ATPL General Navigation Comprehensive Study Notes

THE EARTH: FORM AND GEODETICS

• The Geoid:

  • Defined as the model of the actual shape and size of Earth corresponding with mean sea level (MSL) over oceans, continuing through continents as an imaginary surface defined by spirit level.

  • Topographic heights and ocean depths are measured against this reference.

  • It is irregular, influenced primarily by gravity and centrifugal force.

  • Geodesy: The scientific discipline studying the precise figure of Earth.

• Ellipsoids/Spheroids:

  • Earth is an oblate spheroid (slightly flattened at the poles).

  • Flattening (compression) is approx. $0.3\%$ ($\frac{1}{300}$).

  • Polar Diameter: Approx. $23\text{ NM}$ ($43\text{ km}$) less than the equatorial diameter.

  • WGS-84 (World Geodetic System 1984): The adopted ICAO world standard reference ellipsoid required for geographical coordinates due to Global Positioning System (GPS) and Flight Management System (FMS) accuracy requirements.

  • Circumference: For EASA ATPL calculations, Earth is considered a true sphere with a circumference of approximately $40,000\text{ km}$ or $21,600\text{ NM}$.

EARTH ROTATION AND SEASONS

• Spin Axis: Earth rotates on its axis once a day around the poles. The axis is inclined at $66.5^{\circ}$ to the orbital plane (Plane of the Ecliptic), or $23.5^{\circ}$ to the normal of that plane.

• Plane of the Ecliptic: The orbital plane on which Earth (and other planets) orbits the Sun once per year.

• Seasons: Caused by the tilt of Earth’s axis relative to its orbit.

  • Declination of the Sun: The angle of the Sun above or below the equator. It cycles between $23.5^{\circ}\text{N}$ (Tropic of Cancer) and $23.5^{\circ}\text{S}$ (Tropic of Capricorn).

  • Length of Daylight: Varies with declination. The rate of change is greatest during the equinoxes (March 21 and Sept 21).

POSITION REFERENCE SYSTEMS

• Directional Datums:

  • East: Direction of Earth's spin.

  • North: Direction toward the North Pole when facing East.

  • Sexagesimal System: Measuring directions in $360^{\circ}$.

• Circles on Earth:

  • Great Circle: A circle whose center and radius are those of Earth itself. The shortest distance between two points is the shorter arc of a great circle.

  • Small Circle: A circle whose center and radius are not those of Earth (e.g., parallels of latitude other than the equator).

  • Equator: The great circle at $90^{\circ}$ to the polar axis; the datum for latitude ($0^{\circ}\text{N/S}$).

  • Meridians: Semi-great circles joining the North and South Poles ($0^{\circ}$ to $180^{\circ}$).

  • Prime Meridian: The meridian through Greenwich ($0^{\circ}\text{E/W}$), the datum for longitude.

  • Graticule: The network of meridians and parallels on a map or globe.

• Latitude and Longitude Definitions:

  • Geodetic (Geographic) Latitude: The smaller angle between the normal ($90^{\circ}$) to the meridian at a point and the plane of the equator. Distorted because Earth is an oblate spheroid.

  • Geocentric Latitude: The angle based on the center of the Earth.

  • Longitude: The shorter distance in arc along the equator between the Prime Meridian and the meridian through the point.

  • Calculation of Difference:

    • Same side: Subtract ($100^{\circ}\text{W} - 80^{\circ}\text{W} = 20^{\circ}$).

    • Opposite sides: Add ($20^{\circ}\text{W} + 10^{\circ}\text{E} = 30^{\circ}$).

    • Across the Anti-meridian: Subtract the sum from $360^{\circ}$ to find the shortest arc.

DIRECTION: DATUMS, VARIATION, AND DEVIATION

• True North (TN): The direction to the geographical North Pole. Directions are suffixed with 'T'.

• Magnetic North (MN): The direction indicated by a freely suspended magnet influenced by Earth's field. Suffixed with 'M'.

• Variation (Var): The angular difference between TN and MN.

  • Isogonal: Line joining points of equal variation.

  • Agonic Line: Line of zero variation.

  • Maximum Variation: $180^{\circ}$ at the poles.

  • Changes Over Time: Variation changes due to secular (long-term), annual, and diurnal (daily) cycles, and solar activity (magnetic storms).

• Compass North (CN): The direction indicated by an aircraft's compass, affected by aircraft magnetism. Suffixed with 'C'.

• Deviation (Dev): The angle between MN and CN.

  • Mnemonic: "West is Best" (add West), "East is Least" (subtract East) when going from Magnetic to Compass.

• Conversion Rules:

  • $T \pm Var = M \pm Dev = C$

  • $(C \rightarrow M \rightarrow T):$ East is $(+)$, West is $(-)$.

TRACK, HEADING, AND THE 1 IN 60 RULE

• Heading (Hdg): The direction the aircraft nose is pointing.

• Track (Tr): The actual path over the ground.

• Drift: The difference between Heading and Track. (Aircraft drifts from Heading to Track).

• Wind Correction Angle (WCA): The angle applied to Track to find Heading. Equal to Drift but opposite sign.

• The 1 in 60 Rule: A rule of thumb where $1 \text{ unit off}$ over $60 \text{ units forward}$ equals a $1^{\circ}$ angle.

  • Formula: $\text{Track Error Angle (TEA)} = \frac{\text{Distance Off}}{\text{Distance Flown}} \times 60$

  • Formula: $\text{Closing Angle (CA)} = \frac{\text{Distance Off}}{\text{Distance to Go}} \times 60$

  • Correction to next waypoint: Total Hdg change = $\text{TEA} + \text{CA}$.

