# Units¶

## Introduction¶

Unlike its ancestor FARGO, FARGO3D comes with a variety of unit systems. The reason for this is twofold:

• Working in a different unit system may help reveal bugs. Let us take a simple example to illustrate this: assume that somewhere in an azimuthal derivative calculation, a developer has divided by the angular step rather than the linear one (or vice-versa), i.e. he has forgotten to divide (or multiply) by the cylindrical radius. When working in scale free units, where one usually takes radii commensurable to one, this mistake may remain unnoticed for a long time, especially if it is hidden in a part which has a small impact on the evolution of the flow (such as the viscous stress tensor, for instance). If one switches to MKS or cgs, where the radii have values which are typically 10 to 14 orders of magnitude larger, such errors appear straight away. In fact, we have developed tests to check the dimensional homogeneity of all parts of the code. We run it a first time in a given system of units, then we rerun it in another system of units, and we check that the ratio of the two outputs is flat, to the machine precision. You may have a look at the test named dimp3diso.py in the directory test_suite/. You can run it from the main directory by issuing:

make testdimp3diso

• The other reason for implementing a variety of unit systems comes from the user feed back of the FARGO code. About half of the questions that we received on the code had to do with units. Besides, as the code gains in complexity, by the inclusion of MHD or radiative transfer, it may become desirable to switch to a standard system of units, where constants have a well known value (if you are not convinced, try to work out what is the value of Stefan’s constant in a system of units where the solar mass is the mass unit, one astronomical unit the length unit, such that G, the gravitational constant, has value one, and where the ratio of the ideal gas constant over the mean molecular weight has also value one). Finally, the output of FARGO3D may be used by third party codes, such as radiative transfer codes that produce a simulated image, and having the data in a standard unit system may prove useful.

## Specifying the unit system¶

The unit system must be specified at build time. The different systems are defined in the file named “fondam.h”. The unit system used to build the code depends on whether the preprocessor variable MKS, or CGS, is defined. If none of them is defined, a trivial unit system (dubbed “scale free”) is adopted. From the makefile, activating one or another of these preprocessor variables is done as follows:

make UNITS=MKS


or:

make UNITS=CGS


Finally, to use the scale free system of units, issue:

make UNITS=0


Note

As other build options, the UNITS flag is sticky: is keeps implicitly its previous value until it is changed explicitly.

We note that specifying the unit system in the FARGO3D code is done by giving a numerical value to five constants that have linearly independent powers of $$M$$, $$L$$, $$T$$, $$\theta$$ and $$I$$ (mass, length, time, temperature and electric intensity). These constants are the gravitational constant $$G$$ (G, with units $$M^{-1}L^3T^{-2}\theta^0I^0$$), the central star mass $$M_\star$$ (MSTAR, with units $$M^{1}L^0T^{0}\theta^0I^0$$), a length $$R_0$$ (R0, with units $$M^{0}L^1T^{0}\theta^0I^0$$), the ratio of the ideal gas constant to the mean molecular weight $${\cal R}/\mu$$ (R_MU, with units $$M^{0}L^2T^{-2}\theta^{-1}I^0$$), and the value of the magnetic permeability of vacuum $$\mu_0$$ (MU0, with units $$M^1L^1T^{-2}\theta^{0}I^{-2}$$).

Naturally, if you specify the CGS unit system, your parameter file must provide all real variables in this unit system: YMIN/YMAX must be in centimeters, and so must be ZMAX/ZMIN in cylindrical or Cartesian coordinates, and XMIN/XMAX in Cartesian coordinates. Similarly, NU must be in cm^2/s, the planetary mass in the .cfg file must be in grams, and so on and so forth. There is however an exception to this, when one uses the RESCALE directive, as we explain below.

## Rescaling the input parameters¶

Previous users of FARGO are certainly used to scale free input parameters, in which the central star mass is set to one, the planet’s orbital radius set to one, and the gravitational constant set to one. The orbital period is therefore $$2\pi$$. You may require that FARGO3D be run in a unit system such as cgs or MKS, without editing your scale free parameter file. For this purpose, you must build FARGO3D with the rescale option:

make UNITS=MKS RESCALE=1


Once the parameters are read from the parameter file, they are rescaled using rescaling rules. For instance, the value of YMIN (the mesh minimal radius, in cylindrical or spherical geometry) is multiplied by $$R_0$$. Similarly, the value of SIGMA0 (the disk’s surface density, if your setup uses one) is multiplied by $$M_\star/R_0^2$$, etc. This allows to get an output in a standard unit system while keeping scale free input files, the content of which is probably more intuitive.

## Specifying the scaling rules¶

A scaling rule for a variable is a product of the five dimensionally independent variables ($$G$$, $$M_\star$$, $$R_0$$, $${\cal R}/\mu$$ and $$\mu_0$$), each raised to a specific power, that determines uniquely the dimension of a variable. A scaling rule for a given variable is unique. If it is cast incorrectly, the code will not pass the homogeneity test (if this variable is used in the setup tested).

The scaling rules are required exclusively if you build the code with the RESCALE flag activated, so as to have a dimensional output with scale free input parameters. You can have a look at the file std/standard.units. You can see that each line looks like C code (no ; is required at the end), and the right hand side of the *= symbol has same unit as the left hand side. The scaling rules for some variables is trivial (e.g. SIGMA0, which is a surface density, or PLANETMASS, which is a mass).

During the make process, the python script scripts/unitparser.py is run, which scans all real variables known to the code (that is everything in the setup .par file found to be have a real value). If it finds it in the scaling rules it has access to (those of std/standard.units plus, if any, in setups/SETUP/SETUP.units, in case your setup defines new real variables), it copies that rule in a file made automatically that is called rescale.c, and which contains the rescaling routine called before entering the main loop if you have made a built with the RESCALE option.

If it does not find a scaling rule for a variable it issues a warning asking to check whether this variable is dimensionless. Since the output of this script is found at the very beginning of the make process, and may be unnoticed, it can be a good idea to run the script separately. You have to do that from the src/ directory:

$cd src$ python ../scripts/unitparser.py mri
Warning ! Scaling rule not found for FLARINGINDEX. Is it dimensionless ?
Warning ! Scaling rule not found for SIGMASLOPE. Is it dimensionless ?
Warning ! Scaling rule not found for BETA. Is it dimensionless ?
Warning ! Scaling rule not found for NOISE. Is it dimensionless ?
Warning ! Scaling rule not found for ASPECTRATIO. Is it dimensionless ?


You can verify that each of the variables found by the script is indeed dimensionless. This list is naturally setup dependent, and the above example is for the set mri.

Warning

Upon completion of the manual run of the script as above, you MUST go to the ../bin directory and remove manually the file rescale.c leftover by the script. Otherwise, for dependency reasons, the makefile will not remake it automatically at the next build.