Units

AthenaK code units are dimensionless. The <units> infrastructure is designed to convert physical variables to an equivalent dimensionless quantity in code by a scaling factor, and vice versa. For any variable u, we denote its units as u_0, its value in code units as u_\mathrm{code}, and its value in physical units (e.g., cgs units) as u_\mathrm{phy}. Then we have

    u_\mathrm{code}=u_\mathrm{phy}/u_0 \Longleftrightarrow u_\mathrm{phy}=u_\mathrm{code}u_0

For a unit system, the basic units include:

• length unit, l_0
• time unit, t_0
• mass unit, m_0

Then, other units can be derived from these three units, e.g., velocity unit v_0=l_0/t_0, density unit \rho_0=m_0/l_0^3, energy unit e_0=m_0v_0^2, and pressure unit P_0=e_0/l_0^3. As an option, mean molecular weight, \mu, can also be included in the units system to determine temperature unit. The temperature unit is given by T_0=P_0\mu m_\mathrm{H}/(\rho_0k_\mathrm{B})=v_0^2\mu m_\mathrm{H}/k_\mathrm{B}, where k_\mathrm{B} is the Boltzmann constant and m_\mathrm{H} is the atomic mass unit. In this way, we have P_\mathrm{code}=\rho_\mathrm{code}T_\mathrm{code} in code.

The units system is initialized by the <units> block in the input file, including length, length_cgs, mass, mass_cgs, and time, time_cgs. As an option, mean molecular weight can also be initialized by mu. Default units are cgs units with \mu=1 if <units> block is added.

As an example, for simulations of the interstellar medium, a useful unit system is

<units>
length_cgs  = 3.0856775809623245e+18  # length is 1 pc
mass_cgs    = 6.195900228622575e+31   # number density is 1 cm^-3
time_cgs    = 3.15576e+13             # time is 1 Myr
mu          = 1.27                    # mean molecular weight

General Relativity

In General Relativistic (Magneto)hydrodynamics, the system of equations are scale invariant. In this way, there exists freedom in manipulating, in any one simulation, (1) the density scale \rho_0 to set the density, internal energy, and magnetic field strength in physical units and (2) the black hole mass M to set the physical length scale l_0 = G M/c^2 (and hence timescale t_0 = l_0/c) where G is the gravitational constant and c is the speed of light (see Wong et al. 2022 for a nice discussion).

In General Relativistic Radiation (Magneto)hydrodynamics (see Radiation), however, the system of equations are no longer scale invariant. Thus, one must specify the relevant physical scales in the problem. Rather than specifying a length unit l_0, mass unit m_0, and time unit t_0, as described at the top, it is oft easier to specify a black hole mass M and density unit \rho_0. Given the gravitational constant G, speed of light c, and the black hole mass M, one can derive the length scale l_0 and time scale t_0 in cgs units. Combined with a density unit \rho_0, the mass scale m_0 can then be derived. Finally, given the mean molecular weight \mu (same as above), one can infer the temperature unit T_0. Therefore, when working with General Relativistic Radiation (Magneto)hydrodyanmics, we enable the user to specify

<units>
density_cgs = 1.0e-2  # density unit in g/cm3
bhmass_msun = 10.0    # black hole mass in solar masses
mu          = 0.5     # mean molecular weight (in amu)

**Note: In the above, note that the black hole mass must be specified in solar masses. The conversion of the black hole mass to cgs units is handled by the Units API.

In the codebase, one will notice that no Units member enters in the case of General Relativistic (Magneto)hydrodynamics (which makes sense given the scale invariance). However, in the case of General Relativistic Radiation (Magneto)hydrodynamics, if the <units> block is specified, units are heavily used in the radiation source term (see the Coupling section of Radiation).