Celestial Navigation Intro

Alright, so here's a basic rundown of Celestial Navigation, hopefully shining a little light on some of the terminology used and frames of reference that may be discussed:

Basic Terms (starting at the most base):

Galaxy: a collection of stars

Planets: in Cel Nav, primarily Venus, Mars, Jupiter, and Saturn are used

Astronomical Unit (AU) = the average distance from the Earth to the Sun (~93 million miles)

Light Year = the distance light travels in 1 year (5.87 x 10^12 miles, 63,280 AU)

Reflection: We'll primarily be discussing reflection with reference to the Angle of Incidence, or the angle at which light strikes Êa surface and is then reflected from it (Angle of Return). (And just like in a game of billiards [and ignoring any English on the ball]), Angle of Incidence = Angle of Return

Refraction: Change in direction of light traveling from one medium to another (ex. a straw appears to bend when placed partially submerged in water)

Refraction occurs least at the Zenith, and is greatest at the horizon

Parallax: Apparent shift in position of an object when viewed against a background due to different viewing positions of the observer (ex. hold a finger out at arm's length and using only your left eye, focus on an object beyond your finger. When you switch to your right eye, the object appears to have moved).

Parsec: The distance at which a body, when viewed first from Earth then from the sun will appear to have shifted by 1" of arc

Orbit: One revolution around a body, and in the instance of celestial bodies in motion, the orbital path is typically not a perfect circle, but more ellipsoidal

Tilt: The earth is revolving around the sun, but the Earth's North-South axis is tilted 23.5° from it's orbital path (hence the observed seasons)

Time:

For the most part, Arc and Time are interchangeable
Solar Day: one rotation of the earth with respect to the sun (24 hours)
Lunar Day: one lunar revolution with respect to the Earth (24h 50m)
Sidereal Day: One earthly rotation with respect to the stars (23h 56m). The difference between the solar day and the sidereal day is due to the Earth's revolution around the sun, which places the Earth in a different relative location to distant stars at each instant, and hence, the time required to find the stars in the same apparent location is different.
Apparent time: the time as based on your local meridian
Mean time: the time as based on the passage of the mean sun across your meridian
- Mean and apparent time may differ by as much as 16min
Equation of Time (EQT)= Apparent - Mean (found in the Nautical Almanac)
- Mean Sun Lags = Positive EQT
- Mean Sun Leads = Negative EQT
Atomic Time: Based on the vibrations of cesium atoms
- UTC: Universal time constant
- UT1: Similar to UTC, but adjusted for the irregular motion of the Earth
  - When UTC differs from UT1 by >0.9seconds, a leap second is applied to UTC
  - Your GPS uses atomic time which is not quite equal to UTC or UT1
Chronometer
- Chronometer Error (CE): A well made chronometer will have an error, but the error will be constant or will be increasing/decreasing at a steady rate
- Chronometer Rate: Daily change in Chronometer Error (CE)
- Time Tick: A signal broadcast by the National Institute of Standards and Technology (NIST) for the purpose of determining chronometer error by comparing an observed time to a known time

Precession and Nutation: The Earth wobbles on its axis due to the gravity of other celestial bodies like the moon.
- Precession is not uniform due to the varying positions of the moon (nutation).

Celestial Sphere:
- For navigational purposes, it is assumed that the Earth is in a perfect orbital path at the center of the universe (otherwise there would be so many minute corrections that we'd never figure out anything)
- Equinoctial: the "Celestial Equator," or the Earth's equator projected onto the celestial sphere.
- Geographic Position: The point on the Earth's surface which can be assumed to be directly below an object on the Celestial Sphere. In other words, if you drew a straight line from the Earth's core to the celestial object, it's Geographic Position would be the point at which your line intersected the Earth's surface.

Declination: Angular distance of an object North or South of the Celestial Equator

Ecliptic: The path that the sun appears to trace on the Celestial Sphere over a year.
- The sun cross the celestial equator at Equinoxes and is farthest from the equator during Solstices. The sun's maximum declination is 23.5° (equal to the tilt of the Earth along it's orbital path)

Diurnal Circles: The apparent daily paths of bodies in the sky, centered about Earth's axis/poles

There are 57 "Navigational Stars," and these are the ones you'll find listed on the daily pages in the Nautical Almanac. There are many more that you'll see, some of which can be used, but due to factors like their diminished magnitude, they aren't used in calculations quite as frequently.

Magnitude: The apparent brightness of a celestial body (as compared to others)
- A lower number = Brighter/Better for navigational purposes
- Each step in the scale is a multiplier of 2.5 (like Decibels or the Richter scale). So a magnitude 1 star would be 2.5 times brighter than a magnitude 2, and 6.25 times brighter than a magnitude 3.

