00100	\input basic
00200	\ctrline{\bf Exact expressions for ground range and its derivative}
00300	\ctrline{\bf with respect to elevation angle for a}
00400	\ctrline{\bf Bradley-Dudeney model ionosphere}
00500	\vskip 5pt
00600	\ctrline {\sl L. Ortenburger}
00700	\vskip 10pt
00800	 
00900	By using a quasi-parabolic and quasi-linear approximation to the
01000	variation of electron concentration with height in the
01100	Bradley-Dudeney$↑1$
01200	ionospheric model,
01300	Milson$↑2$ obtained the exact expressions for ground range as a
01400	function of the wave frequency; elevation angle; the F2 region
01500	parameters of maximum electron concentration $(N↓mF2)$, height of
01600	this maximum concentration $(h↓mE)$, and the region (parabolic)
01700	semi-thickness $(y↓mE)$.
01800	
01900	These were derived from the basic ground range equation for a curved
02000	earth-curved ionosphere geometry in which the effects of electron
02100	collisions, positive ions, and the magnetic field are ignored:
     

00100	
00200	$$D=2r↓0\int↓{r↓0}↑{r↓t}{dr \over r\, tan\beta}
00300	=2\int↑{r↓t}↓{r↓0}{r↓0↑2 cos\beta↓0\,dr \over
00400	r\sqrt{r↑2\mu↑2(r)-r↑2↓0cos↑2\beta↓0}}$$
00500	where
00510	$$\eqalign{D ⊗= ground\  range\  (km)\cr
00520	           r↓0 ⊗= radius\ of\ the\ earth\ (km)\cr
00530	           r↓t ⊗= radial\ distance\ to\ the\ ray\ apogee\ (km)
00535	\cr
00540	           \beta ⊗= elevation angle\cr
00550	           \beta ↓0 ⊗= initial angle of the ray at the earth surface\cr
00560	           \mu(r) ⊗= refractive index at a radius of r km\cr}$$
00570	
00580	Milson defined four sections of the ray-path from the ground to the
00590	apogee point:
00600	
00700	\vfill\end