Ly

(a) Stationary telescope

(b) Telescope on earth; in earth frame

(c) Telescope on earth; from stationary frame

Stationary telescope

Stationary telescope

\ Star lig

(b) Telescope on earth; in earth frame

(c) Telescope on earth; from stationary frame

\ Star lig

Figure 4.3. Stellar aberration. The apparent direction of an overhead star is modified due to the speed of the earth in its orbit about the sun. (a) Stationary telescope viewed from a stationary reference frame (b) Telescope in the (moving) reference frame of an observer on the earth. (c) Moving telescope from stationary reference frame.

Figure 4.3. Stellar aberration. The apparent direction of an overhead star is modified due to the speed of the earth in its orbit about the sun. (a) Stationary telescope viewed from a stationary reference frame (b) Telescope in the (moving) reference frame of an observer on the earth. (c) Moving telescope from stationary reference frame.

The measurement of parallax angles is a standard method for the determination of distances to the closer stars (see Section 9.5).

Stellar aberration

There is also a small shift of star positions that arises from the velocity of the earth as it orbits the sun. (Parallax is a consequence of the varying position of the earth.) This effect is called stellar aberration. It should be calculated with special relativity, but since the earth is traveling at a speed much less than the speed of light, a classical derivation yields the correct magnitude of the effect. Consider starlight to be analogous to (vertically) falling rain as one runs through it. In the frame of reference of the runner, the rain does not fall precisely vertically; it appears to come from a slightly different (more forward) angle. It would be advisable for the runner to tilt her umbrella forward a bit. Similarly, starlight appears to come from a different angle, and the telescope must be tilted slightly forward to collect it (Fig. 3b).

The magnitude of the (maximum) aberration angle follows from the geometry of Fig. 3c which shows the situation from a stationary reference frame. The light falls vertically, and the telescope must be tilted at an angle such that light entering the top of the telescope will exit through the eyepiece when it reaches the bottom.

The required angle is given by tan 0aber = uearth/c, or to a very good approximation, vearth 2.979 x 104 m/s 4

The pointing direction of the telescope is the arrival direction of the light according to the moving observer. This results in an annual cyclic variation of the apparent positions of stars on the celestial sphere of up to 20.5'' per year.

The effect is maximal when the star is 90° from the earth's velocity vector, e.g., at the ecliptic pole, and the effect is absent when the star is 0° from the velocity vector. In the course of a year, a star at the ecliptic pole will thus track out a circle of radius 20.5'' on the celestial sphere, and a star on the ecliptic will move back and forth along a line (great circle) of length 41''. At intermediate positions, the star's track will be an ellipse. All stars in the same vicinity will suffer the same motion, so this effect can not be detected simply by taking photographs of a given region of the sky. It differs from parallax in this respect.

The direction and magnitude of aberration depend solely upon the time of year and the direction of the target star. It is easily calculated from appropriate formulae. These corrections are sufficiently small <20.5'' that they traditionally have not been an issue in practical observing where the coordinates of a star are used primarily to point the telescope to the close vicinity of a star, say within ~1'. However, those who create catalogs of precise stellar positions and those who program computers for precise pointing of telescopes must include aberration in their algorithms.

Precession of the earth

Precession of the celestial equatorial coordinate system is due, as noted, to precession of the earth. The latter is rather like a spinning top that precesses about the vertical due to the torque of gravity.

Torque due to a ring of mass

In the case of the earth, the torque is due to the gravitational attraction of the sun, moon, and planets acting on the earth. These bodies could not exert a torque on the earth if it were perfectly spherical. However, the earth has an equatorial bulge; its radius is larger at the equator than at the poles. The sun, moon, and planets (especially Jupiter) are thus able to exert a torque on it. The bulge is due to the centrifugal force in the rotating frame of reference of the spinning earth; the centrifugal force increases with radius from the spin axis, and hence is largest at the equator.

