Math 2.0

Astronomy and astrophysics Astrophysical fluid dynamics and plasma Proposition

Parker solar wind model

Builds on Euler equationsSpeed of soundNewton's law of universal gravitationCritical point

Statement

Let a star of mass MM be surrounded by a steady, spherically symmetric, isothermal flow of gas, described by the Euler equations with Newtonian gravity. Let cs>0c_s > 0 be the constant isothermal speed of sound and GG the gravitational constant. Let v(r)v(r) be the radial velocity and ρ(r)\rho(r) the density at distance rr from the centre. Mass conservation ρvr2=const\rho v r^2 = \text{const} and the momentum equation v v′=−cs2ρ′/ρ−GM/r2v \, v' = -c_s^2 \rho'/\rho - GM/r^2 give (v−cs2v)dvdr=2cs2r−GMr2.\left( v - \frac{c_s^2}{v} \right) \frac{dv}{dr} = \frac{2 c_s^2}{r} - \frac{GM}{r^2}. Let rc=GM/(2cs2)r_c = GM/(2 c_s^2) be the critical radius. Then every solution with v>0v > 0 satisfies, for some constant CC, v2cs2−ln⁡v2cs2=4ln⁡rrc+4rcr+C.\frac{v^2}{c_s^2} - \ln \frac{v^2}{c_s^2} = 4 \ln \frac{r}{r_c} + \frac{4 r_c}{r} + C. Exactly two solutions pass through the critical point (r,v)=(rc,cs)(r, v) = (r_c, c_s), both with C=−3C = -3. One of them, the Parker wind, increases with rr: it is subsonic for r<rcr < r_c, supersonic for r>rcr > r_c, and v(r)∼2csln⁡(r/rc)v(r) \sim 2 c_s \sqrt{\ln(r/r_c)} as r→∞r \to \infty.

Remarks

Dividing the momentum equation by vv and using ρ′/ρ=−v′/v−2/r\rho'/\rho = -v'/v - 2/r gives the first equation; separating variables and integrating gives the second. At r=rcr = r_c the right-hand side vanishes, and a smooth solution can change from subsonic to supersonic flow only there, where the factor v−cs2/vv - c_s^2/v also vanishes. Inserting r=rcr = r_c and v=csv = c_s gives 1−0=0+4+C1 - 0 = 0 + 4 + C, so C=−3C = -3. The other transonic solution decreases with rr and describes spherical accretion onto the star.

For the solar corona, with a temperature of about 10610^6 kelvin, rcr_c is a few solar radii. The model then predicts a supersonic wind of some hundreds of kilometres per second near the Earth. Parker argued that a static isothermal atmosphere would keep a finite pressure at infinity, far above the pressure of the interstellar medium. So the corona cannot be static and must expand. In the wind solution, ρ=const/(vr2)\rho = \text{const}/(v r^2) tends to 00, so the pressure tends to 00 at infinity.

Going deeper

For a student

The model is the simplest example of a transonic flow through a critical point, a structure that appears again in Bondi accretion and in the de Laval nozzle. The steady Euler equations become a single ordinary differential equation that is singular at the sonic point, and the physical solution is the one that crosses this singularity smoothly. More realistic models add rotation and magnetic fields, which lead to the Parker spiral and to the magnetic braking of stars.

Reading path

To understand this page from the ground up, read these 47 pages in order. The foundations come first.

  1. Critical point concept
  2. Derivative concept
  3. Differential equation concept
  4. Euclidean space object
  5. Euclidean vector concept
  6. Euler equations concept
  7. Function concept
  8. Interval concept
  9. Ordinary differential equation concept
  10. Speed of sound concept
  11. Velocity concept
  12. Fréchet derivative definition
  13. Cross product definition
  14. Ordinary differential equation definition
  15. Function of several variables concept
  16. Vector-valued function concept
  17. Classical mechanics concept
  18. Lipschitz condition definition
  19. Momentum concept
  20. Limit of a function of several variables concept
  21. Partial derivative concept
  22. Solution of an ordinary differential equation definition
  23. Inertial frame of reference concept
  24. Continuity in R^n definition
  25. Differentiable function of several variables definition
  26. Partial derivative definition
  27. Initial value problem definition
  28. Newton's laws of motion statement
  29. Differentiable on an open set definition
  30. Gradient definition
  31. Higher partial derivatives definition
  32. Picard–Lindelöf hypotheses definition
  33. Iterated derivative definition
  34. Picard operator definition
  35. C^k function characterised by continuous derivatives statement
  36. Integral equation form of an initial value problem statement
  37. An iterate of the Picard operator is a contraction statement
  38. Vector field definition
  39. Picard–Lindelöf theorem statement
  40. Conservative vector field definition
  41. Classical mechanics definition
  42. Momentum definition
  43. Potential energy definition
  44. Angular momentum definition
  45. Central force definition
  46. Newton's law of universal gravitation statement
  47. Parker solar wind model statement

Where this leads

2 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Solar_wind
About this page
TagGMWEA8Q83B
URIm2://GMWEA8Q83B
Kindstatement, proposition
WritingW0 stub
MSC85A30, 76N15, 35Q85
StatusThis copy of the site has no build data. Run m2 status GMWEA8Q83B.
AboutEuler equations
UsesEuler equations, Speed of sound, Newton's law of universal gravitation, Critical point
Changed3 November 2026

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