While most stars sweep by as the Earth rotates, a star that is aligned with our planet’s rotation axis, at the north celestial pole, seems to remain placed in an unchanging location at 90°north declination. The Earth’s northern rotation axis, for example, now points close to Polaris, also known as the North Star or the Pole
10 1 Observing the Universe
Star, which would lie approximately overhead when viewed from the Earth’s geographic North Pole. The latitude of any location in the Earth’s Northern Hemisphere is equal, within about 1°, to the angular altitude of Polaris. The uncertainty is due to the fact that Polaris is not exactly at the north celestial pole, where the north end of the Earth’s rotation axis pierces the night sky.
We can locate Polaris by following the line joining the two stars farthest from the handle of the Big Dipper, which accounts for the phrase ‘‘follow the drinking gourd’’ used by southern slaves escaping to the northern parts of the United States.
Mariners have also used the North Star for navigation, to find the direction of north and the latitude of their ship.
Nevertheless, everything in the universe is in a state of perpetual change, and the locations of the so-called fixed stars on the celestial sphere are no exception.
Their change in position is related to the Earth’s elongated shape (Focus 1.2), which has sent the Earth into a wobbling rotation that resembles a spinning top.
This causes a very slow change of the celestial positions of the north celestial pole, the Pole Star and all the other stars, called precession. The changing positions of bright stars on the celestial sphere were first observed by Hipparchus, a Greek astronomer who lived in the second century BC (Hipparchus, 125 BC); the tele- scope was not invented until 17 centuries after Hipparchus established the stellar positions using his eyes.
Focus 1.2 The elongated shape of the Earth
The Earth isn’t precisely spherical in shape. It has a slight bulge around its equatorial middle and is flattened at its poles, with a shape more like an egg than a marble or billiard ball. This elongated, oblate shape is caused by the Earth’s rapid rotation. The outward force of rotation opposes the inward gravitational force, and this reduces the pull of gravity in the direction of spin. Since this effect is most pronounced at the equator, and least at the poles, the solid Earth adjusts into an oblate shape that is elongated along the equator.
An ellipse of eccentricity,e,and major axis,ae, which is rotated about the polar axis, defines the Earth’s reference ellipsoid at sea level. The planet’s equatorial radius isae, and its polar radius, ap. They are given by:
ap ¼aeð1fÞ ¼aeð1e2Þ1=2;
where the flattening factorf=(ae-ap)=aeis related to the eccentricity,e, bye2=2f-f2.
The mean surface radius of the Earth,hai,is given by:
h i ¼a a2eap13
6:371 106m;
which is the radius of a sphere of volume equal to the Earth ellipsoid.
Geophysicists use another definition of mean radius given by (2ae?ap)=3.
1.5 The Locations of the Stars are Slowly Changing 11
The radius, r, of the surface of the Earth geoid at any latitude/is given by r¼aeð1fsin2/Þ:
Two of the primary constants of the International Astronomical Union are (Kaplan2005):
Equatorial radius of the Earth¼ae¼6:3781366 106m;
and
Flattening factor for Earth¼f ¼0:0033528197¼1=298:25642:
These values of ae and f give a polar radius for the Earth of ap=6.3567529106m, and the difference between the equatorial and polar radius is 21,385 m or about 21 km.
The world geodetic system, which is the basis of terrestrial locations obtained from the Global Positioning System, or GPS for short, uses an Earth ellipsoid withae=6.3781379106m andf=1=298.257223563.
The changing locations of celestial objects are caused by the gravitational action of the Moon, Sun and planets on the spinning, oblate Earth. As a result of this gravitational torque, the Earth’s rotation axis is constantly changing with respect to a space-fixed reference system.
The precessional motion of the Earth’s rotation axis is caused by the tidal action of the Moon and Sun on the spinning Earth. That is, because the Moon and the Sun lie in the ecliptic plane, which is inclined by 23.5° to the plane of the Earth’s Equator, they exert a gravitational force on the Earth’s equatorial bulge. This causes the rotation axis to sweep out a cone in space, centered at the axis of the Earth’s orbital motion and completing one circuit in about 26,000 years (Fig.1.4).
So the Earth is not placed firmly in space; instead it wobbles about causing the identity of the Pole Star to gradually change over time scales of thousands of years.
The northern projection of the Earth’s rotation axis is currently within about 0.75°
of Polaris and will move slowly toward it in the next century. After that, the north celestial pole will move away from Polaris and, in about 12,000 years, the Earth’s rotation axis will point to within 5°of the bright star Vega.
The slow conical motion of precession carries the Earth’s Equator with it; as that Equator moves, the two intersections between the celestial equator and the Sun’s path, or ecliptic, move westward against the background stars. One of these intersections is the Vernal, or Spring, Equinox, from which right ascension is measured. This equinox point moves forward (westward) along the ecliptic at the rate of about 50 s of arc, denoted 5000, per year, which is equivalent to 3.33 s per year.
