Simulations, Data Analysis and Algorithms

# Table 1 Reference values for different types of stars

Protostar CTTS Brown dwarf White dwarf Neutron star
M ($M ⊙$) 0.8 0.8 0.056 1 1.4
$R ∗$ $2 R ⊙$ $2 R ⊙$ $0.1 R ⊙$ 5,000 km 10 km
$R 0$ (cm) 2.81011 2.81011 1.41010 109 2106
$v 0$ (cm s−1) 1.95107 1.95107 1.6107 3.6108 9.7109
$P ∗$ 1.04 days 5.6 days 0.13 days 89 s 6.7 ms
$P 0$ 1.04 days 1.04 days 0.05 days 17.2 s 1.3 ms
$B ∗$ (G) 3.0103 103 2103 106 109
$B 0$ (G) 37.5 12.5 25.0 1.2104 1.2107
$ρ 0$ (g cm−3) 3.710−12 4.110−13 1.410−12 1.210−9 1.710−6
$n 0$ (cm−3) 2.21012 2.41011 8.51011 71014 1018
$M ˙ 0$ ($M ⊙ yr − 1$) 1.810−7 210−8 1.810−10 1.310−8 210−9
$E ˙ 0$ (erg s−1) 2.11033 2.41032 2.51030 5.71034 61036
$L ˙ 0$ (erg s−1) 3.11037 3.41036 1.71033 1.61035 1.21033
$T d$ (K) 2,290 4,590 5,270 1.6106 1.1109
$T c$ (K) 2.3106 4.6106 5.3106 8108 5.61011
1. We choose the mass M, radius $R ∗$, equatorial magnetic field $B ∗$ and the period $P ∗$ of the star and derive the other reference values. The reference mass $M 0$ is taken to be the mass M of the star. The reference radius is taken to be twice the radius of the star, $R 0 =2 R ∗$. The surface magnetic field $B ∗$ is different for different types of stars. The reference velocity is $v 0 = ( G M / R 0 ) 1 / 2$. The reference time-scale $t 0 = R 0 / v 0$, and the reference angular velocity $Ω 0 =1/ t 0$. We measure time in units of $P 0 =2π t 0$ (which is the Keplerian rotation period at $r= R 0$). In the plots we use the dimensionless time $T=t/ P 0$. The reference magnetic field is $B 0 = B ∗ ( R ∗ / R 0 ) 3 / μ ˜$, where $μ ˜$ is the dimensionless magnetic moment which has a numerical value of 10 in the simulations discussed here. The reference density is taken to be $ρ 0 = B 0 2 / v 0 2$. The reference pressure is $p 0 = B 0 2$. The reference temperature is $T 0 = p 0 /R ρ 0 = v 0 2 /R$, where is the gas constant. The reference accretion rate is $M ˙ 0 = ρ 0 v 0 R 0 2$. The reference energy flux is $E ˙ 0 = M ˙ 0 v 0 2$. The reference angular momentum flux is $L ˙ 0 = M ˙ 0 v 0 R 0$. The poloidal magnetic field of the star (in the absence of external plasma) is an aligned dipole field.