In [1]:
%matplotlib inline
import matplotlib.pyplot as plt
import numpy as np
import scipy.interpolate as scint
Surface magnetic field strengths are approximated by assuming the magnetic field is in thermal equipartition with the ambient plasma at an optical depth $\tau_{\rm ross} = 1$. Thermal equipartition implies that \begin{equation} B_{{\rm eq},\, \tau=1} = \left(8\pi P_{\rm gas}\right)^{1/2}. \end{equation} Gas pressures for the ambient stellar plasma are estimated using detailed stellar model atmospheres. Combinations of $\log g$ and $T_{\rm eff}$ necessary to determine the surface gas pressure are taken from a standard stellar evolution model isochrones at a given age. Since we're interested in a range of ages for comparison, we'll select multiple ages and compare values of equipartition magnetic field strengths. Provided equipartition values don't change appreciably, predictions from magnetic models computed with a single equipartition magnetic field strength will be robust.
We'll start by importing a series of standard model isochrones at 1, 5, 10, and 15 Myr.
In [2]:
iso_01 = np.genfromtxt('../../evolve/models/smc/std/dmestar_00001.0myr_z-0.50_a+0.00_phx.iso')
iso_05 = np.genfromtxt('../../evolve/models/smc/std/dmestar_00005.0myr_z-0.50_a+0.00_phx.iso')
iso_10 = np.genfromtxt('../../evolve/models/smc/std/dmestar_00010.0myr_z-0.50_a+0.00_phx.iso')
iso_15 = np.genfromtxt('../../evolve/models/smc/std/dmestar_00015.0myr_z-0.50_a+0.00_phx.iso')
Next, we'll load the atmospheric boundary condition table used to compute the isochrones. The tables are compiled using PHOENIX AMES-COND model atmospheres with a solar composition from Grevesse & Sauval (1998). While interior structure models rely on prescribing the gas pressure and temperature at $\tau_{\rm ross}=10$, we need to choose the table for $\tau_{\rm ross}=1$.
In [3]:
atm = np.genfromtxt('../../evolve/phoenix/ames-cond/tab/Zm0d5.ap0d0_t001.dat')
Atmosphere tables are structured so that $T_{\rm eff}$ values are listed in the first column, but $\log g$ is not tabulated; it must be prescribed.
In [4]:
teffs = atm[:,0]
loggs = np.arange(-0.5, 5.6, 0.5)
The rest of the table is structured as a set of values (P, T) at each $\log g$. It'll be easier to create new arrays for each of these.
In [5]:
temps = np.empty((len(teffs), len(loggs)))
press = np.empty((len(teffs), len(loggs)))
for i, teff in enumerate(atm[:, 1:]):
for j, prop in enumerate(teff):
if j%2 == 0:
press[i, j/2] = prop
else:
temps[i, j/2] = prop
With the individual tables formed, we can construct an interpolation surface using a 2D interpolation routine. Note that we only really care about the pressure table, as that sets the equipartition magnetic field strengths. A linear interpolation will suffice for now, but there is structure to the temperature-pressure relation that requires a shape preserving interpolation algorithm.
In [6]:
pgas_surface = scint.interp2d(loggs, teffs, press, kind='linear')
We are now ready to compute equipartition magnetic field strengths along each of the isochrones. It'll be easiest to quickly define a generic function.
In [7]:
def computeBeq(iso):
Beq = np.empty((len(iso)))
for i in range(len(iso)):
Beq[i] = np.sqrt(8.0*np.pi*pgas_surface(iso[i, 2], 10**iso[i, 1]))
return Beq
And compute equipartition magnetic field strengths along each isochrone.
In [8]:
Beq_01 = computeBeq(iso_01)
Beq_05 = computeBeq(iso_05)
Beq_10 = computeBeq(iso_10)
Beq_15 = computeBeq(iso_15)
Now to compare how equipartition magnetic field strengths vary with age.
In [9]:
fig, ax = plt.subplots(1, 1, figsize=(9., 6.))
ax.set_xlabel('${\\rm Mass}\ (M_{\\odot})$', fontsize=20)
ax.set_ylabel('$\\langle {\\rm B}f \\rangle_{\\rm eq}$', fontsize=20)
ax.grid()
ax.plot(iso_01[:,0], Beq_01, '-', lw=4, color='#333333', alpha=0.7)
ax.plot(iso_05[:,0], Beq_05, '-', lw=4, color='#0099cc', alpha=0.7)
ax.plot(iso_10[:,0], Beq_10, '-', lw=4, color='#b30000', alpha=0.7)
ax.plot(iso_15[:,0], Beq_15, '-', lw=4, color='#e68a00', alpha=0.7)
Out[9]:
There is significant variation between 1 and 15 Myr at nearly every mass, with the exception of stars in the vicinity of $1 M_{\odot}$. However, if we choose a single set of equipartition magnetic fields (say at 10 Myr), we will know where the effects from magnetic fields will either be over and/or under-estimated.
Let's take, for a start, equipartition magnetic field strengths at 10 Myr.