Bert's blog

## Units and unit conversions

Units are very important in science, as they provide a mapping from the abstract realm of maths to the reality of physics. Maths tells us exactly how to compute the circumference of a circle with a radius of $2$, but it does not tell us the actual size of that circle compared to any other size that we know. If the radius is $2$ cm, then the circle is very small and probably drawn on a piece of paper as part of a calculation. If the radius is $2$ km, then the circle is rather large and might be part of a particle accelerator. And who knows what the size of the circle is if the radius of the circle is $2$ miles?

All of this is pretty well established and sounds incredibly trivial. Nonetheless, units are still a very big issue in computational sciences (again, I can only really judge computational astrophysics). With good reasons.

First of all, computers are pretty much very basic but fast mathematicians: they can do a lot of calculations (with a very limited number of operations), but they do not attempt to interpret any of those. For a computer, just like for a mathematician, the radius of the circle is $2$, and it is up to you as a user to interpret this value with appropriate units. The computer itself does not care about units, neither does it keep track of them. Second, this inevitable transfer of responsibility from the computer to the user means that the user somehow has to keep track of units throughout a computation, i.e. has to actively do something to make sure the input and output of the computation makes physical sense. I probably have complained before about the lack of proper documentation in many astrophysical simulation codes; since code documentation is the most straightforward place to provide the necessary unit information, it is clear to see the issue. Third and last, there seems to be a lack of interest in units within the community, maybe because most test cases for simulation codes use “code units” and unit handling is never properly tests, or because scientists don’t like to admit they have problems with something as trivial as units, or because units are just a distraction of the real science, which tends to be captured by complicated equations.

The fact is that units are a problem; I have personally struggled with simulation units way to often, and I have experienced at least one occasion during which a senior scientist messed up unit conversions and caused a lot of problems for a student that was trying to replicate that scientist’s work.

So in this post I intend to discuss how I deal with units, and what I think should be the minimum requirements for unit handling in any scientific simulation code.

# The basics: quantities

While units and especially the conversion from one unit into another can be incredibly painful, the basics underlying any set of consistent units (or unit system) are incredibly simple. In physics, there are only seven independent quantities: length, time, mass, temperature, electric current, luminous intensity, and amount of substance. As an astronomer, I prefer to also add plane and solid angle to this list, as angles are typically treated as independent quantities. A unit system is consistent if all these quantities have a unit, and if the unit for each quantity is the same whenever that quantity appears in a derived quantity (e.g. a velocity: length divided by time).

Whenever you do a computation on a computer, you have to choose a consistent unit system, as the equations that are evaluated during the computation are agnostic about the conversions between different units for the same quantity. This means that all quantities you input at the start of the computation should be expressed in the unit system of choice, and also that all quantities that come out of the computation (your result) will be expressed in that same unit system.

So before you start worrying about units, you should be aware of the quantities that are involved in your calculation. In many problems that involve dynamics, there will only be a limited number of these: length, time and mass. If temperatures are involved, they are usually only used in conjunction with Boltzmann’s constant $k$ and converted into energies, which again only involve length, time and mass. Electric current is only used in problems that involve electromagnetic forces, and again, are usually converted into forces by means of appropriate physical constants. Luminous intensity is not used for dynamics. And amount of substance is pretty much a counting exercise, which again has no impact on dynamics. So length, time and mass are your most likely candidate quantities. In which case you have to know only three units to fix your unit system.

# Fixing a unit system

So how do you fix your unit system? And what does it actually mean to fix your unit system? Let’s look at the first question first. For simplicity, I will assume a (hydro)dynamical simulation that only involves length, time and mass. To fix the length scale, you typically need to choose a spatial scale for your simulation. Many simulations involve a simulation box that is a cube or rhombus with specific dimensions. Choosing the length scale then boils down to defining what physical size scale this box has, and what number to use for that same physical size scale. If your box is for example a cube with a side length of $1$ km and you choose $2$ as the box length in your simulation program, then this fixes the length unit to $\left(\frac{1}{2}\right)$ km (because $2$ times the length unit equals $1$ km).

Fixing the mass unit is very similar: you choose some mass scale of interest (if you are simulating the solar system, then the mass of the Earth or Sun could do) and choose a value for this same mass within the computation.

The time unit can in principle also be fixed like this, although this is not what usually happens. And this is where things get more interesting. When choosing internal (computational) values for quantities, a scientist usually tries to choose values that are close to $1$, as these values are more accurately represented within the finite precision of a floating point variable. This is especially true if a computation uses single precision floats, as they have a limited precision. This means that $1$ is a good size for a simulation box (or $100$ or $10,000$, but definitely not $10^{50}$). Similarly, $1$ is a very good average mass for all masses in the computation. But for the time unit, there is not really such a choice.

The problem is that dynamical simulations try to solve for the time evolution of a system. This means that you know the state of the system at one time (in arbitrary time units), and want to evolve it forward in time to an arbitrary later time, but preferably a time that is sufficiently far away from the initial time so that interesting dynamics can happen in between. Since you don’t necessarily know what that later time should be, it is not so clear how to choose the time unit so that the size of the total time interval is small when represented on a computer.

