Polytropic Gas Effects in Parker's Solar Wind Model and Coronal Hole Flows

A detailed and systematic investigation of polytropic gas effects in Parker's solar wind model and coronal-hole flows is given. We present a viable equation governing the acceleration of solar wind of a polytropic gas and give its exact analytical and numerical solutions and deduce its asymptotic analytic properties (i) near the sun, (ii) far away from the sun, (iii) near the Parker sonic critical point (where the wind speed is equal to the speed of sound in the wind). We proceed to give a detailed and systematic investigation of coronal-hole polytropic gas outflows which contribute to bulk of the solar wind. We will model coronal-hole outflow by considering a single radial stream tube and invoke phenomenological considerations to represent its rapidly-diverging flow geometry. We give analytical and numerical solutions for this outflow and deduce its asymptotic analytic properties in the three flow regimes above. We find that, in general, the polytropic effects cause the Parker sonic critical point to move closer to the sun than that for the case with isothermal gas. Furthermore, the flow acceleration is found to exhibit (even for an infinitesimal deviation from isothermality of the gas) a power-law behavior rather than an exponential-law behavior near the sun or a logarithmic-law behavior far away from the sun, thus implying a certain robustness of the power-law behavior. The Parker sonic critical point is shown to continue to be of X-type, hence facilitating a smooth transition from subsonic to supersonic wind flow through the transonic regime. Our analytical and numerical solutions for coronal-hole outflows show that the super-radiality of the stream tube causes the Parker sonic critical point to move further down in the corona, and the gas to become more diabatic (the polytropic exponent $\gamma$ drops further below 5/3), and the flow acceleration to be enhanced further.