Since the precession is very slow with a period of 25 770 yr, the sun might be viewed as circulating around the earth 25 770 times per precession period. If one runs this as a speeded up movie, the sun would appear as a ring of mass around the ecliptic (Fig. 4a). The moon's orbit lies only 5.1° from the ecliptic so its mass

Figure 4.4. Precession of the earth's spin axis about the normal to the ecliptic plane with period 25 770 yr. (a) Time-averaged distribution of mass in the ecliptic plane due to the many cycles of the sun (and moon and Jupiter) which acts gravitationally on the earth's equatorial bulge to produce an instantaneous torque N and hence precessional motion. (b) Earth alignment relative to celestial sphere on 2000 Mar 21, with the ecliptic as the equator. The motions of the spin axis and the associated celestial equator (light solid line) are indicated with arrows. The vernal equinox moves to the left as the precession proceeds, i.e., in the direction of lower right ascension. The sun follows the ecliptic (heavy line); it is moving into the northern hemisphere on Mar 21. The situation after one-half precession cycle is shown with dashed lines.

Figure 4.4. Precession of the earth's spin axis about the normal to the ecliptic plane with period 25 770 yr. (a) Time-averaged distribution of mass in the ecliptic plane due to the many cycles of the sun (and moon and Jupiter) which acts gravitationally on the earth's equatorial bulge to produce an instantaneous torque N and hence precessional motion. (b) Earth alignment relative to celestial sphere on 2000 Mar 21, with the ecliptic as the equator. The motions of the spin axis and the associated celestial equator (light solid line) are indicated with arrows. The vernal equinox moves to the left as the precession proceeds, i.e., in the direction of lower right ascension. The sun follows the ecliptic (heavy line); it is moving into the northern hemisphere on Mar 21. The situation after one-half precession cycle is shown with dashed lines.

also contributes to the ring. Similarly, Jupiter's inclination is only 1.3° so it too contributes to the ring of mass. The two small arrows labeled F on the earth in the figure represent the approximately horizontal forces that "pull" the bulge toward the ring of suns. Together these forces create a (vector) torque N that is directed more or less out of the paper. The resultant angular momentum change dl ("dee ell") would be parallel to the torque (from N = dl/dt). This causes the spin vector of the earth to precess as shown. The result is that the earth's spin axis traces out a circle on the celestial sphere with a half-cone angle of 23.45°. The center of the circle is in the direction of the ecliptic pole.

Take a closer look at the precessional motion. Figure 4b shows precession relative to the celestial sphere. The spin axis of the earth and the celestial equator in 2000 are shown. Recall that the celestial equator is the projection of the plane of the earth's equator onto the celestial sphere. The ecliptic is a fixed track among the stars because the earth's orbit is quite rigidly fixed in space. The sun's position on 2000 March 21 is shown; it is in the constellation Pisces and moving to the right on the ecliptic. Note that it is just crossing the celestial equator from south to north in its annual motion; it is at the vernal equinox, the zero of right ascension (in 2000).

As the earth begins to precess away from the 2000 position, the spin axis moves toward the reader (arrow Al) and the celestial equator slides to the left (see arrows). The intersection of the celestial equator and the ecliptic (the vernal equinox) thus also moves to the left or the west, the precession of the equinoxes. This intersection is the zero of right ascension by definition so it too moves to the west. A given star in this region will thus find itself farther and farther east of the vernal equinox. Since right ascension increases to the east, the star will have an increasing right ascension from year to year. The declination will also change. For this reason, as noted in Section 3.2, one always must specify the epoch (year) of the coordinate system being used.

As the precession continues for —12 885 yr, the spin axis will have precessed 180° to the position shown as the dashed line. The new celestial equator is also shown as a dashed line. An intersection of the celestial equator and the ecliptic is again at the same position among the stars, in Pisces, but when the sun passes this point it will be moving from north to south relative to the earth, i.e., the first day of fall in the northern hemisphere! Thus this position will be assigned right ascension a = 12 h, and the a = 0 position will be on the opposite side of the sphere.

Rate of precession

The rate in arcsec per year at which the vernal equinox slides around the ecliptic is

Rate = 360° x 1 /(25 770) x 3600 (Motion of the vernal (4.11)

or 42' in 50 yr, a sizable motion. The shifts in coordinates of a celestial object can be of this magnitude; they can also be much lower depending on the location of the star on the celestial sphere. The vernal equinox used to be in the constellation Aries; it was named the first point of Aries in the time of Hipparchus (~ 135 BCE). It is now in Pisces, and in about 600 yr, it will be in Aquarius, the Age of Aquarius!

