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# Calculating the Differential Equations

The basic equations for the gas are simple. The equation of motion is

With the self-similar functions (), this becomes

In keeping with the expected scaling law, Taylor lets

which gives an equation for A:

This is the first of the final equations of motion.

The second equation comes from the continuity equation, expressing mass conservation,

or

The final equation derives from the equation of state. At this point, a further approximation is needed; the gas must have a polytropic equation of state, . This approximation is often very accurate. The number is the ratio of specific heats of the gas, , and it represents the number n of degrees of freedom available, according to . For a monoatomic gas, there are only the three translational degrees of freedom, so . For a diatomic gas, there are two additional rotation axes with nonzero moments of inertia, so n=5, and . For air, which is 99% diatomic (with N and O ), the measured value of is 1.40. However, there are other values of interest. The interstellar medium contains mostly atomic hydrogen gas, H , so the monoatomic value of is more useful there. (In fact, Taylor did numerical calculations for both these values, as well as for and )

For whatever value of is appropriate, the equation of state may be written in the alternative form that Taylor uses

With the equations above, this can be written in the (rather more complicated) form

This is the last of the three dynamical equations.

Taylor's last step in analyzing these equations is to nondimensionalize and , using the resting sound speed . The dimensionless functions f and are then

This substitution entirely eliminates A from the equations of motion. (If A did not fall out, it would imply the existence of some other dimensional quantity in the problem.) The nondimensional equations of motion finally become

This system of differential equations can be solved by an ordinary procedure, steppping through values of in either direction.

Next: Solving the Differential Equations Up: The formation of a Previous: The Basic Scaling Argument

brenner@math.mit.edu