Engines

Athelas supports methods for artificially driving a supernova called engines. These engines are packages which may implement one or more source terms and other options. Currently, we implmenent a ThermalEngine to drive a supernova. This is often referred to as a “thermal bomb.”

Thermal Engine

The ThermalEngine model for driving an explosion injects a fixed amount of energy through a time dependent heating source term over a fixed interval in mass coordinate and in time. The energy injection profile is an exponential decay in mass and in time, described delow.

We chose the energy per unit time injected into the star to be given by:

(1)\[Q_{\text{engine}}(t) = d' e^{-c' t}\]

We set the ratio

(2)\[\frac{Q_{\text{engine}}(t_{\text{start}})}{Q_{\text{engine}}(t_{\text{end}})} = R_t,\]

where R_t controls the steepness of the exponential decay in time and is set to 100.0 in engines/thermal.hpp. Note that the start time of energy injection is taken to be time 0.0. Then, using this ratio and integrating over the injection timescale,

(3)\[c' = \frac{\ln R_t}{t_{\text{end}} - t_{\text{start}}} \quad d' = \frac{c' E_{\text{tot}}}{e^{-c' t_{\text{start}}} - e^{-c' t_{\text{end}}}}\]

where E_tot is the total energy to be deposited. The energy injected per unit time per unit mass is then of the form

(4)\[Q_{\text{engine}, i}(m_i) = b' e^{-a' m_i}.\]

The above is the source term implemented in the ThermalEngine package. Similarly to above, we control the steepness of the decay in mass coordinate

(5)\[\frac{Q_{\text{engine}, m_i}(m_{\text{start}})}{Q_{\text{engine}, m_i}(m_{\text{end}})} = R_m,\]

where R_m is again set to 100.0 in engines/thermal.hpp. we obtain the final coefficients

(6)\[a' = \frac{\ln R_m}{m_{\text{end}} - m_{\text{start}}} \quad b' = \frac{d' e^{-c' t}}{\sum_i e^{-a' m_i} \Delta m}\]

where the summation in the expression for b’ is performed over the mass extent of the injection.

Note that to determine d’ we perform an integration assuming a continuous time coordinate. This approach is valid because the timestep at the beginning of the calculation is much smaller than the duration of energy injection by the engine. However, individual cell masses are generally not small enough to justify a continuum approximation, so we instead use a sum over discrete mass cells.

Once the injection timescale has passed the ThermalEngine package is disabled and the source term no longer computed or applied.

Timestep Restriction

The ThermalEngine requires that the energy injection is spread out over at least 2500 timesteps.

Configuration

The ThermalEngine has a few relevant controls. The extent of injection in time and mass space can be controlled. Most importantly, the mode of calculating the injected energy can be set. The direct mode specifies the amount of energy to be injected and uses that for E_tot. This may not be sufficient energy to unbind the star, resulting in a failed supernova. More useful is asymptotic where the desired final energy is specified. In this mode the total energy of the stellar model is calculated and the injection energy is adjust such that the energy of the ejecta reaches the desired energy after unbinding the star. If for some reason the calculated energy is determined to be less than the binding energy of the progenitor the code will crash.

Note

We do not at present allow runtime configuration of the steepness parameters R_t and R_m. This is simple to add.

Input Deck

The ThermalEngine is controlled in the input deck as follows:

[physics]
engine = true

[engine.thermal]
enabled = true
mode = "asymptotic" # or "direct"
energy = 1.5e51 # erg
tend = 0.1 # seconds
mend = 0.1 # Msun