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Let $f_*$ and $f_0$ be the observed flux of a star and a reference flux in a particular spectral band, and let $m_*$ and $m_0$ be their respective apparent magnitudes. Then the star's magnitude is given by $m_*-m_0=-2.5log(f_*/f_0)$.

*Encyclopædia Britannica* maintains that

All magnitude systems must have a reference, or zero, point.

Why is this? It seems to me that one could just say $m_*=-2.5log f_*$ and move on, no?

Because you can't take the logarithm of something with dimensions only the log of a number.

Your magnitude would depend on what you measured f in. In other words, by choosing a set of units, you choose a zero point! Whether that unit is Jy, W/m$^2$ or the flux of some other object.

## Thread: question about absolute magnitude scale

I realize that 5 points in the absolute magnitude scale represents a 100x change in luminosity. I also realize that the reference distance for a stellar object is 10 parsecs.

But why is the Sun's magnitude set at 4.85? Sol is often used as a reference point for stars, whether it be stellar radii or mass (and why not temperature?).

It just seems like having Sol's magnitude benchmarked at 5 instead of 4.83 would be so much simpler, especially since the scale is based on 5-point intervals.

So how did the 4.85 come about?

Is that it's *actual* luminosity, rather than visual? I don't think the greeks had ways of determining those kinds of things thousands of years ago (I think they were the ones who devised a magnitude of 1 - 6 for visible stars).

You can compute a star's absolute magnitude with the following formula:

A slightly larger number could have been chosen instead of 10pc to make the Sun exactly 5, but that would be sloppier than leaving it at 10pc and having the Sun just under mag 5.

*Apparent* magnitudes came first, before we knew how far away the stars were. The apparent magnitude scale was tied to various reference stars, firstly just "of first magnitude" of "second magnitude", and so on. Once photographic techniques were introduced, the apparent magnitude of these reference stars was refined to fit in with Pogson's formalization of the magnitude scale (5 grades = 100-fold change). Meanwhile, we were also sorting out the distances to stars, and coming up with the idea of absolute magnitude. By that time, with the apparent magnitude scale already well established, I guess the choice would have been either to choose a non-integral standard distance, so that the Sun came out with an integral magnitude, or to choose an integral standard distance, in which case the Sun came out with a non-integral standard magnitude.

Since we're generally converting between absolute and apparent magnitude *for a given star*, the integral-standard-distance route had a marginal advantage, because it slightly simplified the sums. I guess there is also the possibility that no-one had nailed down the apparent magnitude of the sun accurately enough at the time the decision was made . it's a bit of a stretch comparing the daytime sun to a standard reference star.

*Apparent* magnitudes came first, before we knew how far away the stars were. The apparent magnitude scale was tied to various reference stars, firstly just "of first magnitude" of "second magnitude", and so on. Once photographic techniques were introduced, the apparent magnitude of these reference stars was refined to fit in with Pogson's formalization of the magnitude scale (5 grades = 100-fold change). Meanwhile, we were also sorting out the distances to stars, and coming up with the idea of absolute magnitude. By that time, with the apparent magnitude scale already well established, I guess the choice would have been either to choose a non-integral standard distance, so that the Sun came out with an integral magnitude, or to choose an integral standard distance, in which case the Sun came out with a non-integral standard magnitude.

Since we're generally converting between absolute and apparent magnitude *for a given star*, the integral-standard-distance route had a marginal advantage, because it slightly simplified the sums. I guess there is also the possibility that no-one had nailed down the apparent magnitude of the sun accurately enough at the time the decision was made . it's a bit of a stretch comparing the daytime sun to a standard reference star.

Indeed. Magnitudes are generally used for nighttime astronomy, so it was more important to get a scale which was useful for nighttime astronomy than one which was useful for solar observations.

