Space Pyhsics Course

Magnetopause

Shape: A paraboloid (a rotated parabola about the x-axis),
Functional form: The bow shock is formed at the points ( $x_{mp}$ , $ρ_{mp}$ ) where
$x_{mp} = x_{o} + \frac{{λ_{o}}^{2}-μ^{2}}{2}$
$y_{mp} = λ_{o} . μ . cos(φ)$
$z_{mp} = λ_{o} . μ . sin(φ)$
$ρ_{mp} = \sqrt{{y_{mp}}^{2} + {z_{mp}}^{2}}$
Here λ, μ and φ are parabolic coordinates and x_o is the focus. The surface of the magnetopause is on the λ = λ_o = constant value. The object is at (x,y,z) = (0,0,0) and the x-axis points from the center of the object toward the Sun.
Derivation of the free parameters: The subsolar point of the magnetopause, i.e. the value of x at the point where the magnetopause crosses the x-axis is derived from the pressure balance equation by assuming that the dynamic pressure, P_dyn,
$P_{dyn} = κ.m_{p}.{U_{sw}}^{2}.n_{sw}$ , is balance by the magnetic pressure in the magnetosphere, $p_{B} = \frac{B^{2}}{2. μ}$ that is,
$P_{dyn} = p_{B}$
Here kappa, $κ$ , is a given dimensionless parameter which its value is close to one, m_p is the mass of a proton, U_SW is the speed of the solar wind, and n_SW is the density of the solar wind.
MORE TEXT HERE

Bowshock

Shape: A cone,
Functional form: The bow shock is formed at the points ( $x_{BS}$ , $ρ_{BS}$ ) where
$r_{BS} = \frac{L_{BS}}{1+{ecc}_{BS}.cos(θ)}$
$x_{BS} = r_{BS} . cos(θ) + {x_{o}}_{BS}$
$y_{BS} = r_{BS} . sin(θ)$
$ρ_{BS} = \sqrt{{y_{BS}}^{2} + {z_{BS}}^{2}}$
Here r_B is the distance from the focus at x = x_o on the x-axis which points from the center of the object toward the Sun. $θ$ is the angle between the x-axis and the direction of the point on the bow shock. The ecc_BS is the eccentricity and and L_BS is the semi-latus rectum.
Derivation of the free parameters:
MORE TEXT HERE

Reference: Kallio, E., and H. Koskinen, A semiempirical magnetosheath model to analyze the solar wind-magnetosphere interaction, J. Geophys. Res, 105, 27,469-27,479 , 2000