## Perturbations of Coordinates

Even if a star remains fixed with respect to the Sun, its coordinates can change, due to several disturbing effects. Naturally its altitude and azimuth change constantly because of the rotation of the Earth, but even its right ascension and declination are not quite free from perturbations.

Precession. Since most of the members of the solar system orbit close to the ecliptic, they tend to pull the equatorial bulge of the Earth towards it. Most of this "flattening" torque is caused by the Moon and the Sun.

Fig. 2.16. Due to precession the rotation axis of the Earth turns around the ecliptic north pole. Nutation is the small wobble disturbing the smooth precessional motion. In this figure the magnitude of the nutation is highly exaggerated

Ecliptic north pole

Celestial pole

### Ecliptic north pole

Fig. 2.16. Due to precession the rotation axis of the Earth turns around the ecliptic north pole. Nutation is the small wobble disturbing the smooth precessional motion. In this figure the magnitude of the nutation is highly exaggerated

But the Earth is rotating and therefore the torque cannot change the inclination of the equator relative to the ecliptic. Instead, the rotation axis turns in a direction perpendicular to the axis and the torque, thus describing a cone once in roughly 26,000 years. This slow turning of the rotation axis is called precession (Fig. 2.16). Because of precession, the vernal equinox moves along the ecliptic clockwise about 50 seconds of arc every year, thus increasing the ecliptic longitudes of all objects at the same rate. At present the rotation axis points about one degree away from Polaris, but after 12,000 years, the celestial pole will be roughly in the direction of Vega. The changing ecliptic longitudes also affect the right ascension and declination. Thus we have to know the instant of time, or epoch, for which the coordinates are given.

Currently most maps and catalogues use the epoch J2000.0, which means the beginning of the year 2000, or, to be exact, the noon of January 1,2000, or the Julian date 2,451,545.0 (see Sect. 2.15).

Let us now derive expressions for the changes in right ascension and declination. Taking the last transformation equation in (2.23), sin 8 = cos e sin ¡i + sin e cos ¡i sin X , and differentiating, we get cos 8 d8 = sin e cos ¡i cos X dX .

Applying the second equation in (2.22) to the right-hand side, we have, for the change in declination,

By differentiating the equation cos a cos 8 = cos i cos X , we get

— sin a cos 8 da — cos a sin 8 d8 = — cos ¡i sin X dX ;

and, by substituting the previously obtained expression for d8 and applying the first equation (2.22), we have sin a cos 8 da = dX(cos ¡i sin X — sin e cos2 a sin 8) = dX(sin 8 sin e + cos 8 cos e sin a — sin e cos2 a sin 8).

Simplifying this, we get da = dX(sin a sin e tan 8 + cos e). (2.26)

If dX is the annual increment of the ecliptic longitude (about 50"), the precessional changes in right ascension and declination in one year are thus d8 = dX sin e cos a , da = dX(sin e sin a tan 8 + cos e).

These expressions are usually written in the form d8 = n cos a , da = m + n sin a tan 8 , where m = dX cos e , n = dX sin e

are the precession constants. Since the obliquity of the ecliptic is not exactly a constant but changes with time, m and n also vary slowly with time. However, this variation is so slow that usually we can regard m and n as constants unless the time interval is very long. The values of these constants for some epochs are given in

Table 2.1. Precession constants m and n. Here, "a" means a tropical year dá = dX sin e cos a .

Table 2.1. Precession constants m and n. Here, "a" means a tropical year

Epoch |
m |
n | |

1800 |
3.07048 s/a |
1.33703 s/a |
= 20.0554"/a |

1850 |
3.07141 |
1.33674 |
20.0511 |

1900 |
3.07234 |
1.33646 |
20.0468 |

1950 |
3.07327 |
1.33617 |
20.0426 |

2000 |
3.07419 |
1.33589 |
20.0383 |

Table 2.1. For intervals longer than a few decades a more rigorous method should be used. Its derivation exceeds the level of this book, but the necessary formulas are given in *Reduction of Coordinates (p. 38).

Nutation. The Moon's orbit is inclined with respect to the ecliptic, resulting in precession of its orbital plane. One revolution takes 18.6 years, producing perturbations with the same period in the precession of the Earth. This effect, nutation, changes ecliptic longitudes as well as the obliquity of the ecliptic (Fig. 2.16). Calculations are now much more complicated, but fortunately nuta-tional perturbations are relatively small, only fractions of an arc minute.

Parallax. If we observe an object from different points, we see it in different directions. The difference of the observed directions is called the parallax. Since the amount of parallax depends on the distance of the observer from the object, we can utilize the parallax to measure distances. Human stereoscopic vision is based (at least to some extent) on this effect. For astronomical purposes we need much longer baselines than the distance between our eyes (about 7 cm). Appropriately large and convenient baselines are the radius of the Earth and the radius of its orbit.

Distances to the nearest stars can be determined from the annual parallax, which is the angle subtended by the radius of the Earth's orbit (called the astronomical unit, AU) as seen from the star. (We shall discuss this further in Sect. 2.10.)

By diurnal parallax we mean the change of direction due to the daily rotation of the Earth. In addition to the distance of the object, the diurnal parallax also depends on the latitude of the observer. If we talk about the parallax of a body in our solar system, we always mean the angle subtended by the Earth's equatorial radius (6378 km) as seen from the object (Fig. 2.17). This equals the apparent shift of the object with respect to the background stars seen by an observer at the equator if (s)he observes the object moving from the horizon to the zenith. The parallax of the Moon, for example, is about 57', and that of the Sun 8.79".