  • Double Track Error Method: Turn toward track by twice the TEA to regain track in the same time it took to get off track.

DISTANCE CALCULATIONS

• Units:

  • $1\text{ NM} = 1.852\text{ km} = 6,080\text{ ft}$.

  • $1\text{ Statute Mile (SM)} = 5,280\text{ ft}$.

  • $1\text{ km} = 1,000\text{ metres} = 3,280\text{ ft}$.

• Latitude/Longitude Distance:

  • $1' \text{ of Latitude} = 1\text{ NM}$.

  • $1^{\circ} \text{ of Latitude} = 60\text{ NM}$.

  • $1' \text{ of Longitude} = 1\text{ NM}$ at the equator only.

• Departure: The distance in NM between two meridians at a specific latitude. Formula: $\text{Departure (NM)} = \text{Change of Longitude (mins)} \times \cos(\text{Latitude})$.

TRUE AIRSPEED (TAS) AND SPEED CALCULATIONS

• Airspeed Definitions:

  • IAS (Indicated): Read from the instrument.

  • CAS (Calibrated): IAS corrected for instrument and position error.

  • TAS (True): CAS corrected for density and compressibility.

• Density Correction: A rule of thumb is to add $2\%$ to CAS for every $1,000\text{ ft}$ of altitude to find TAS.

• Mach Number (M): The ratio of TAS to the Local Speed of Sound (LSS).

  • Formula: $M = \frac{\text{TAS}}{\text{LSS}}$.

  • Formula: $\text{LSS} = 38.95 \times \sqrt{T (\text{Kelvin})}$.

• TAS/GS/Wind Relationship:

  • $\text{Ground Speed (GS)} = \text{True Airspeed (TAS)} \pm \text{Wind Component}$.

  • $\text{Nautical Air Miles (NAM)} = \text{Distance (NM)} \times \frac{\text{TAS}}{\text{GS}}$.

TRIANGLE OF VELOCITIES (TOV)

• Components:

  • Air Vector: Heading ($1 \text{ arrow}$) and TAS.

  • Wind Vector: Direction ($3 \text{ arrows}$) and Speed (blown from).

  • Ground Vector: Track ($2 \text{ arrows}$) and Ground Speed (GS).

• Solution Methods: Mechanical Computer (CRP-5) or Scale Drawing.

VERTICAL NAVIGATION: GRADIENT AND RATE OF CLIMB/DESCENT

• Climb Gradient: Vertical distance divided by horizontal distance (percentage or degrees).

  • $\text{Gradient (\%)} = \frac{\text{Vertical Distance (ft)}}{60 \times \text{Ground Distance (NM)}}$.

  • $\text{Gradient (}^{\circ}) = \frac{\text{Vertical Distance (ft)}}{100 \times \text{Ground Distance (NM)}}$.

• Rate of Descent (ROD):

  • $\text{ROD (ft/min)} = \text{Glide Path Angle (}^{\circ}) \times \text{GS (NM/min)} \times 100$.

  • $\text{ROD (ft/min)} = \text{Gradient (\%)} \times \text{GS (kt)}$.

  • For a $3^{\circ} \text{ glide path}: \text{ROD} \approx 5 \times \text{GS (kt)}$.

VFR NAVIGATION TECHNIQUES

• Checkpoints: Ideal points are large, unique, have vertical extent, and provide contrast.

• Map Reading: Orientation of the map (direction of flight up) is essential.

• Lost Procedure:

  1. Aviate, Navigate, Communicate.

  2. Check Hdg/Speed.

  3. Climb for better horizon.

  4. Dead Reckon (DR) from last known fix.

  5. Circle of Uncertainty ($10\%$ of distance flown since fix).

  6. Fly toward a line feature (coast/motorway).

CHART PROJECTIONS

• Direct Mercator:

  • Cylindrical projection touching at equator.

  • Properties: Rhumb lines are straight; Great circles are curved (concave to equator); Scale expands as $\sec(\text{Lat})$.

  • Correct convergence only at the equator (Chart convergence = $0$).

• Lambert's Conformal Conic:

  • Conical projection crossing Earth at two Standard Parallels.

  • Properties: Great circles are approximated by straight lines; Scale correct at standard parallels, smallest at the parallel of origin.

  • Constant of the Cone ($n$): $\sin(\text{Parallel of Origin})$.

  • $\text{Chart Convergence} = \text{Change of Longitude} \times n$.

• Polar Stereographic:

  • Plane (Azimuthal) projection touching at a pole.

  • Properties: Meridians are straight lines radiating from the pole; Parallels are concentric circles; Scale expands away from the pole.

  • $n = 1$ (Chart convergence = Change of Longitude).

TIME

• Days:

  • Mean Solar Day: Average length of apparent solar day, constant $24 \text{ hrs}$.

  • Sidereal Day: Measured against stars, approx. $23 \text{ hrs } 56 \text{ mins}$.

• Local Mean Time (LMT): Time based on the Sun's transit of the local meridian.

• UTC (Coordinated Universal Time): LMT at the Greenwich Meridian.

• Conversion: $15^{\circ} \text{ Longitude} = 1 \text{ hour}$; $1^{\circ} \text{ Longitude} = 4 \text{ minutes}$.

  • "Longitude East, UTC Least; Longitude West, UTC Best."

• International Date Line: Boundary separating calendar dates ($180^{\circ} \text{ meridian}$). Cross West to gain a day.

• Twilight: Period of illumination before sunrise/after sunset.

  • Civil Twilight: Sun between $0^{\circ}50'$ and $6^{\circ}$ below the sensible horizon.