Nutation

The earth also exhibits a small nutation (wobble) of 9'' with a period of 18.6 yr. This nutation is also due to gravitational effects of the moon and sun. The period is the same as that for the moon's orbit to precess one complete cycle with respect to the stars. See further discussion below (solar eclipses). The pole and equator of the equatorial coordinate system follow this motion also.

Calendar

What happens to the calendar and the seasons as the earth precesses? Consider a future date 12 885 yr from now. The sun will pass annually through Pisces as it always does, but as noted above it will be passing from north to south in earth coordinates. It thus will be the beginning of fall in the northern hemisphere; winter will be on the way. The location of the sun along the ecliptic (or equivalently, the location of the earth in its orbit) is thus not an indicator of the season because of precession. The season is indicated by the position of the sun in its track relative to the (precessing) vernal equinox. The right ascension of the sun will thus always indicate the season.

Since the adoption in 46 BCE of the Julian calendar by Julius Caesar, the calendar has been tied to the seasons. Thus March 21 should always take place at the beginning of spring when the sun crosses from south to north (at the vernal equinox), even though the vernal equinox keeps shifting relative to the stars. Stated otherwise, the Julian calendar is based on the tropical year, the time it takes the sun to travel from vernal equinox to vernal equinox.

The Roman calendar which preceded the Julian calendar was based on lunar cycles. The Julian calendar basically is the calendar now in use. It consists of a 365-d year interspersed with a leap year of 366 d every 4 yr, on the year numbers divisible evenly by 4. Thus, the Julian year is exactly 365.2500 d long, a standard Caesar adopted from the Egyptians. The Julian calendar was also set so that the vernal equinox fell on March 25 in 46 BCE, the traditional date at the time. This required a 3 month adjustment which caused great confusion (46 BCE contained 445 d).

The actual duration of a tropical year, 365.242 189 d, is slightly shorter than both the sidereal year (4) (see Fig. 4b) and also the Julian year. Thus, by the time of Pope

Gregory XIII, in 1582, the date of the vernal equinox had slipped back to March 11. Motivated by the drift of Easter Sunday later and later into the spring (Easter was referenced to the calendar date Mar. 21), Gregory instituted the Gregorian calendar which set March 21 to the vernal equinox. To keep it close to that date, he also adjusted the average length of the year by declaring that three out of four "century years" would not be leap years. In this scheme, 1700,1800, and 1900 would not be leap years, but year numbers evenly divisible by 400, e.g., 1600 and 2000, would still be leap years as they were in the Julian calendar. This brings the calendar into good synchrony with the tropical year; the error is one day in ~3000 yr.

Catholic countries adopted the Gregorian calendar in 1582 but the British Isles and the United States did not do so until 1752 when parliament decreed that the day after Sept. 2 would be Sept. 14. This led to great consternation wherein some people demanded that their 11 d be returned to them. The czars of Russia never adopted the Gregorian calendar. Only after the 1917 revolution was it adopted by the Soviet Union (including Russia), in 1918. Astronomers still use the Julian year of 365.25 d x 86400 s/d = 31 557 600 s and the Julian century of 36525 d as sometimes useful time intervals.

This discussion traces the development of the Gregorian calendar, currently the international standard. Other important systems are (or were) the Mayan-Aztec, Hebrew, Asian Indian, Islamic, and Chinese calendars.

Zodiac

A great deal of attention is paid to the signs of the zodiac by astrologers. If you were born on December 7, as I was, your sign would be Sagittarius. The sun is supposedly located in the so-named constellation on that date. However, since the calendar follows the earth's precession, the sun is no longer in Sagittarius on Dec. 7; it is in the adjacent constellation of Ophiuchus. It was also there on the Dec. 7 (1930) when I was born. Thus, I can not understand why I am instructed to read the horoscope for Sagittarius. However, it doesn't really matter because there is no physical basis for the predictions therein.