This can however be estimated from other variables. (Hydro)dynamics is all about movements: the current state of a system gives rise to some net forces that accelerate material and cause it to evolve into a different state. The time scale of these movements will depend on the size of the net forces, and can usually be characterised in terms of a dynamical time scale or a characteristic velocity. In purely gravitational problems like the movement of the Earth around the Sun, the dynamical time scale could e.g. be the orbital period of the Earth. In problems involving hydrodynamics, the sound speed provides a characteristic velocity; this sound speed depends on the temperature of the medium.

If a dynamical time scale can be derived for your problem, then it makes sense to use that as simulation time unit, and then the conversion factor can be determined as before. If on the other hand you only have a characteristic velocity, then some more thinking is required. Knowing the characteristic velocity, you can compute how long it will take for something moving with that velocity to cross the side of your simulation box (because you already fixed the length unit). You can use that value as your internal time unit, and use the technique from before to fix your length unit. Note that in some cases this technique can be somewhat counter-intuitive. If your characteristic velocity is for example the sound speed, then fixing the time unit will require you to know the temperature of the system and the size of the simulation box!

Now: what does it mean to fix the unit system? At the end of the procedure described above, you will have internal units for all three basic quantities that are involved. These internal units are nothing more than conversion factors from internal simulation values to real physical values. Note that the internal units themselves have units, while the internal quantities do not. You do need to keep track of what the units of the internal units are!

At the start of the simulation, you need to convert from physical units to internal units, which means you need to divide all input quantities by an appropriate internal unit (or combination of internal units). Whenever the simulation produces an output, you then need to convert back to physical units by multiplying with the same (or appropriately different) internal units.

# What can go wrong?

Many things. An obvious mistake that everyone makes at some point is mixing up divisions and multiplications. Things are fine as long as you realise that simulation units - as conversion factors - have units themselves, because then it is clear that you need to divide physical quantities by them and multiply simulation quantities with them. But it is very easy to forget to remember of keep track of the units of a conversion factor, and only remember its value. Especially if the original physical unit is some reciprocal (e.g. cm$^{-3}$), then this can lead to confusion (while $1$ m is definitely larger than $1$ cm, $1$ m$^{-3}$ is a lot less than $1$ cm$^{-3}$).

Another thing that can go wrong is the determination of your internal unit system. As pointed out before, many simulations only use three basic quantities, which means you can only fix three units (all the others are derived from those). If you still try to fix more than three units, you end up with inconsistencies which will either cause a lot of trouble during your simulation, or will result in output having different units than you expect. A typical example is a hydrodynamical simulation where the time unit depends on the sound speed and hence the temperature. While temperature as a quantity has its own unit, fixing the time unit using the temperature is completely independent of the temperature unit, and does only affect the time unit. After setting the time unit, you can no longer set any other units that might affect the time unit (like e.g. the velocity unit)!

Another typical problem with units is the use of physical constants. As I already pointed out before, physical constants can be used to get rid of some quantities and convert them into more dynamically relevant quantities. If you multiply a temperature with Boltzmann’s constant, you get an energy (mass times length squared divided by time squared). Similarly, if you multiply a mass with Newton’s constant, G, you get a volume acceleration (length cubed divide by time squared). If you choose to set the value of a physical constant to something other than its actual value in internal units (typically $1$), then this automatically sets one unit in the internal unit system. This is a very subtle way to overfix your unit system without realising it.

# What should we do?

Personally, I have experienced the pain of unit conversions enough during my scientific career, and I have developed some ways of avoiding a lot of the problems that I encountered in the past. I am not saying that these are perfect solutions, but I have seen some of these being used by other simulation codes, and I know that they work for me.

The first and most obvious way of dealing correctly with units is being aware where units are important: for simulation input and simulation output. Every value that gets fed into a simulation code should have a unit, and it should be clear what that unit is. This is also true for values that are read in from parameter files or even from the command line. In my recent simulation codes (e.g. CMacIonize) every value in the parameter file has to have an explicit unit (e.g. you need to write 1 m and not just 1 for a length quantity), and the same is true for all quantities read from other input files (unless units are provided in a different way). It took quite a lot of effort to write the code that deals with reading in these units and automatically converting quantities from input units to internal units, but all of this code only needs to run at the start of the simulation, so it has no significant impact on the simulation run time. And it has made dealing with units a lot easier.

Making sure units are present in simulation output is even easier: you just need to make sure that whatever output format your simulation code uses has a way of attaching some metadata to output fields in which you explicitly specify the units.

For code developers, dealing with units is - as already mentioned above - pretty much only a matter of providing sufficient documentation in the code. If a function takes a physical quantity as input and outputs another physical quantity, you should either document the units of both quantities, or (if internal units are used) specify the basic quantities for both values. This makes it possible to rigorously check that all units inside the function are correct.

So in conclusion: keeping track of units is pretty much just a matter of providing information to whoever uses or further develops your code, so that everyone knows what goes in and comes out of the code.

Professional astronomer.