for reporting anomalies. The navigators did not know until long after launch that the spacecraft sent home its own record of each firing’s velocity change; comparing it with the tracking data might have exposed the unit problem [Reference 2]. And the board found a widespread feeling that “Orbiting Mars is routine” [Reference 2].

NASA’s head of space science, Edward Weiler, put it this way a week after the loss: “The problem here was not the error, it was the failure of NASA’s systems engineering, and the checks and balances in our processes to detect the error” [Reference 14].

In its second report, the board judged that relying on Earth-based radio tracking of the spacecraft alone, with no independent cross-check, carried considerable risk, because unnoticed pushes could pile up over a long cruise [Reference 17]. According to the JPL history of delta-DOR, both the project and the Deep Space Network soon concluded that with delta-DOR the mission “would most certainly have been saved” [Reference 12]. That is their judgment, not the board’s. In December 1999 NASA lost a second spacecraft, Mars Polar Lander, which was due to land on 3 December [Reference 2] and was never heard from [Reference 18]. After these two failures, NASA quickly adopted delta-DOR; it supported Mars Odyssey, launched in April 2001, and several Mars missions after it [Reference 12].

Open questions

• What happened to the spacecraft? The board left two possibilities open: destruction in the atmosphere, or a skip back out into orbit around the Sun [Reference 2]. No observation has settled it.
• What altitude was “planned”? The board gives 226 km as the TCM-4 target [Reference 2]. On the day of the loss JPL said it had planned about 150 km (and estimated about 60 km, with an 85-km survival limit); NASA’s catalog still says 140–150 km [Reference 19,Reference 16]. Possibly, though no document we found confirms it, the 150-km figure reflects the final week’s estimates (150–170 km). We follow the board.

Five Sigma, “One wrong unit: how we navigate to Mars, and how we once lost a spacecraft,” Five Sigma (12 October 2026), https://thefivesigma.com/articles/mars-one-wrong-unit. Version of 12 October 2026.