Proper motion

The other important motion of the stars is that due to their diverse individual motions on the celestial sphere. Their transverse angular velocities relative to the solar system observer are known as proper motions, which we mentioned in Section 3.2. Since proper motions are angular motions, radial motion is not included. This projected angular velocity arises from the actual motions of the stars in inertial space relative to the barycenter (center of mass) of the (moving) solar system.

The effects of the earth's motion relative to the barycenter are subtracted from the quantities measured from the earth.

Proper motion is due in part to the movement of stars, including the sun, about the center of the Galaxy. The stars closer to the center of the Galaxy rotate with greater angular velocity than those farther out. Random motions of stars due to gravitational interactions also contribute to the proper motion of a given star.

Motion on celestial sphere Proper motion is a rate of change of angular position, f = d9/dt (radians/s) on the celestial sphere, relative to the distant galaxies. It is often given in the mixed units of milliarcsec per year (mas/yr) by astronomers. The direction of movement is given by specifying the rate of movement in the directions of right ascension a and declination 8, that is (d9/dt)a and (d9/dt)8. These angular displacements are great-circle angles, the angles you would obtain by measuring along two axes on a photograph of that small part of the sky using the appropriate scale factor for the camera to convert from mm to arcsec. The photograph is a projection onto a plane tangent to the celestial sphere at the position of the star.

One should be cautious about the relation between the great circle angles measured and the associated changes in right ascension and declination. In declination, there is a direct equivalence because declination is measured along the great-circle meridians. Thus a proper motion of 1''/yr in declination would lead to a change in declination of 1.0" in one year, e.g., from 35° 00' 00" to 35° 00' 01".

In contrast, a proper motion of 1''/yr in the direction of right ascension (a) could lead to a larger change in a, if specified in terms of angle, because the lines of constant a (the meridians) converge toward the celestial poles. For example, at declination 45°, a great-circle angle of 1'' would subtend 1"^/2 = 1.4" of right ascension. Since a is actually measured in h m s with 24 h corresponding to 360° on the equator, it follows that 1 h corresponds to 15 ° and that 1 s (of time) corresponds to 15'' (of angle), and conversely 1'' corresponds to 1/15 s of time. Thus our = 1.4'' of right ascension (at 8 = 45°) would correspond to Aa = (1/15) x -J2 = 0.094 s of time. In one year the right ascension of the star in question might thus change from 23 h 00 m 00 s to 23 h 00 m 00.094 s. Alternatively, a catalog may give the proper motion in RA as a direct correction to a in seconds of time per year, s/yr.

The proper motion of each star is unique. Thus catalogs of precise positions must list the magnitude of this motion for each star in each coordinate, a and 8, on the celestial sphere. Only a few hundred stars have proper motions greater than 1''/yr. The largest value 10.25''/yr is for Barnard's star. Its close proximity to the sun, only 6 LY, leads to this large angular velocity. Measured proper motions range down to <1 mas/yr.

Peculiar motion and local standard of rest

The velocity of a star in three-dimensional space relative to the barycenter of its neighbors is called its peculiar motion. It is likely due to gravitational interactions with other stars at some time in the past. Stars that approach close to one another experience mutual gravitational attraction and hence accelerations. The distances between stars are normally so great that their motions at present can be assumed to be straight lines over the mere —100 yr of modern astronomy. The peculiar motions of stars in the solar neighborhood are measured with respect to the frame of reference that moves with their barycenter (center of mass), the local standard of rest. Their peculiar motions are typically of order 10 km/s with a few reaching 100 km/s. They contribute to the observed angular velocities relative to the celestial coordinate system, i.e., to the proper motions.

Solar motion

Measurements of the radial and transverse motions of stars in the vicinity of the sun, and in all directions about it, indicate that the sun is moving (in the local standard of rest) toward the bright star Vega in the constellation Lyra (a = +18 h, 8 = +30°) at a speed of 19.5 km/s (4.1 AU/year). On the average, spectra of stars in this direction show a blue Doppler shift, and those in the opposite direction show a red shift. The celestial coordinate toward which the sun moves is called the apex of solar motion; the position from which it recedes is called the antapex. This motion is distinct from the general motion of the sun with its neighboring stars about the center of the Galaxy. See Section 9.5 ("Secular and statistical parallaxes") for more on this.

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