x = r / r_{s}

and

y = u / c_{s}

, mass conservation and part (c) give

\frac{ρ}{ρ _{0}} = \frac{e ^{3/2}}{x ^{2} y} .

Substitution into the Bernoulli function, together with

GM / r = 2 c_{s}^{2} / x

, eliminates every dimensional parameter and gives the transcendental equation

\frac{y ^{2}}{2} - ln y - 2 ln x + \frac{3}{2} - \frac{2}{x} = 0 .

Equivalently,

y^{2} - ln (y^{2}) = \frac{4}{x} + 4 ln x - 3,

y^{2} e^{- y^{2}} = x^{- 4} e^{3 - 4/ x} .

The transonic branch passes through

(x, y) = (1, 1)

 Ancestors (10)

 Discussion (0)

There are no discussions about this article yet.

There are currently no matching articles.