Lecture 9: Variations in Celestial Coordinates

Causes:

  1. Motion of the coordinate systems with respect to the stars and with respect to the solid earth

  2. Physical phenomena

  3. Relative motion of the stars with respect to each other in space

Variations due to the Motion of the Coordinate System

  • Why gravity affects Earth’s rotation?

    • Earth is not a perfect sphere. It bulges at the equator because of the rotation. This bulge is what the Sun and Moon pull on.

    • If the Earth were a perfect sphere, external bodies could not exert torques on it, and the axis would not change direction. But since it has a bulge, the gravitational attraction is uneven —> this causes torques —> which changes the orientation of Earth’s axis —> which moves the celestial poles and equator.

Three major coordinate-system motions:

  1. General Precession

  2. Astronomic Nutation

  3. Polar Motion

  • “The sun is situated in the direction S and attracts the opposite bulges with forces F1 and F2 where F1 > F2.”

    • Earth’s equator bulges north and south

      • When the bulge is on the side closer to the Sun, the gravitational attraction is stronger —> F1

      • When the bulge is farther from the Sun, attraction is weaker —> F2

    • This difference in attraction creates a net torque trying to tip the Earth

  • “The centrifugal forces at the activity centers of the bulges… are C1 and C2 where C1 < C2.”

    • Because Earth revolves around the Sun, it experiences small centrifugal effects

    • These oppose the gravitational forces slightly

    • Again, the near bulge and far bulge have different values

    • Combined:

      • Gravity (stronger on near side)

      • Centrifugal effect (slightly stronger on far side)

  • “Resolving forces into components perpendicular and parallel to the equator…the perpendicular components cause a moment…”

    • The forces acting on each bulge can be broken down into:

      • Parallel components - do not affect tilt

      • Perpendicular components - try to tilt the Earth’s equator toward the ecliptic

    • These perpendicular components are the important ones.

    • They do not success in tipping Earth over (Earth’s rotation resists this). Instead, they produce a twisting force (torque) on the equatorial bulge.

    • Because Earth is spinning, that torque does not tilt the axis directly—instead, it causes a slow, circular wobbling of the axis, exactly like:

      • A spinning top under gravity

      • Its axis traces a cone instead of pointing in a fixed direction

      • This is precession

  • "The moment + rotational momentum of the Earth results mainly in a motion similar to a top termed ‘PRECESSION’

    • Torque from the Sun + Earth’s rotational angular momentum = precession of the equinoxes

    • This makes:

      • The celestial poles trace a circle in the sky

      • The equinox move westward along the ecliptic 

      • Star coordinates slowly change over centuries

The Effects of the Sun and the Moon are Resolved into 2 Components:

  • Lunisolar Precession (1st component)

    • Moves the celestial pole around the ecliptic pole with a period of about 25,800 years and an amplitude equal to the obliquity of the ecliptic, about 23.5°, resulting in the westerly motion of the equinox on the equator of about 50.3” a year.

  • Astronomic Nutation (2nd Component)

    • Due partly to the periodic motion of the Earth around the Sun, and of the moon around the Earth, partly to the fact that the moon’s orbit does not coincide with the ecliptic, and partly to other causes.

    • A relatively short periodic motion of the celestial pole superimposed on the luni-solar precession with a maximum amplitude of about 9” and a main period of about 18.6 years.

The Planets Affect the Position of the Mean Orbital Plane of the Earth, the Ecliptic:

  • Planetary Precession

    • Consists of a slow rotation of the ecliptic (0.5” a year) about a slowly moving axis of rotation (λ = 174°) and it results in the westerly motion of the equinox of about 12.5” per century and a decrease in the obliquity of the ecliptic of about 47” per century.

General Precession = lunisolar + planetary precessions considered together

  • Since both the general precession and the nutation affect the positions of the celestial equator and the ecliptic, thus the equinoxes, the orientation of most primary and secondary reference planes defining the various celestial coordinate systems change with time

Polar Motion 

  • The relative motion of the solid mass of the Earth, with respect to the rotation of the axis

  • makes the geographic coordinates of the observer change with time

  • The resultant of two components:

    • Revolution of the true pole around the principal moment of inertia axis, counterclockwise as viewed from the north, with a period of 1.2 yrs, which is due to the free-force precession (Euclerian Motion) disturbed by the elastic yielding of the Earth

    • Revolution in the same direction with an annual period, which is caused by the continuous redistribution of the mass in the meteorological and geophysical processes

    • Context:

      • The Eulerian Motion (Free Nutation)

        • Free oscillation (no external forces needed)

        • Because the Earth is not a perfect rigid body.
          It has:

          • oceans

          • atmosphere

          • mantle that deforms slowly

          • elastic crust

          The Earth can wobble, similar to a spinning top that’s not perfectly symmetric.

      • Annual Motion (Forced Motion)

        • Because these processes occur regularly throughout the year, they produce a forced motion of the Earth’s pole.

Variation due to the Physical Effects

  1. Aberration

  2. Parallax

  3. Astronomic Refraction

Aberration

  • The apparent displacement of a celestial object caused by the finite velocity of light propagation in combination with the relative motion of the observer and the object

  • The apparent direction of an object seen by an observer in motion is displaced from the true direction in which it would be seen if the observer is motionless by an amount equal in linear measure to observer’s motion at a constant speed during the time interval the light propagated from the object to the observer.

  • The direction of displacement is that of the observer’s motion at the moment of observation

  • Planetary Aberration

    • The angular displacement of the geometric direction between the object S and the observer at the instant of light emission T0 from the geometric direction at the instant of observation T.

