hydrostatic corona (inconsistency)
Definition: The unphysical result that a sufficiently hot static corona retains finite pressure at very large radius instead of matching interstellar conditions.
Why it matters here: A naive hydrostatic corona (using Lecture 01’s scale-height framework at the corona’s real temperature, T=1.5e+06 K) predicts a finite, non-zero gas pressure even at an infinite distance from the Sun -- an unphysical result, since the corona must eventually meet the near-vacuum of interstellar space, and this contradiction is precisely what led Eugene Parker in 1958 to propose that the corona instead flows outward continuously as a wind, reaching supersonic speeds well before 1 AU.
Relationship: Use this term alongside Parker solar wind; the lecture's quantitative model is summarized below.
\[ u^2 - \ln(u^2) = 4\ln\!\left(\frac{r}{r_c}\right) + \frac{4 r_c}{r} - 3, \qquad r_c = \frac{GM_\odot}{2c_s^2},\ u \equiv \frac{v}{c_s} \]
Boundary or common confusion: Assuming the Parker wind equation has a simple algebraic closed-form solution for v(r) at all radii. Only the critical point itself (r=r_c, v=c_s) has a simple closed form; the full transonic profile requires numerically solving a transcendental equation (as this lecture’s figure does), and naively rearranging the equation algebraically for v(r) is not possible in general.
Parker solar wind
Definition: The steady hydrodynamic outflow obtained when a hot corona passes smoothly from subsonic to supersonic speed.
Why it matters here: Parker Solar Probe (Unit 5) has directly sampled the solar wind acceleration region within a few tens of solar radii, providing direct in-situ tests of the transonic wind theory this lecture derives.
Relationship: Use this term alongside critical point; the lecture's quantitative model is summarized below.
\[ u^2 - \ln(u^2) = 4\ln\!\left(\frac{r}{r_c}\right) + \frac{4 r_c}{r} - 3, \qquad r_c = \frac{GM_\odot}{2c_s^2},\ u \equiv \frac{v}{c_s} \]
Boundary or common confusion: Assuming the Parker wind equation has a simple algebraic closed-form solution for v(r) at all radii. Only the critical point itself (r=r_c, v=c_s) has a simple closed form; the full transonic profile requires numerically solving a transcendental equation (as this lecture’s figure does), and naively rearranging the equation algebraically for v(r) is not possible in general.
critical point
Definition: The radius where the Parker wind speed equals the sound speed and the differential equation becomes singular unless its numerator also vanishes.
Why it matters here: Parker (1958) instead solved the steady-state, spherically symmetric momentum equation for an isothermal outflow, finding that physically acceptable solutions exist only along specific branches of a family of curves, with the unique physically realized solution (matching low speed near the Sun and continuing to accelerate outward, becoming supersonic) passing through a critical point at r_c=GM_sun/(2 c_s^2), where the flow speed equals the isothermal sound speed c_s.
Relationship: Use this term alongside transonic solution; the lecture's quantitative model is summarized below.
\[ u^2 - \ln(u^2) = 4\ln\!\left(\frac{r}{r_c}\right) + \frac{4 r_c}{r} - 3, \qquad r_c = \frac{GM_\odot}{2c_s^2},\ u \equiv \frac{v}{c_s} \]
Boundary or common confusion: Assuming the Parker wind equation has a simple algebraic closed-form solution for v(r) at all radii. Only the critical point itself (r=r_c, v=c_s) has a simple closed form; the full transonic profile requires numerically solving a transcendental equation (as this lecture’s figure does), and naively rearranging the equation algebraically for v(r) is not possible in general.
transonic solution
Definition: The unique physically continuous wind branch that crosses the critical point from subsonic to supersonic flow.
Why it matters here: The full nonlinear equation for this transonic solution, u^2 - ln(u^2) = 4 ln(r/r_c) + 4 r_c/r - 3 (u=v/c_s), has no closed-form algebraic solution for v(r) and must be solved numerically (root-finding on the correct physical branch) at each radius -- exactly the numerical approach this lecture’s worked example and figure use, rather than an algebraic shortcut.
Relationship: Use this term alongside isothermal sound speed; the lecture's quantitative model is summarized below.
\[ u^2 - \ln(u^2) = 4\ln\!\left(\frac{r}{r_c}\right) + \frac{4 r_c}{r} - 3, \qquad r_c = \frac{GM_\odot}{2c_s^2},\ u \equiv \frac{v}{c_s} \]
Boundary or common confusion: Assuming the Parker wind equation has a simple algebraic closed-form solution for v(r) at all radii. Only the critical point itself (r=r_c, v=c_s) has a simple closed form; the full transonic profile requires numerically solving a transcendental equation (as this lecture’s figure does), and naively rearranging the equation algebraically for v(r) is not possible in general.
isothermal sound speed
Definition: The characteristic pressure-wave speed c_s equal to the square root of kT divided by mean-particle mass for an isothermal gas.
Why it matters here: Parker (1958) instead solved the steady-state, spherically symmetric momentum equation for an isothermal outflow, finding that physically acceptable solutions exist only along specific branches of a family of curves, with the unique physically realized solution (matching low speed near the Sun and continuing to accelerate outward, becoming supersonic) passing through a critical point at r_c=GM_sun/(2 c_s^2), where the flow speed equals the isothermal sound speed c_s.
Relationship: Use this term alongside terminal wind speed; the lecture's quantitative model is summarized below.
\[ u^2 - \ln(u^2) = 4\ln\!\left(\frac{r}{r_c}\right) + \frac{4 r_c}{r} - 3, \qquad r_c = \frac{GM_\odot}{2c_s^2},\ u \equiv \frac{v}{c_s} \]
Boundary or common confusion: Assuming the Parker wind equation has a simple algebraic closed-form solution for v(r) at all radii. Only the critical point itself (r=r_c, v=c_s) has a simple closed form; the full transonic profile requires numerically solving a transcendental equation (as this lecture’s figure does), and naively rearranging the equation algebraically for v(r) is not possible in general.
terminal wind speed
Definition: The asymptotic or far-heliosphere bulk speed approached by a solar-wind stream, not a universal constant for all streams.
Why it matters here: A naive hydrostatic corona (using Lecture 01’s scale-height framework at the corona’s real temperature, T=1.5e+06 K) predicts a finite, non-zero gas pressure even at an infinite distance from the Sun -- an unphysical result, since the corona must eventually meet the near-vacuum of interstellar space, and this contradiction is precisely what led Eugene Parker in 1958 to propose that the corona instead flows outward continuously as a wind, reaching supersonic speeds well before 1 AU.
Relationship: Use this term alongside hydrostatic corona (inconsistency); the lecture's quantitative model is summarized below.
\[ u^2 - \ln(u^2) = 4\ln\!\left(\frac{r}{r_c}\right) + \frac{4 r_c}{r} - 3, \qquad r_c = \frac{GM_\odot}{2c_s^2},\ u \equiv \frac{v}{c_s} \]
Boundary or common confusion: Assuming the Parker wind equation has a simple algebraic closed-form solution for v(r) at all radii. Only the critical point itself (r=r_c, v=c_s) has a simple closed form; the full transonic profile requires numerically solving a transcendental equation (as this lecture’s figure does), and naively rearranging the equation algebraically for v(r) is not possible in general.