Semi-Diameter: Radius of a body as viewed from Earth, described in minutes of arc.
- Typically used in determining error as it is difficult to determine the exact center of celestial objects like the Moon, which though they aren't very big, are very close, making them appear larger

First Point of Aries (♈): The point at which the sun is at its vernal equinox (the point at which it crosses the Celestial Equator in the spring)
- This is important because this point serves as a stationary reference in an orbiting universe, and many angles use this as a reference point

Celestial Coordinate Systems
- Terrestrial systems use Latitude (L) and Longitude (&lambda)
- For celestial navigation, we add 2 more systems
- Celestial Equator System:
  - Based on the Celestial Equator as the Primary Great Circle
  - Earth is assumed to be fixed at the center of the universe, and rotates from East to West
  - Celestial Poles: The earth's north-south axis extends to intersect the celestial sphere
  - Celestial Meridian: The Earth's meridians projected onto the celestial sphere
    - Greenwich Celestial Meridian is still the primary meridian, just like in terrestrial navigation
  - Hour Circle: The great circle which begins and ends at the poles, but passes through a celestial object
  - Declination: Angular distance of an object North or South of the equator
  - Hour Angle: The angular distance measured WEST from the celestial meridian and the hour circle of a body.
    - Greenwich Hour Angle (GHA): The angular distance between the Greenwich celestial meridian and hour circle of a body(similar to terrestrial longitude).
    - Local Hour Angle (LHA): The angular distance between the observer's celestial meridian and the hour circle of a celestial body
      - LHA = GHA + (East Longitude of Observer)
      - LHA = GHA - (West Longitude of Observer)
    - Meridian Angle (&tau): The angular distance East or West between the observer's celestial meridian and the hour circle of the celestial body
    - Sidereal Hour Angle (SHA): The angular distance between the hour circle of the First Point Of Aries and the hour circle of a celestial body.
      - Always measured WEST
    - Right Ascension (RA): The angular distance measure EAST from the hour circle of Aries and the hour circle of the body
    - All angles are measured in Degrees, Minutes, and Tenths (of minutes)
  - Time Diagram: A visual representation of items described using the Celestial Coordinate System
    - Essentially it's a diagram of the Earth as observed from a point above the South Pole.
    - M = the upper branch of the observer's meridian
    - G = the upper branch of the Greenwich meridian
    - Q = the upper branch of the Sun's meridian
    - Lower case titles and dotted lines indicated the lower branch of a meridian (and vice-versa)
    - Bodies are labeled based on LHA, which on this diagram is the angle as measured counter-clockwise
- Celestial Horizon System: based on a body as observed by you.
  - Zenith: On an axis running straight up from the Earth's center through you, the Zenith is the highest point, directly above you
  - Nadir: The opposite point of the Zenith, this one would be directly below you
  - Celestial Horizon: The halfway point between the Zenith and Nadir, and which cuts through the center of the Earth
  - Sensible Horizon: The theoretical horizon as a projected plane perpendicular to the observer's eye level. Basically, if you look straight ahead, you're looking at your sensible horizon, so at no point does the plane of that horizon run through the Earth.
  - Geoidal Horizon: A plane tangential to the Earth at the surface, (touching at one point and perpendicular to the surface).
  - Geometrical Horizon: A plane that is tangent to the Earth at Eye Level, without accounting for refraction.
  - Visible Horizon: The horizon that is tangent to the Earth at Eye Level, and does account for refraction.
  - Vertical Circle: A great circle line between the Zenith and Nadir
  - Principal Vertical Circle: A vertical circle that passes through the poles
  - Prime Vertical Circle: A vertical circle that passes through the poles, and is perpendicular to the Principal Vertical Circle at the poles (like 90° longitude)
  - Elevated Pole: The celestial pole above your horizon (an equivalent in t-nav would be North in North Lat, South in South Lat)
  - Depressed Pole: The celestial pole below your horizon (an equivalent in t-nav would be South in North Lat, North in South Lat)
  - Altitude (h): The angular distance from the horizon to the body (always 0-90°)
  - Co-Altitude (aka Zenith Distance [ZD]): The angular distance from the Zenith to a body (ZD = 90- h)
  - Azimuth (Zn): Horizontal direction of a point on the celestial sphere measured against the Principal Vertical Circle (North) in True°, running from 0-360°. The terms True Bearing and Azimuth can be thought of as synonymous, though they're used in different systems of navigation.
  - Azimuth Angle (Z): The inside angle, bearing 0-180°, from either North or South (whichever is closer), and measured either East or West. This comes into play when pulling numbers from Pub 229. You'll see the following again, but just so things are a little clearer at present:
    - In North Lat:
      - Where LHA>180°, Zn=Z
      - Where LHA<180°, Zn=360°-Z
      - In South Lat:
        
        Where LHA>180°, Zn=180°-Z
        
        Where LHA<180°, Zn=180°+Z
All 3 systems (Lat/Long, Celestial Horizon, Celestial Coordinate) have a great circle in common: your meridian.

