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| -rwxr-xr-x | calc_mass_potential.py | 58 |
1 files changed, 53 insertions, 5 deletions
diff --git a/calc_mass_potential.py b/calc_mass_potential.py index 48a20d4..46fe777 100755 --- a/calc_mass_potential.py +++ b/calc_mass_potential.py @@ -2,20 +2,63 @@ # # Weitian LI # Created: 2016-06-24 -# Updated: 2016-06-24 +# Updated: 2016-06-25 # # Change logs: +# 2016-06-25: +# * Update documentation # 2016-06-24: # * Update method 'gen_radius()' # """ Calculate the (gas and gravitational) mass profile and gravitational -potential profile from the electron number density profile. -The temperature profile is required. +potential profile from the electron number density profile, under the +assumption of hydrostatic equilibrium. + +The electron density profile and temperature profile are required. + +Assuming that the gas is in hydrostatic equilibrium with the gravitational +potential and a spherically-symmetric distribution of the gas, we can +write the hydrostatic equilibrium equation (HEE) of the ICM as +(ref.[1], eq.(6)): + derivative(P_gas, r) / rho_gas = - derivative(phi, r) + = - G M_tot(<r) / r^2 +where, + phi: gravitational potential; + G: gravitational constant; + rho_gas: gas mass density: + rho_gas = mu * m_atom * n_gas + P_gas: gas pressure: + P_gas = rho_gas * k_B * T_gas / (mu * m_atom) = n_gas * k_B * T_gas + mu: mean molecular weight in a.m.u (i.e., m_atom) (~ 0.6) + m_atom: atom mass unit + n_gas: gas number density; sum of the electron and proton densities + k_B: Boltzmann constant + T_gas: gas temperature + +Solve the above equation, we can get the total mass of X-ray luminous +galaxy clusters (ref.[1], eq.(7)): + M_tot(<r) = - (k_B * T_gas(r) * r) / (mu * m_atom * G) * + (derivative(log(T_gas), log(r)) + + derivative(log(n_gas), log(r))) + +Note that the second part (the derivatives) is DIMENSIONLESS, since + d(log(X)) = d(X) / X +Also note that ('R' is a ratio constant): + d(log(n_gas)) = d(log(R*n_e)) = d(log(n_e)) + +Note that 'kT' has dimension of energy. Therefore, if the gas temperature +is given in 'keV', then the 'kT' should be substitute as a whole. + +For example: + (1.0 keV) * (1.0 kpc) / (0.6 * m_atom * G) ~= 3.7379e10 [ Msun ] +which is consistent with the formula of (ref.[2], eq.(3)) + References: -[1] Ettori et al, 2013, Space Science Review, 177, 119-154 +[1] Ettori et al., 2013, Space Science Review, 177, 119-154 +[2] Walker et al., 2012, MNRAS, 422, 3503 Sample configuration file: @@ -40,11 +83,16 @@ t_profile = t_profile.txt t_unit = "pixel" # number of data points for the output profile calculation -num_dp = 100 +num_dp = 1000 # output gas mass profile m_gas_profile = mass_gas_profile.txt +# output total (gravitational) mass profile +m_total_profile = mass_total_profile.txt + +# output gravitational potential profile +potential_profile = potential_profile.txt ------------------------------------------------------------ """ |