Lin Model

Logs : Content#Fitting the Lin model

General expression of the Lin model

r = r_{0} \cdot {\cos \frac{θ}{2} + m \cdot \sin (2 θ) \cdot [1 - \exp (- θ)]}^{β} + Q

where

β = β_{0} + β_{1} \cdot \cos φ + β_{2} \cdot \sin φ + β_{3} \cdot (\sin φ)^{2},

Q = c_{n} \cdot \exp (d_{n} \cdot ψ_{n}^{e_{n}}) + c_{s} \cdot \exp (d_{s} \cdot ψ_{s}^{e_{s}}),

ψ_{n} = \arccos [\cos θ \cdot \cos θ_{n} + \sin θ \cdot \sin θ_{n} \cdot \cos (φ - φ_{n})],

ψ_{s} = \arccos [\cos θ \cdot \cos θ_{s} + \sin θ \cdot \sin θ_{s} \cdot \cos (φ - φ_{s})] .

azimuth angle (φ): the angle between the projection of r in the Y‐Z plane and the direction of the positive Y axis from 0 and 2p in clockwise looking from the Earth to the Sun

Coordinate transformations:

x = c o s θ

y = r s i n θ c o s ϕ

z = r s i n θ s i n ϕ

Pasted image 20250310160137.png

Parametrized model

After the fitting processes above, all coefficients for equation (19) are finally determined and listed in Table 9. On the basis of the above fittings, a new three‐dimensional asymmetric magnetopause model has finally been constructed and parameterized by the solar wind dynamic and magnetic pressures (Pd + Pm), the interplanetary magnetic field (IMF) Bz, and the corrected dipole tilt angle ()
[[Lin model.pdf#page=8&annotation=193R]]

The paper parametrizes the #General expression of the Lin model with (Pd, Pm, Bz, Dφ) - pressure magnetic field and tilt. The final surface r(Pd, Pm, Bz, Dφ) should be described by these 4 parameters and given by equation:

r = r_{0} \cdot f (θ, φ, ϕ) + c_{n} \cdot \exp (d_{n} \cdot ψ_{n}^{e_{n}}) + c_{s} \cdot \exp (d_{s} \cdot ψ_{s}^{e_{s}}),

where

r_{0} = a_{0} \cdot (P_{d} + P_{m})^{a_{1}} \cdot [1 + a_{2} \frac{\exp (a_{3} \cdot B_{z}) - 1}{\exp (a_{4} \cdot B_{z}) + 1}],

f (θ, φ, ϕ) = {\cos \frac{θ}{2} + a_{5} \cdot \sin (2 θ)}^{β_{0} + β_{1} \cdot \cos φ + β_{2} \cdot \sin φ + β_{3} \cdot (\sin φ)^{2}} \cdot [1 - \exp (- θ)],

β_{0} = a_{6} + a_{7} \cdot \frac{\exp (a_{8} \cdot B_{z}) - 1}{\exp (a_{9} \cdot B_{z}) + 1},

β_{1} = a_{10}, β_{2} = a_{11} + a_{12} \cdot ϕ, β_{3} = a_{13},

c_{n} = c_{s} = a_{14} \cdot (P_{d} + P_{m})^{a_{15}},

d_{n} = a_{16} + a_{17} \cdot ϕ + a_{18} \cdot ϕ^{2},

d_{s} = a_{16} - a_{17} \cdot ϕ + a_{18} \cdot ϕ^{2},

ψ_{n} = \arccos [\cos θ \cdot \cos θ_{n} + \sin θ \cdot \sin θ_{n} \cdot \cos (φ - \frac{π}{2})],

ψ_{s} = \arccos [\cos θ \cdot \cos θ_{s} + \sin θ \cdot \sin θ_{s} \cdot \cos (φ - \frac{3 π}{2})],

θ_{n} = a_{19} + a_{20} \cdot ϕ, θ_{s} = a_{19} - a_{20} \cdot ϕ, e_{n} = e_{s} = a_{21} .

And the corresponding final coefficient values after fitting for the in situ observations:

\begin{array}{cr} Coefficient & Value \\ a_{0} & 12.544 \\ a_{1} & - 0.194 \\ a_{2} & 0.305 \\ a_{3} & 0.0573 \\ a_{4} & 2.178 \\ a_{5} & 0.0571 \\ a_{6} & - 0.999 \\ a_{7} & 16.473 \\ a_{8} & 0.00152 \\ a_{9} & 0.382 \\ a_{10} & 0.0431 \\ a_{11} & - 0.00763 \\ a_{12} & - 0.210 \\ a_{13} & 0.0405 \\ a_{14} & - 4.430 \\ a_{15} & - 0.636 \\ a_{16} & - 2.600 \\ a_{17} & 0.832 \\ a_{18} & - 5.328 \\ a_{19} & 1.103 \\ a_{20} & - 0.907 \\ a_{21} & 1.450 \\ σ (r) & 1.033 \end{array}

CMEM model

In the CMEM slides they have used some extra parameters to describe the cusps:

r = p_{0} r_{0} {[\cos (\frac{θ}{2}) + m \sin (2 θ) (1 - \exp (- θ))]}^{p_{1} β} + p_{2} Q

In this way we can fit the $p_{0}, p_{1}, p_{2}$ parameters to better describe the simulation's magnetopause.