## Why must all magnitude systems have a reference point? - Astronomy

Lecture 1: Introduction to Astronomy 250

Astronomical Coordinate Systems:

Astronomers base their measurement of positions of objects on the concept of the*celestial sphere* upon which all objects are assumed to lie regardless of their true distances. The celestial poles and equator are the projections of the Earth's poles and equator onto the sky. The**meridian** is the circle running from one pole to the other through a point directly overhead for an observer. The point directly overhead is called the**zenith** (and the point 180º away is called the nadir).

Several different coordinate systems are useful depending on the situation:

**Equatorial coordinates:** An Earth-based system useful for pointing telescopes with axes that are parallel to the Earth's polar axis and equator (called equatorial mount telescopes). This system was the first used in compiling stellar catalogues, and the two coordinates used to define the location of an object, right ascension (abbreviated often as or RA ) and declination ( or DEC), are in common use today.

Right ascension is analogous to longitude, is usually measured in units of time: hours, minutes, seconds. The zero-point for right ascension is the Vernal Equinox (also called the Aries Point in the text), location on the celestial equator of sunrise on the first day of spring. The total range of right ascension is 24 hrs = 360 deg / 15 deg/hr. The 15 deg/hr conversion factor arises from the rotation rate of the Earth.

Declination is analogous to latitude and is measured as north or south of the celestial equator. Declination is usually expressed in degrees, minutes of arc, and seconds of arc.

1 degree = 1 º = 60 arc minutes = 60' = 3600 arc seconds = 3600"

Note that because right ascension is measured in time units, before performing calculations you need to multiply by 15 degrees/hr. Another complication with right ascension arises from the changing angular size of circles of constant right ascension when moving from the celestial equator towards the celestial pole, the circles shrink by a factor of cos(DEC) which must be taken into account.

1' = 1/15 min = 60/15 secs = 4 secs 15' = 1 minute

It is very important to keep minutes of time and arc minutes clear!

Examples: A star on the celestial equator with right ascension 6 hrs lies 6 hrs x 15 deg/hr = 90 degrees from the Vernal equinox.

A star at 60 deg declination and right ascension 6 hrs lies 6hrs x 15 deg/hr x cos(60) = 45 degrees from a point at 60 deg declination and 0 hrs right ascension.

When using equatorial coordinates, the**H.A.**or**hour angle** of an object is also useful to know. The hour angle is the distance in time units that object lies east or west of the meridian. It is related to time as measured by the stars called sidereal time via the relation

H.A. = Local sidereal time - right ascension

If the H.A. for an object is negative, it lies east of the meridian and will cross the meridian, reaching its highest point in the sky, when the sidereal time equals its right ascension. This illustrates how the equatorial system accounts for the rotation of the Earth and explains why early astronomers chose to use time units for right ascension.

In addition to RA and DEC, astronomers use a number of other coordinate systems depending on the circumstances. Coordinates based on a telescope's location can be advantageous as can coordinates based on the plane of the Solar System or on the plane of the Milky Way galaxy.**Spherical trigonometry**is handy for converting between coordinates systems.

Positions of stars and other objects in the sky can be used for navigation (now laregely made obsolete by Global Positioning Systems). For example, the elevation of the North Star above an observer's horizon is equal to the observer's latitude:

The Earth's rotation is used a basic unit of time measure -- the day. If you keep track of the intervals between successive times of maximum elevation of the Sun above the horizon (e.g. noon), you will discover a complicated pattern with the differences between these intervals and an average day called the equation of time.

These differences result from two causes -- first, we measure time with respect to the Earth's rotation axis and hence equator while the Sun's motion is along the ecliptic, and second, the Earth's orbit is elliptical and the Earth moves faster along its orbit when closer to the Sun.

We also use the time that it takes the Earth to orbit the Sun as a time unit, the year. Note that several different lengths of a year can be defined depending on what reference is used:

Sidereal year = length of time for the Earth to return to the same position with respect to the stars

Tropical year = length of time between Vernal equinoxes

** Seasons on our Home Planet**

Seasons result from the 23.5º inclination of the Earth's polar axis with respect to the plane of the Earth's motion around the Sun (called the ecliptic). The polar axis points in an invariant* direction in space so that for example, the north pole alternately points towards the Sun (summer) or away from the Sun (winter). The seasons are obviously opposite in the Earth's two hemispheres.