Celestial Triangle

Uses lines from all 3 systems
Each triangle has 3 points (vertices)
- The elevated pole
- Observer's Zenith
- Celestial Body
Each triangle has 3 legs (connecting the vertices)
- Co-Latitude (90°-Latitude)
- Co-Declination (90°-Declination or 90°+Declination, depending on the situation)
- Co-Altitude (90°-Altitude)
Each triangle has 3 angles (between each of the legs)
- Azimuth (z): Angle between the Co-Lat and Co-Alt
- Meridian Angle (t): Angle between the Co-Lat and Co-Dec
- Parallactic Angle: Not frequently used in calculations, but the angle between the Co-Dec and Co-Alt
The full system is referred to as the Navigational Triangle if the points/lines can be assumed to be on the Earth's surface
- The 3 points become the Elevated Pole (Pn), Geographic Position (GP) of the Body, and your location (M)

Latitude and Declination

When both are the same "sign" (either both North or South), they are referred to as being "Same Name"
- Conversely, when the two signs are different (one south and one north), they are referred to as "Contrary Name"
- This comes in to play particularly when using Pub 229, as the actual page you will use relies on this fact
There are only 4 possible situations for the Navigational Triangle
- 1: North Lat, Body to the East (LHA>180°, t=E, Z=Zn)
- 2: North Lat, Body to the West (LHA<180°, t=W, Zn=360°-Z)
- 3: South Lat, Body to the East (LHA>180°, t=E, Zn=180°-Z)
- 4: South Lat, Body to the West (LHA<180°, t=W, Zn=180°+Z)

Celestial Lines of Position (LOP):

The simplest form of a LOP is a circle of equal altitude
Circle of Equal Altitude
- See Plotting High Altitude LOPs
- Basically, it's a circle drawn onto the Earth, where at any point along that circle, a body is observed at the same altitude, ie. You observed a body at 87°30' at one point, you can move to any other point on this circle and observe this same altitude.
- Radius = Co-altitude of the body
- The only useful altitudes are ≥87°, referred to as a "high altitude sight"
- How this comes into play is that if the GP of the body is assumed to be directly beneath the celestial body, each 1' of altitude observed (below 90°) is equal to 1nm away from the GP.
  - For example: You observe a body at 89°, you can assume that you are somewhere on a circle of equal altitude, 60nm from the GP of the body.
Typically, we work with sights much lower than 87°. It would be impractical to plot a circle of equal altitude as you wouldn't be able to fit most on a single plotting sheet.
- So instead, we just draw a segment of the Circle of Equal Altitude, and for the scales that we work with, we can just assume that the segment is a straight line. (See Plotting Celestial LOPs)

Diagram on Plane of Celestial Meridian

Based on your longitude and the meridian 90° to yours
See more details here: GlobalMaritime.com

Altitude Intercept Method

Developed in 1874 by Marcq St. Hilaire
Rather than drawing an entire Azimuth line and the full navigational triangle, you only draw small segments
You use your Assumed Position (AP), Computed Altitude of a body (Hc), Zn, and the Actual Altitude (Ho) of the body
The basics:
- Your Assumed Position (AP) is a position in which you are assuming that you are within 30 minutes of your DR position
  - Place your AP on the nearest whole ° of lat (ie. If your DR is at 43°23' N, use 43°00' N as your lat)
  - For your longitude, round down and use the nearest whole ° of LHA, (ie. If the LHA at your DR position is 138°35', use 138°)
  - Plot the elevated pole (Pn) and GP of the body
  - Your Altitude Intercept (a) is the difference, in nm, between your Calculated Co-Altitude and your Observed Co-Altitude
  - Your actual position is found to be a distance away from your AP equal to your Altitude Intercept, along your Zn (though either toward or away from the body as below):
    - Ho Mo To: Essentially, if Ho (Observed Altitude) is More than Hc (Calculated Altitude), your position is Toward the GP of the body

Sight Reduction Form: Bowditch Figure 2005

Hs: Height observed, through a sextant, above the visible horizon
Ha: Apparent altitude, after corrections are made to the Hs for sextant errors and dip
Ho: Height observed above the visible horizon, after corrections are made to Ha for meteorological conditions and the refraction of light in the atmosphere
Hc: Calculated altitude
Sextant: Tool used to measure the angle between 2 objects (<90° apart)
- Measures Hs, which provides Ho when corrected
Errors (which must be accounted for in calculations):
- Dip: the difference between the horizontal plane tangent to the Earth's surface below you, as compared to the actual visible horizon
  - Correction found on the inside covers of the Nautical Almanac (and is always subtracted)
  - Is equivalent to -0.97 x (square root of: Height of Eye)
- A2: Refraction of light in the atmosphere
- Horizontal Parallax: Difference between the surface and center of the Earth
  - Essentially, it's where you can't visible measure the center of a body because it appears so large that the center is indeterminable
- Semi-Diameter (SD): The observed radius of a body
- Instrument Error: Caused when the (Horizon) mirror and the (Index) glass on your sextant are not parallel to each other
- Index Error: As measured when the sextant is set to 0°, this is the observed "break" between the right and left halves of a body or the horizon as observed through the sextant.
  - If the error is "on the arc," the sextant is reading too high
  - If the error is "off the arc," the sextant is reading too low
- Altitude Corrections: lumped together in the yellow pages of the Nautical Almanac