The Moon landings

Contrary to urban myth, NASA did use the metric system for the Apollo Moon landings. SI units were used for arguably the most critical part of the missions – the calculations that were carried out by the Lunar Module’s onboard Apollo Guidance Computer (AGC) during the computer-controlled phases of the spacecraft’s descent to the surface of the Moon, and for the journey of the Ascent stage of the craft during its return to lunar orbit, where it would rendezvous with the Command and Service Module (CSM).

Apollo Guidance Computer
Apollo Guidance Computer (AGC) and DSKY (Display and Keyboard)

As is the case in the UK with road signage, the use of metric units in the USA is often hidden from public view. The Apollo Guidance Computer is a good example of this. The computer display readouts were in units of feet, feet per second, and nautical miles – units that the Apollo astronauts, who had mostly trained as US Air Force pilots, would have been accustomed to using. Internally, however, the computer’s software used SI units for all powered-flight navigation and guidance calculations, and values such as altitude and altitude rate were only converted to imperial units when they needed to be shown on the computer’s display.

Apollo Guidance Computer – use of measurement units
Quantity Internal Displayed
distance metres feet, nautical miles
time centiseconds minutes, seconds
altitude metres feet
altitude rate metres per centisecond feet per second
acceleration metres per centisecond squared
mass kilograms
fuel burn rate kilograms per centisecond
thrust newtons
impulse newton centiseconds
momentum newton centiseconds

Source code for the Apollo Guidance Computer program has been released into the public domain. The following extracts highlight examples of the use of SI units in the software.

Apollo11_AGC1

Apollo11_AGC2

Apollo11_AGC3
Source code extracts from Luminary 1A build 099

Physics equations – SI versus imperial

NASA’s mathematicians used Newton’s laws of motion in their space flight calculations. To demonstrate why using SI units for such calculations is simpler and clearer than using imperial units, it is a useful exercise to take Newton’s second law as an example to compare the differences.

Newton’s second law of motion states that the rate of change of momentum of a body is directly proportional to the force applied, and this change in momentum takes place in the direction of the applied force. This can be expressed in the following equation:

force is proportional to mass × acceleration
force is proportional to mass × acceleration

or
where k is the proportionality constant
where k is the proportionality constant

Using SI units

In the International System of Units (SI), the newton (symbol N) is the unit of force. It is a coherent derived unit, which means that it is defined from SI base units such that the proportionality constant is one. Therefore, using SI units, the above equation can be simplified as:

force = mass × acceleration
force = mass × acceleration

A force of 1 newton on a mass of 1 kilogram produces an acceleration of 1 metre per second squared.

Using SI units, calculations involving Newton’s second law are straight forward. e.g. The force required to accelerate a 5 000 kg spacecraft at 3 m/s2 is 15 kN.

Using imperial units

In contrast to SI, the use of imperial units is cumbersome and confusing. Imperial generally has more than one unit for a given quantity, so before any calculation can be done, a choice of units has to be made (working with miles, yards, feet and inches at the same time is not even a consideration here). Conventionally feet, pounds and seconds would be used. To further complicate matters, the names used for imperial units of force and mass are the same*. For physics calculations, their use has to be explicitly stated as pound-force (lbf) and pound-mass (lbm). The definition of the pound-force also has the effect of obfuscating the difference between mass and weight (a force).

One pound-force is defined as the force due to gravity acting on a mass of one pound at sea level on Earth.

For Newton’s second law, if units of pounds and feet are used in the equation F = k m a, the proportionality constant k does not equal one. The units pound-force, pound-mass, and foot per second squared do not form a coherent system. This is a crucial difference from SI. In order to complete the calculation, the value of the proportionality constant needs to be known. It also means that, using imperial units, two multiplication operations are needed to perform the same task that needs only one multiplication using SI units.

In what amounts to an admission that imperial units are unfit for purpose, there is an alternative approach that involves defining novel units for either mass or force, such that the proportionality constant k is equal to one. For mass there is the slug, and for force there is the poundal. Their use is mutually exclusive, and applicable only when acceleration is defined in units of ft/s2.

A slug is defined as the mass that is accelerated by 1 ft/s2 when a force of 1 pound (lbf) is exerted on it.

1 slug = 14.593903 kg

A poundal is defined as the force necessary to accelerate a mass of 1 pound (lbm) at 1 ft/s2.

1 poundal = 0.138254954376 N exactly.

Using the first of these methods, to calculate the force required to impart a certain acceleration to a spacecraft, it would be necessary to know the spacecraft’s mass in slugs, a unit that has never been in general use**.

1960s computer technology

The Apollo Guidance Computer (AGC) was one of the first computers to use integrated circuits. In the 1960s it was the state-of-the-art. Its performance was comparable to a first generation home computer from the late 1970s.

By choosing to use SI, NASA’s software engineers removed the need for the extra multiplication calculations that working with imperial units would have entailed. The consequent reduction in the number of mathematical operations required by the navigation and guidance programs contributed to the task of making the most efficient use of the onboard computer’s limited processing power and memory capacity.

*It makes no sense for different units in a measurement system to share the same unit name. However, imperial has more than one instance of this confusing practice. For mass and force, there is the pound; and for mass, force and volume, there is the ounce. When used for volume, the ounce (or fluid ounce) also has a different magnitude depending on whether it is of the US or imperial variety.

**Slugs have only ever been used for physics calculations involving obsolete imperial units. For better or worse, in 1969 we never got to see the news headline, “A spaceship of more than 1000 slugs has landed on the Moon“. Similarly, fruit and vegetables have never been authorised for sale by the slug (any requirement for the word slugs to appear on the contents label of a packet of lettuce would probably have made that a non-starter).