Please advise formula for calculating rising/dipping ranges if tables are not available or the height of light is beyond the scope of the tables.

Question

Please advise formula for calculating rising/dipping ranges if tables are not available or the height of light is beyond the scope of the tables.

ie Ht of light 155m/ ht of eye 3m

Thanks

Mike

Answer 1

If you assume a spherical Earth, and if you assume a true horizon without refraction, then the problem boils down to what is the horizon distance for each of the altitudes (eye and light). The maximum range is when the horizons for each is tangent, ie,the horizon range for the eye adds to the horizon range for the light.

Here is the development for horizon range for a satellite. If you add a second point, the eye, and put the horizon circles tangent, you have the eye-light range.

See equations 9.2 and 9.8.

http://en.wikipedia.org/wiki/Belfast_Peace_Lines

So, for the eye to light, range is given by

range=eyehorizon + lighthorizon
=Re/tanRhoeye + Re/tanRoelight

Where the angles are defined by

sinRho=Re/(Re+H) equation 9.2

Now, the radius of earth as a variation with latitude, for this I would ignore it, as the assumption of no refraction will cause greater error. Refraction bends the light around the surface, giving greater range. Refraction is largely unpredictable, due to humidity and temperature gradients above the Sea Surface. The greatest refraction I have seen was in springtime, in the Carribean. This is in my mind the cause of the legendary "ghost ships"appearing and disappearing. If you want to include Re as a variation of latitude, it is easily done with the Cosine function of latitude (but it is just clutter in the calculations).

Answer 2

Opps, wrong link. Sorry.

http://books.google.com/books?id=HfnGxKhcQWUC&pg=PA418&lpg=PA418&dq=spherical+geometry+horizon&source=web&ots=lFC48ZfxZg&sig=iRpF0g3nZCxsSilom9iZIh2X5mc

Answer 3

Thanks Bob,

Got it sorted.

2.08 x Sq root of the height gives horizon in NM.

Answer in my question is 29.5M

Mike

Answer 4

To calculate the rising or dipping ranges when tables are not available or when the height of the light is beyond the scope of the tables, you can use the formula:

Range (in nautical miles) = 1.17 * √(Ht of light + Ht of eye)

Here's how you can calculate the rising/dipping ranges using this formula, given the following values:

Ht of light = 155m (height of the light in meters)
Ht of eye = 3m (height of the eye in meters)

Step 1: Convert the height of the light and eye from meters to nautical miles:
Using the conversion factor 1 nautical mile = 1852 meters, divide the height of the light and eye by 1852:
Ht of light = 155m / 1852 = 0.0837 nautical miles
Ht of eye = 3m / 1852 = 0.0016 nautical miles

Step 2: Apply the formula to calculate the range:
Range = 1.17 * √(0.0837 + 0.0016) ≈ 1.17 * √0.0853 ≈ 1.17 * 0.2923 ≈ 0.342 nautical miles

So, the calculated rising/dipping range is approximately 0.342 nautical miles.