Cosmic Ray (Particle) Ionisation rate¶

A potential ionisation of the gas by cosmic rays is included by default in ProDiMo and the chemistry. The main ionisation parameter (ionisation rate per $\mathrm{H_2}$) can be controlled by setting the parameter

1.36E-17     ! CRI         [1/s]   : cosmic ray ionisation per H2 molecule

to the desired value (the given value is the default). With this setting alone it is assumed that the CR irradiation is isotropic and constant throughout the disk.

In the following we describe several ways to control the cosmic ray ionisation rate within the disk (i.e. not constant) and other potential particle ionisation sources.

Using a cosmic ray absorption cross-section¶

The absorption of cosmic rays in ProDiMo is controlled by the following two parameters

3.5e-26     ! CRCS [cm^2]          : the cross section for cosmic ray absorption (per hydrogen atom)
1           ! ABS_CRI              : method for the calculation of the CR absorption

The default for CRCS is zero, hence no absorption (constant ionisation rate).

If CRCS is non-zero, ABS_CRI controls the method which is used for the calculation of the CR-Absorption.

• 1 is the default and refers to Eq. 49 in Fujii+ 2011. This equation is based on Umebayashi & Nakano 1980.

• 2 uses an improved Eq. from Umebayashi & Nakano 2009 (Appendix). This Eq. also considers the stronger radial absorption of CR.

• 3 see CR energy spectra and absorption for details.

The cross-section CRCS is also taken from Umebayashi & Nakano (1980,2009) where they give a value of $\mathrm{CRCS}=96\,\mathrm{g/cm^2}$.

CRCS refers to 1/CHI_CR and is not given in $\mathrm{g/cm^2}$ but in $cm^2$ (vertical column density of hydrogen). $3.5\times10^{-26}\,\mathrm{cm^2}$ is equivalent to $2\times \frac{1}{(96\mathrm{g/cm^2})}$ (considering the unit conversion).

Revision: In ProDiMo, we define the mean gas mass as the sum of the hydrogen + He with the appropriate number ratio. With a mean mass of $2.1856 \times 10^{-24}\,\mathrm{g}$, the $\mathrm{CRCS}=2.27\times10^{-26}$. It should be kept in mind that the CRCS value is pretty uncertain.

CR energy spectra and absorption¶

There are also options to indirectly consider different energy spectra of the cosmic rays. For this we use fits to detailed cosmic ray particle transport methods, see Padovani+ 2009, Padovani+ 2013, Cleeves+ 2013. We implemented Eq. 1 of Padovani+ 2013. For usage in ProDiMo models see Rab+ 2017 and Rab+ 2018).

Padovani 2009/2013¶

This can be activated by setting.

3           ! ABS_CRI              : method for the calculation of the CR absorption

In that case default fitting parameters are used:

--- W98 Padovani+ (2009/2013) (Default)
--- The so called W98 CR spectrum plus absorption in the disk
2.0E-17     ! CRP_low
2.1E-2      ! CRP_a
2.6E-18     ! CRP_high
244.d0      ! CRP_s0

Those can be adapted for example for the so-called solar maximum particle spectrum (results in a lower cosmic ray ionisation rate by about a factor of 100 compare to the standard ISM value):

--- Solar Max Cleeves+ (2013)
2.0E-19    ! CRP_low 
-1.0E-2    ! CRP_a 
8.0E-19    ! CRP_high 
230.d0     ! CRP_s0 

We note, these are just two examples for the various fitting formula parameters. In principle one can use any combination, however, we do not recommend that but rather use values that are derived from detailed particle transport models.

Padovani 2018¶

Some new calculations and a new fitting formula for cosmic ray absorption in disks are published in Padovani+ (2018, Appendix F).

Those can be activated by setting

4    ! ABS_CRI  CR Spectrum L from Padovani+ 2018

For the H spectrum of Padovani+ 2018 use 5 and with 6 one can use (L+H)/2 spectrum (not really physically motivated).

Particle ionisation originating from the star¶

It is also possible to model the so called stellar energetic particle irradiation (similar to solar cosmic rays) of a star. In that case the origin of the particle emission is the star and the particles enter the disk along radial rays. Please see Rab+ 2017, where this approach and its limitations are explained in detail. This option can be activated in ProDiMo by setting:

*** Active Tauri STCRP
1.06E-7    ! STCRP_low
8.34E-2    ! STCRP_high
-0.61      ! STCRP_a
-2.61      ! STCRP_b

Ionisation rate as function of x and z¶

There is also a possibility to provide a particle ionisation rate as function of the spatial coordinates x and z. In that case one has to provide an input file called zetaSP.in in the model directory and set

3     ! STCRP_method   Use input file zetaSP.in

The zetaSP.in has the ionisation per molecular hydrogen for each grid point of the model. Such a file can be created by running first a model without STCRP and then read in this model with prodimopy to e.g. get the dimensions for the spatial grid (i.e. use a numpy array) and set the desired ionization rates for each grid point. Then one can write this numpy array to a file (i.e with np.savetxt). Note the array is directly read into an fortran array in ProDiMo so check your dimensions and your results by plotting the quantity zetaSTCR.

Please note those ionization rates are simply added to the CR ionization rate. If you do not want that simply set the CRI parameter to zero. So although we set here a STCRP ionisation rate in prodimo one can also use that for setting a CR ionisation rate. If this method is used the code does not know what it is. This approach was used in Brunn+ (2024).