In astronomy parallax may also refer to distance in general, even if it is not measured using the shift in the observed direction.

Aberration. Because of the finite speed of light, an observer in motion sees an object shifted in the direction of her/his motion (Figs. 2.18 and 2.19). This change of apparent direction is called aberration. To derive

Fig. 2.19. A telescope is pointed in the true direction of a star. It takes a time t = l/c for the light to travel the length of the telescope. The telescope is moving with velocity v, which has a component v sin 0, perpendicular to the direction of the light beam. The beam will hit the bottom of the telescope displaced from the optical axis by a distance x = tv sin 0 = l(v/c) sin 0. Thus the change of direction in radians is a = x/l = (v/c) sin 0

Fig. 2.19. A telescope is pointed in the true direction of a star. It takes a time t = l/c for the light to travel the length of the telescope. The telescope is moving with velocity v, which has a component v sin 0, perpendicular to the direction of the light beam. The beam will hit the bottom of the telescope displaced from the optical axis by a distance x = tv sin 0 = l(v/c) sin 0. Thus the change of direction in radians is a = x/l = (v/c) sin 0

the exact value we have to use the special theory of relativity, but for practical purposes it suffices to use the approximate value a = - sin 0 , [a] = rad, c

Let the true zenith distance be z and the apparent one, Z. Using the notations of Fig. 2.20, we obtain the following equations for the boundaries of the successive layers:

sin z = nk sin zk , where v is the velocity of the observer, c is the speed of light and 0 is the angle between the true direction or of the object and the velocity vector of the observer. The greatest possible value of the aberration due to the orbital motion of the Earth, v/c, called the aberration constant, is 21". The maximal shift due to the Earth's rotation, the diurnal aberration constant, is much smaller, about 0.3".

n2 sinz2 = ni sinzi , ni sinz1 = n0sin Z , sin z = n0 sin Z .

Refraction. Since light is refracted by the atmosphere, the direction of an object differs from the true direction by an amount depending on the atmospheric conditions along the line of sight. Since this refraction varies with atmospheric pressure and temperature, it is very difficult to predict it accurately. However, an approximation good enough for most practical purposes is easily derived. If the object is not too far from the zenith, the atmosphere between the object and the observer can be approximated by a stack of parallel planar layers, each of which has a certain index of refraction n (Fig. 2.20). r _ Outside the atmosphere, we have n = 1.

When the refraction angle R = z — Z is small and is expressed in radians, we have n0 sin Z = sin z = sin( R + Z)

Thus we get

The index of refraction depends on the density of the air, which further depends on the pressure and temperature. When the altitude is over 15°, we can use an approximate formula

where a is the altitude in degrees, T temperature in degrees Celsius, and P the atmospheric pressure in hectopascals (or, equivalently, in millibars). At lower altitudes the curvature of the atmosphere must be taken into account. An approximate formula for the refraction is then

P 0.1594 + 0.0196a + 0.00002a2 = 273 + T 1 + 0.505a + 0.0845a2 .

These formulas are widely used, although they are against the rules of dimensional analysis. To get correct values, all quantities must be expressed in correct units. Figure 2.21 shows the refraction under different conditions evaluated from these formulas.

Altitude is always (except very close to zenith) increased by refraction. On the horizon the change is about 34', which is slightly more than the diameter of the Sun. When the lower limb of the Sun just touches the horizon, the Sun has in reality already set.

Light coming from the zenith is not refracted at all if the boundaries between the layers are horizontal. Under some climatic conditions, aboundary (e. g. between cold and warm layers) can be slanted, and in this case, there can be a small zenith refraction, which is of the order of a few arc seconds.

Stellar positions given in star catalogues are mean places, from which the effects of parallax, aberration and nutation have been removed. The mean place of the date (i. e. at the observing time) is obtained by cor-

\ | |

s |
----P = 1050 hPa, T= - -30 °C |

v \ |
- P = 950 hPa, r=+30°C |

\ V \ V |
P= 700 hPa, r= 0°C |

Fig. 2.21. Refraction at different altitudes. The refraction angle R tells how much higher the object seems to be compared with its true altitude a. Refraction depends on the density and thus on the pressure and temperature of the air. The upper curves give the refraction at sea level during rather extreme weather conditions. At the altitude of 2.5 kilometers the average pressure is only 700 hPa, and thus the effect of refraction smaller (lowest curve)

Fig. 2.21. Refraction at different altitudes. The refraction angle R tells how much higher the object seems to be compared with its true altitude a. Refraction depends on the density and thus on the pressure and temperature of the air. The upper curves give the refraction at sea level during rather extreme weather conditions. At the altitude of 2.5 kilometers the average pressure is only 700 hPa, and thus the effect of refraction smaller (lowest curve)

recting the mean place for the proper motion of the star (Sect. 2.10) and precession. The apparent place is obtained by correcting this place further for nutation, parallax and aberration. There is a catalogue published annually that gives the apparent places of certain references stars at intervals of a few days. These positions have been corrected for precession, nutation, parallax and annual aberration. The effects of diurnal aberration and refraction are not included because they depend on the location of the observer.

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