9.1 Celestial Sphere, Equatorial/Horizon Coordinate Systems, and Astronomical Time Systems (UT, ST, LMT)

9.1 Celestial Sphere, Equatorial/Horizon Coordinate Systems, and Astronomical Time Systems (UT, ST, LMT)

Geodetic astronomy establishes the fundamental relationship between terrestrial survey observations and celestial reference frames. By observing stars, the Sun, or artificial satellites relative to the Earth's gravity field (plumb line) and rotation axis, geodetic engineers can determine true astronomical azimuth, latitude, and longitude independent of terrestrial control networks.

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1. The Celestial Sphere Geometry & Fundamental Concepts

The celestial sphere is an imaginary sphere of arbitrary or infinite radius, concentric with the Earth (or observer), onto which all celestial bodies are projected along lines of sight. Although stars exist at vastly different physical distances, projecting them onto this spherical surface enables the application of spherical trigonometry to celestial navigation and geodetic positioning.

Key Elements of the Celestial Sphere

• Zenith ($Z$): The point on the celestial sphere directly overhead the observer, defined by the upward extension of the local plumb line (direction of gravity).
• Nadir ($Z'$): The point directly opposite the Zenith, beneath the observer's feet ($180^\circ$ from Zenith).
• Celestial Equator: The great circle formed by projecting Earth's equatorial plane onto the celestial sphere. It lies in a plane perpendicular to the Earth's rotational axis.
• Celestial Poles ($NCP$ and $SCP$): The intersection points of Earth's rotational axis extended to the celestial sphere. The North Celestial Pole ($NCP$) lies near Polaris in the northern hemisphere.
• Ecliptic: The great circle representing the apparent annual path of the Sun across the celestial sphere relative to the background stars.
• Obliquity of the Ecliptic ($\epsilon$): The inclination angle between the celestial equator and the ecliptic, approximately $\epsilon \approx 23^\circ 26' 21.4''$ ($23.44^\circ$).
• Vernal Equinox ($\gamma$ or First Point of Aries): The point where the Sun crosses the celestial equator moving from south to north (declination $\delta = 0^\circ$, right ascension $\alpha = 0^h$). It serves as the primary origin for the equatorial coordinate system.
• Autumnal Equinox ($\Omega$): The point where the Sun crosses the celestial equator moving from north to south ($\alpha = 12^h$).

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2. The Horizon (Alt-Azimuth) Coordinate System

The Horizon coordinate system (also known as the Alt-Azimuth or topocentric system) is defined relative to the local observer's position and plumb line.

Coordinate Parameters

1. Altitude ($h$): The angular distance of a celestial body measured above the local horizon along a vertical circle passing through the body and the zenith.
   • Range: $0^\circ \le h \le +90^\circ$ (above horizon) and $-90^\circ \le h < 0^\circ$ (below horizon).
2. Zenith Distance ($z$): The angular distance from the Zenith to the celestial body measured along the vertical circle. $\text{Zenith Distance } z = 90^\circ - h$
3. Azimuth ($A$): The horizontal angular distance measured clockwise along the horizon plane from the reference direction (typically True North in modern geodetic engineering) to the vertical circle containing the object.
   • Range: $0^\circ \le A < 360^\circ$.

Principal Reference Circles

• Observer's Meridian (Principal Meridian): The great circle passing through the Celestial Poles, Zenith, and Nadir. It intersects the horizon at True North and True South.
• Prime Vertical: The great circle passing through Zenith and Nadir perpendicular to the observer's meridian, intersecting the horizon at True East and True West.

Key Distinction: The Horizon coordinate system is strictly local and time-dependent. Because Earth rotates continuously from west to east, the altitude $h$ and azimuth $A$ of every fixed star change every second.

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3. The Equatorial Coordinate System

The Equatorial coordinate system is an observer-independent celestial coordinate frame tied to the celestial equator and the vernal equinox ($\gamma$).

