# Personal equation

The personal equation for star observations in astronomy and geodesy is the average reaction time of the observer. Depending on the measurement method and the experience of the observer, it is between a few hundredths and tenths of a second.

In contrast to the random time errors , which mostly stand out when averaging many measurements, the personal equation as a systematic error always works in the same direction. It therefore influences the measurement results noticeably, so that they have to be determined by reference measurements or special experiments (e.g. artificial star ) and taken into account in the calculation.

Even with automated measurements (using optoelectronic sensors or the like), there are small systematic effects that are analogously called instrumental equations . For example, a measurement accuracy of 0.1 "has the time at telescopes on the Earth's surface to the hundredths of seconds (see 0.01) are measured because of the star, according to the Earth's rotation by about 5 to 15" move per second.

## Overview: average values

The personal equation has - as the name suggests - for each observer a characteristic value, which is usually quite stable over periods of many months. It also changes relatively little due to tiredness or external circumstances. This value can therefore be determined very reliably by reference measurements (in which the target result is known) and subtracted from the measured times. In addition, the personal equation depends on the experience of the observer. The following typical values ​​are given in the specialist literature:

• Visual star passages (telescope with at least 30x magnification):
• Experienced observers 0.05 s to 0.20 s, although the value can fluctuate by around 0.03 s
(For some people, the personal equation can also be negative , which means a reaction in the range of 0.1 s too early )
• Less experienced observers between 0.1 and 0.4 s with fluctuations of about ± 0.05 s.
(Between the 2nd and 5th night of measurement, however, the value stabilizes at a level typical of the person)
• On the recording micrometer (manual tracking) between 0.01 and 0.15 s (fluctuation by approx. 0.02 s)
• With automatic tracking and manual correction under 0.10 seconds
• When the stars are covered by the moon, an average of 0.3 seconds.

Such short reaction times may seem implausible to the layman, as they are far below the so-called shock moment . A star passage, however, is not surprising , but precisely predictable.

If, on the other hand, an event is really unexpected - for example a falling star - even an experienced astronomer must expect a longer delay, which can be up to 1 second. To a certain extent, you can estimate it in retrospect (in your imagination, preferably with your eyes closed).

## Visual measurement of star passages

Numerous methods of astrometry and astrogeodesy are based on the measurement of star passages through a crosshair or a reticle of a suitable telescope . Mention should be made, among other things, of the precise determination of sidereal time and star locations (right ascension, declination), geographical coordinates (more precisely: astronomical latitude and longitude ) - especially for plumbing deviation and geoid determination - as well as for determining the direction ( azimuth ) and the size of celestial bodies .

When measuring a star passage in the field of view of a theodolite or passage instrument, the observer registers the point in time at which the star is located exactly behind the thread or is "bisected" by it. This can be done using a digital stopwatch , a hand button and chronograph or the eye-to-ear method . About the tracking on the moving thread see below.

The mean accuracy that can be achieved by the observer is called the throughput error. Apart from errors in the time system (e.g. clock errors , time signals ) and when setting up the instrument , the passage through the star - or a whole series of "thread approaches" - results in the superposition of two personal influences:

• a delayed reaction around a person-typical value (in extreme cases everyday language says "long lead")
• a somewhat asymmetrical view of bisection.

The first influence is inevitable, but constant for about 0.03 s. The second can be minimized through calm attention or with increasing experience and can be eliminated through special measuring arrangements, for example through an inverted prism or the observation of symmetrical pairs of stars .

## Measurements on the recording micrometer

In the second half of the 19th century, the instrument maker Johann Adolf Repsold invented the recording micrometer named after him . A thread that is movably arranged in the thread network can follow the star and is connected to a measuring spindle that closes electrical contacts at precisely defined intervals. If one takes the mean of the times registered in this way, this corresponds to the star passage on the central thread of the visual field.

The Repsold micrometer is also called the impersonal micrometer , although it does not completely eliminate the personal equation . However, it significantly reduces the systematic measurement errors . Analogous to the above, they can be broken down into two parts, the bisection error and the tracking error.

1. Bisection error means that the observer does not hold the moving thread exactly on the "star center" ( diffraction disk ), but always something to the right or always a little to the left of the center. It depends on the apparent stellar speed and for a star near the celestial equator is between 10 and 30 milliseconds (0.01 to 0.04 s) for most observers . It can be eliminated if the sighting telescope according to the center folded and the moving direction of the star is reversed. In the case of a “broken telescope” (for example universal instruments of the T4 or DKM3 type ), this happens automatically , while a straight telescope requires an “ocular reversing prism”.
2. Tracking error means that the moving thread is allowed to precede or follow the star slightly. If the bisection error is eliminated, the tracking error is identical to the personal equation . It can be determined with a special test arrangement (" artificial star ") and is (according to Steinert) between 20 and 40 ms.

## Measurements with motorized tracking, photography or sensors

In order to relieve the observer from the concentrated, even turning of the recording micrometer, a motor-driven tracking system has been designed for some larger instruments ( meridian circle , Danjon astrolabe, circumzenital ), which only needs to be adjusted slightly to the speed of the star. It reduces the personal equation again, whereby the remaining amounts are very constant and can be easily determined by measuring at reference stations . With the Danjon astrolab, the observer has to hold two opposing constellations with a handwheel at the same height, which is achieved in about 0.01 s.

The personal equation can only be avoided entirely through automatic observation methods:

However, this is paid for with instrumental errors, some of which are difficult to model mathematically.

## Newer Methods of Cosmic Geodesy

When the aim was to get into the millisecond accuracy range in the 1970s , these small residual errors were an obstacle that was difficult to overcome. Therefore, the described measuring methods have gradually been replaced by other, fully automatable measuring principles:

As before, however, methods are in use in which the personal equation plays a minor role, but still plays a role, such as the precise determination of astronomical differences in length (so-called length compensation via continental surveying networks ) for reference stations of the absolute vertical deviation for a continental "centimeter geoid " (especially with the Ni2 astrolabe in high mountains ) and for some special purposes.

Since the personal equation of a visual observer can be eliminated arithmetically in almost all cases (residual error, depending on the method and effort, less than 0.01 to 0.03 s), the question of further automation methods remains one of the cost-benefit ratio . For monitoring, for example, the rotation of the earth and the polar movement , she switched to modern satellite and quasar methods around 1980 (see Cosmic Geodesy and IERS ), while astrogeodesy continues to offer suitable methods for expeditions and some geoid projects.

## history

In 1796 the astronomer Royal Nevil Maskelyne dismissed his assistant Kinnebrooke, because he had recorded star approaches in the micrometer on average 0.8 seconds later than Maskelyne himself. The effect was initially forgotten until the Königsberg astronomer Friedrich Wilhelm Bessel in 1816 on the story Kinnebrookes noticed.

Until 1821, Bessel made systematic investigations into this effect, which he carried out with his assistant Walbeck . He found that there was more than a second difference between Walbeck's observations and his own. Bessel later continued the research in collaboration with other astronomers. He is considered to be the discoverer of the personal equation.