Earth Surface Distance Calculations

Overview

The lecture covers calculating distances between two points on the Earth's surface at the same latitude, focusing on departure and related formulas.

Great Circles and Small Circles

• Meridians (lines of constant longitude) and the equator are great circles on Earth.
• All other lines of latitude (parallels) are small circles, except the equator.
• One minute of latitude always equals one nautical mile, everywhere on Earth.
• One minute of longitude equals one nautical mile only at the equator.

Departure: Definition and Formula

• Departure is the distance between two meridians measured along a parallel of latitude.
• Departure is always measured along a rhumb line (line of constant heading) on a parallel.
• Maximum departure occurs at the equator; it decreases to zero at the poles.
• The formula for departure:
  Departure = change in longitude (minutes) × cosine(latitude).
• Cosine values decrease from 1 at 0° to 0 at 90° latitude.
• Cosine key values:
  0° = 1, 30° = 0.86, 45° = 0.7071, 60° = 0.5, 75° = 0.259, 90° = 0.

Worked Examples

• To solve for departure, convert longitude change to minutes and multiply by cos(latitude).
• To find position after eastward/westward travel:
  1. Convert distance to nautical miles if needed (1 NM = 1.852 km).
  2. Rearrange the departure formula to solve for the unknown (longitude, latitude, or departure).
  3. For example, after flying 165 km east at 60°N, the new longitude is calculated using the formula and cosine value.
• To find latitude for a given departure and longitude change:
  1. Convert longitude change to minutes.
  2. Rearrange formula to solve for latitude using the inverse cosine.

Useful Ratio Formula

• When comparing departures at different latitudes:
  Departure at latitude A / Departure at latitude B = cos(latitude A) / cos(latitude B).
• This ratio formula speeds up calculations when departure values at different latitudes are related.

Key Terms & Definitions

• Great Circle — the largest possible circle that can be drawn on a sphere, dividing it into two equal halves.
• Small Circle — any circle on the sphere's surface that is not a great circle (e.g., most parallels of latitude).
• Departure — distance between two meridians measured along a parallel of latitude.
• Rhumb Line — a line crossing all meridians at the same angle, representing a constant compass direction.

Action Items / Next Steps

• Practice converting between kilometers and nautical miles.
• Solve example problems using the departure formula.
• Memorize key cosine values for standard angles.
• Review the new ratio formula for quick departure comparisons.