Coordinate Parameters

1. Declination ($\delta$): The angular distance of a celestial body north ($+$) or south ($-$) of the celestial equator, measured along the star's hour circle.
   • Range: $-90^\circ \le \delta \le +90^\circ$. Analogous to terrestrial latitude ($\phi$).
2. Right Ascension ($\alpha$): The angular distance measured eastward along the celestial equator from the Vernal Equinox ($\gamma$) to the star's hour circle.
   • Units: Expressed in sidereal time units ($0^h$ to $24^h$), where $1^h = 15^\circ$. Analogous to terrestrial longitude ($\lambda$).
3. Local Hour Angle ($t$ or LHA): The angular distance measured westward along the celestial equator from the observer's upper meridian to the star's hour circle.
   • Range: $0^h$ to $24^h$ (or $-180^\circ$ to $+180^\circ$).

Fundamental Hour Angle Identity

The Local Sidereal Time ($\text{LST}$) at any instant equals the hour angle of the Vernal Equinox. This establishes the fundamental relationship linking time, right ascension, and hour angle:

$\text{LST} = \alpha + t \quad \implies \quad t = \text{LST} - \alpha$

When converting between Greenwich and local observations, the Greenwich Hour Angle ($\text{GHA}$) and Greenwich Sidereal Time ($\text{GST}$) relate to local parameters via terrestrial longitude ($\lambda$):

$\text{LST} = \text{GST} + \lambda_E \quad \text{or} \quad \text{LST} = \text{GST} - \lambda_W$ $\text{LHA} = \text{GHA} + \lambda_E \quad \text{or} \quad \text{LHA} = \text{GHA} - \lambda_W$

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4. Astronomical Time Systems

Geodetic astronomy utilizes distinct time systems based on Earth's rotation relative to celestial reference points or atomic standards.

Equation of Time ($E$)

The Equation of Time ($E$) is the difference between Apparent Solar Time (AST) and Mean Solar Time (MST) at any instant:

$E = \text{Apparent Solar Time} - \text{Mean Solar Time}$

Angular and Time Unit Conversion Rules

Because Earth completes a full $360^\circ$ rotation in $24$ hours, conversions between arc units and time units follow strict proportionality:

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5. Worked Computational Example

Problem: A geodetic engineer in Cebu ($\lambda = 123^\circ 53' 00'' \text{ E}$) observes a star with Right Ascension (\alpha = 18^h 45^m 12^s). If the Greenwich Sidereal Time ($\text{GST}$) at the instant of observation is $21^h 10^m 30^s$, calculate:

1. The Local Sidereal Time ($\text{LST}$) at Cebu.
2. The Local Hour Angle ($t$) of the star.

Solution:

Step 1: Convert Longitude to Time Units $\lambda = 123^\circ 53' 00'' \text{ East}$ $\lambda_{\text{hours}} = \frac{123^\circ}{15} + \frac{53'}{60 \times 15} + \frac{00''}{3600 \times 15} = 8^h 15^m 32^s \text{ East}$

Step 2: Calculate Local Sidereal Time ($\text{LST}$) Since Cebu is East of Greenwich, add longitude in time: $\text{LST} = \text{GST} + \lambda_E = 21^h 10^m 30^s + 8^h 15^m 32^s = 29^h 26^m 02^s$ Subtract $24^h$ for full revolution: $\text{LST} = 29^h 26^m 02^s - 24^h 00^m 00^s = 05^h 26^m 02^s$

Step 3: Calculate Local Hour Angle ($t$) $t = \text{LST} - \alpha = 05^h 26^m 02^s - 18^h 45^m 12^s$ Add $24^h$ to $\text{LST}$ to avoid negative results: $t = (05^h 26^m 02^s + 24^h 00^m 00^s) - 18^h 45^m 12^s = 29^h 26^m 02^s - 18^h 45^m 12^s = 10^h 40^m 50^s$

Converting $t = 10^h 40^m 50^s$ to arc: $10^h \times 15 = 150^\circ$ $40^m \times 15' = 600' = 10^\circ$ $50^s \times 15'' = 750'' = 12' 30''$ $t = 160^\circ 12' 30'' \text{ West of meridian}$