    • 2 Components of Planetary Aberration:

      • Stellar Aberration

        • The exact position where a star appears in the sky does not only depends on the coordinates of the source observed, but also on the observer’s relative velocity. The observer velocity is responsible for a phenomenon called “Bradley aberration” or “Stellar aberration”.

        • Stellar aberration is a well known phenomenon among astronomers. It was discovered by the astronomer James Bradley in 1727. It is claimed to be caused by the relative transverse motion between the Earth and the star emitting the photons.

        • Based on the principle of invariance, Einstein’s relativity predicts that absolute motion does not exist. Consequently, there should be no difference between a star having a velocity with respect to an observer and an observer having a velocity with respect to a star. In the case of stellar aberration, this prediction appears contrary to observations. It is shown that the description of stellar aberration, in terms of relative transverse velocity between the star and an observer on Earth should be corrected, because it is an erroneous interpretation of Einsten’s relativity.

        • Stellar aberration is a correction, which is absolutely needed, in order to get a logical system of coordinates for stars and galaxies, which is valid at any time all year round and even at any epoch. Without stellar aberration, it is impossible to establish a coherent system of coordinates in the universe. It takes into account the velocity of the observer due to the Earth rotation and also its translation around the Sun.

        • 3 Components of Stellar Aberration:

          • Diurnal Aberration - due to the rotation of the Earth

          • Annual Aberration - due to the revolution of the Earth around the center of mass of the solar system

          • Secular Aberration - due to the motion of the solar system in space around the center of galaxy

      • Correction for Light Time

        • Due to the motion of the object in the inertial frame of references during the interval while the light was propagating

    • In geodetic astronomy, when the objects for observations are stars, the corrections for light-time are of no importance and may be assumed to be zero, since the secular aberration is practically constant for each star and therefore ignored, only the diurnal and annual aberrations are considered.

  • Parallax / Parallactic Displacement

    • The angle between the direction of the celestial object as seen from the observer and from some standard point of reference

    • 2 Kinds of Parallax:

      • Geocentric Parallax

        • The angle of the object A between the direction of the observer and the direction of the center of the Earth

        • Owing to the rotation of the Earth about its axis, the topocentric direction from the observer to a celestial object changes

        • To reduce an observed or topocentric direction to the geocenter, a correction to this must be applied

      • Annual or Stellar Parallax

        • The angle subtended at an object by the radius of the Earth’s orbit

        • Owing to the revolution of the Earth around the center of mass of the solar system, the geocentric direction must be reduces to the center of mass of the solar system

        • In geodetic astronomy, when the objects of observations are stars whose distances from the Earth are very large, the effect of the geocentic parallax is always neglected, while the correction from the annual parallax must sometimes be applied.

  • Astronomic Refraction

    • The apparent displacement of the celestial object outside the atmosphere that results from light rays being bent in passing through the atmosphere

    • As light passes through the atmosphere, the variation of air density along the path causes a continuous change in direction of propagation

    • Scattering and absorption by the gases of the atmosphere and by the dust particles cause attenuation and change of the spectral composition

    • The general effect is that the light rays bend downward; thus the celestial object appears to be at higher altitude than it is in reality

    • The magnitude of this displacement depends upon the zenith distance of the object and upon the atmospheric conditions (i.e. temperature, barometric pressure, etc.)

None of the above-mentioned phenomena affects the frame of reference, but cause changes in the direction in which the celestial object is observed.

The corrections must therefore be calculated with respect to a particular reference system and applied to the objects position in the same system

Variation due to the Proper Motion of the Stars

  • Proper Motion

    • Each star appears to have a small motion of its own

    • The resultant of the actual motion of the star in space and its apparent motion due to the changing direction arising from the motion of the Sun with the planets

Summary of the Reduction of Star Positions and Secondary Effects

  • Observed Place

    • The actual position of the celestial object determined by means of the direct readings on some instrument corrected for systematic instrumental errors

  • Apparent Place

    • The position referred to the true equator and equinox of date, but as seen from the geocenter (i.e., corrected for atmospheric refraction, diurnal aberration, geocentric parallax). In short: move from topocentric observed to geocentric apparent by removing refraction, diurnal aberration, and geocentric parallax.

    • This is the position where the object would actually be seen by a theoretical observer at the center of the Earth, referred to the true equator and equinox of the date. It includes all relevant corrections, such as: annual aberration, annual parallax

  • True Place

    • The barycentric position of an object, referred to the true equator and equinox of date when the object actually is observed at the equator.

    • Differs from the apparent place by the effects of the annual aberration and annual parallax corrections (shifts geocentric apparent to barycentric true)

    • Place of the object as viewed by an imaginary observer situated at the center of mass of the solar system in the coordinate system affected by the precession and the nutation

    • This is the position of an object relative to the center of the Earth at a specific date, corrected for the long-term effects of precession and the short-term periodic effects of nutation (wobbles in Earth's axis). It still assumes an observer at the Earth's center and an idealized, transparent Earth and atmosphere.

  • Mean Place

    • The barycentric position of an object, referred to some specified mean equator and equinox

    • Differs from the true place by the removing nutation (so the reference is a mean equator/equinox rather than a true one).

    • Place of the object as viewed by an imaginary observer situated at the center mass of the solar system (practically at the center of the Sun) in the coordinate system affected by the precession

    • It is a long-term average position that is corrected for an object's actual motion (proper motion) over time but does not account for short-term and periodical shifts caused by the Earth's orbit and rotation.