As the Earth’s rotational axis precesses, declinations also change, through a range of 47°, or twice 23.5°, over 26,000 years.
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Because the long period, 26,000 year, conical motion of the Earth’s rotation axis is caused by the gravitational action of the Moon and Sun, it is called lunisolar precession. Simon Newcomb (1835-1909) derived the detailed theory for com- puting the corrections to astronomical coordinates for this precession (Newcomb 1895).
In addition to this steady, progressive motion, there are small, periodic varia- tions in both precessional speed and axial tilt caused by the gravitational action of the planets on the Earth’s equatorial bulge. The most important term in this nutation, first observed by James Bradley (1693-1762), induces an 18.6 year periodic wobble in the precessional motion with a size of 1700 in the direction of precession and 900 perpendicular to it (Bradley 1748).
Because of positional changes caused by precession and nutation, the equinox, or reference date, must be given when specifying the right ascension or declination of any cosmic object. The standard epoch that is now in used for celestial positions is:
J2000:0¼2000 January 1:5¼JD2451545:0;
where JD denotes Julian Date and the prefix J denotes the current system of measuring time in Julian centuries of exactly 36,525 days in 100 years, with each day having a duration of 86,400 s.
The combination of lunisolar and planetary precession is called general pre- cession, and the astronomical constants at standard epoch J2000.0 include (Kaplan 2005):
General precession in longitude¼q¼5,028.79619500per Julian century:
To Polaris/
North Pole Star
Equator 23.5°
23.5°
26,000 Year Precession
To Vega
Fig. 1.4 Precession The Earth’s rotation axis traces out a circle on the sky once every 26,000 years, sweeping out a cone with an angular radius of about 23.5°. The Greek astronomer Hipparchus (c. 146 BC) discovered this precession in the second century BC. The north celestial pole, which marks the intersection of this rotation axis with the northern half of the celestial sphere, now lies near the bright star Polaris. However, as the result of precession, the rotational axis will point toward another bright north star, Vega, in roughly 13,000 years. This motion of the Earth’s rotational axis also causes a slow change in the celestial coordinates of any cosmic object
1.5 The Locations of the Stars are Slowly Changing 13
The nutation term,N, for that epoch is
Constant of nutation¼N¼9:205233100:
Kaplan (2005) provides modern formulas for precession and nutation. They describe the transformation of celestial coordinates from one date to another, as a function of time since a reference epoch.
The astronomical constants also include an aberration constant, j, which accounts for the observed position shift of an astronomical object in the direction of the Earth’s motion (Focus 1.3). The aberration constant at the standard epoch J 2000.0 is
Constant of aberration¼j¼20:4955200:
Focus 1.3 Stellar aberration
As the Earth orbits the Sun, the stars all appear to be shifted in the direction of motion, a phenomenon called stellar aberration, described by James Bradley (1693-1767) in 1728. Because the speed of light is finite, the apparent direction of a celestial object detected by a moving observer is not the same as its geometric direction at the time. For stars, the normal practice is to ignore the correction for the motion of the celestial object, and to compute the stellar aberration due to the motion of the observer. This also gave Bradley a means to improve on the accuracy of previous estimates for the speed of light (Bradley1728).
The magnitude Dh of stellar aberration depends on the ratio of the velocity of the observer,V,to the speed of light,c, and the angle,h, between the direction of observation and the direction of motion. The displacement, Dh, in the sense of apparent minus mean place, is given by
DhV
csinh1 2
V c
2
sin2hþ V c
3
sinhcos2h0:33sin3h þ:::
As the Earth orbits the Sun, it is moving at a velocity of approximately 30 km s-1, and the speed of light c&300,000 km s-1, so the term of order V=c is 10-4rad or 20 s of arc, denoted 2000, and the term (V=c)2 has a maximum value of 0.001 s of arc. Bradley (1728) used aberration obser- vations to determine the speed of light as approximately 183,000 miles per second or 294,500 km s-1.
The constant of aberration,j, at standard epoch J2000.0 is given by Constant of aberration¼j¼2pa
Pc ð1e2Þ12¼V
c ¼0:9365104rad,
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which is equivalent to:
j¼20:4955200
Here the constantp&3.14592654, the mean distance between the Earth and the Sun is a =1 AU=1.4959891011 m, known as the astronomical unit, e=0.01671 is the mean eccentricity of the Earth’s orbit,P=3.15589107s is the length of the sidereal year,c=2.9979259108m s-1is the speed of light, and one radian=2.062648910500.
When the observer is moving directly at the star,h is zero and there is no aberration shift at all. The shift achieves its greatest value of about 20.500 when the observer’s motion is perpendicular to the direction of the star, withh=90°.