Martin, Euclid was pretty good with that geometry stuff but he was weak on refraction. The rule of thumb I stated takes refraction into account, which is important to observers near the Earth's surface. Hence the factor of 3.84 rather than the value of 3.56 that Euclid would have predicted. For radar ranges to the horizon, add another 2 % to 3 %, especially for the lower frequency radars.
This is of course of great interest to celestial navigators, who can even fudge it a bit more based on extreme weather conditions. There are look-up tables available to add an additional correction to sextant readings that are based on barometric pressure and temperature. Of course, navigators add the range to the horizon based on their height of eye as well when calculating visibility ranges to lights. The two tangents to Earth's surface are co-linear. If one is calculating the expected sighting of a low light both distances to the horizon are important. In the case of lightning, where the upper portion is at least a few kilometers above the Earth's surface, one may omit the few kilometers the observer's horizon is from him or her. In fact, at that great distance above the Earth's surface, the geometry and the allowance for refraction break down a bit. The rule of thumb is just for estimating, of course, which is why I didn't mind rounding the factor off to 4. How well can the observer determine the height of the bolt? I suggest Dutton's _Navigation_and_Piloting_ and Bowditch's _American_Practical_Navigator_ as references. A more terse explanation can be found in the _Nautical_Almanac_ each year, which also has the corrections to sextant readings for extreme weather conditions. Bowditch can now be downloaded for free online; at least it was available at one time and I suppose it still is. As one who has done and who has taught celestial navigation I find it quite interesting. A really nifty technique is that of shooting an amplitude of the Sun and that depends on the refraction correction. It's a great way to do a quick calculation for a compass check, though! And of course, refraction lets us see the Sun before it actually rises above the geometric horizon and for a few moments after it has set below the geometric horizon. I fondly recall using metric navigation charts in the '70s, by the way. So, there. We're still on topic. Jim On Thursday 22 December 2005 16:20, Martin Vlietstra wrote: > Just to be pedantic, I calculated the rule of thumb as being 3.56*sqrt(h). > > It is quite easy to demonstrate using Euclidean Geometry: > > Consider a circle and a point X that lies outside the circle. > Construct a line XAB such that AB is the diameter of the circle. > Construct a line XC such that XC is a tangent to the circle. > There is a theorem in Euclidean Geometry that states: > XC^2 = XA*XB > > If XA is small compared to AB, then we can replace XB by AB. > Set XB to the Earth's diameter (40000/pi) > Set XA to the height of the observation station (or object) above the > Earth's surface > Solving for XC gives > XC = sqrt((XA/1000) * 40000/pi) (the factor of 1000 is to convert metres > to km) > Simplifying gives XC=3.56*sqrt(XA) > > It should be noted of course that the Earth's Circumference is as close as > makes no difference 40,000km as everybody who knows anything about the > metric system should know.. > > ----- Original Message ----- > From: "James R. Frysinger" <[EMAIL PROTECTED]> > To: "U.S. Metric Association" <[email protected]> > Sent: Thursday, December 22, 2005 5:05 PM > Subject: [USMA:35440] Re: Lightning and thunder > > > On Wednesday 21 December 2005 15:42, Pat Naughtin wrote: > > > Dear All, > > > > .... > > > > > >From what distance can lightning be seen? > > > > A rough rule of thumb for estimating distance to the horizon for visible > > light is based on the height of the source. If that is given in meters as > > h, > > > then the distance d is given in kilometers by d=3.84sqrt(h). -- James R. Frysinger Lifetime Certified Advanced Metrication Specialist Senior Member, IEEE http://www.cofc.edu/~frysingj [EMAIL PROTECTED] [EMAIL PROTECTED] Office: Physics Lab Manager, Lecturer Dept. of Physics and Astronomy University/College of Charleston 66 George Street Charleston, SC 29424 843.953.7644 (phone) 843.953.4824 (FAX) Home: 10 Captiva Row Charleston, SC 29407 843.225.0805