* Not strictly invariant -- remember precession! The polar direction is slowly changing due to precession (direction takes 26,000 yrs to complete one circuit) but the angle between the polar axis and the ecliptic is fixed and invariant.

We also tend to identify various constellations with the seasons because the stars visible at night change throughout the year. The ecliptic passes through the constellations of the zodiac.

The Month: Time Unit Based on the Moon

28 days to orbit the Earth with the time ranging from 27.32 days if measured with respect to the stars (**sidereal month**) or 29.53 days if measured with respect to the Sun (**synodic month**). The phases of the moon repeat with a month time scale as the Moon orbits the Earth.

** Positional Astronomy**

Astrometry: the science of measuring stellar positions very accurately.

- A century ago was done with a special purpose telescope which can point only at the meridian.
- Satellites in space are beginning to do this job far more accurately than can be done with observations through the Earth's shimmering atmosphere

When making precise measurements of stellar positions, various effects must be taken into account:

- Aberration -- an object's position will be shifted slightly because of the finite speed of light and the observer's motion

- Refraction -- light gets bent (refracted) by the Earth's atmosphere, a star's altitude is increased by refraction. When the Sun is setting, it is actually beneath the horizon when its lower limb just appears to touch the horizon -- in other words, the refraction amounts to

By making accurate position measurements, we have discovered a number of interesting effects such as precession which is caused by the torque of the moon on the Earth. The direction that the Earth's polar axis is pointing in space is slowly changing and describes a circle on the sky (and it takes the polar direction 26,000 years to travel this circle once). The location of the Vernal Equinox shifts by 50" per year. Precession also means that positions of astronomical objects are tabulated for a stated time such as 2000.0 meaning the beginning of 2000 and need to be corrected for the date when you will be observing.

Eclipses occur when the Earth, Sun, and Moon lie along a line such that either the shadow of the Earth falls on the Moon (lunar eclipse) or the Moon's shadow falls on the Earth (solar eclipse). *Note that lunar eclipses can only occur at full moon and conversely, solar eclipses can occur only at new moon.*

Why don't eclipses occur every month?

The line connecting the two intersection points between the Moon's orbit and the ecliptic plane is called the line of nodes. Only when the line of nodes is pointing towards the Sun can eclipses occur. The calendar time of this "eclipse season" slowly changes because of gravitational forces exerted by the Sun on the Moon. The direction of the line of nodes makes a complete 360º circuit in 18.6 years (called the Saros cycle).

The existence of solar eclipses is a lucky happenstance -- the apparent diameters of the Sun and Moon are nearly equal.

Because the Moon's orbit around the Earth is an ellipse, the apparent size of the Moon varies during a month. If a solar eclipse occurs when the Moon is at a distant part of its orbit, it will be too small to cover the disk of the Sun fully and an annular eclipse occurs. The Moon's orbit is slowly increasing in size so someday there will be no total solar eclipses.

## Absolute Magnitude

The apparent brightness of a star is how bright it seems when viewed from the Earth, but a large, bright star can appear dim if it is a long way from the Earth and a dim star can appear to be bright if it is close to the Earth. Therefore, the apparent magnitude has no bearing on the distance from the Earth.

To give an acurate measurement of the brightness of a star we need to make an absolute magnitude scale. The absolute magnitude is how bright a star is when viewed from a set distance. Stars being rather large objects, a distance of 10 parsecs was chosen.

## Answers and Replies

Hello,

It is well know that the moment of a force ##F## depends on:

a) the force magnitude ##|F|##

b) the choice of the moment reference point ##P##

c) the distance (lever arm) from the point ##P## to the point of application of the force ##Q##.

That said, an object with a single force applied to it will experience a moment which will vary in magnitude and sign with difference choices of the moment reference point ##P##. However, physically, the object will move in one specific and unique way under that same force (rotation+translation). How do different values of the moment ##M## produce the same physical situation?

The same linear momentum counts as a different amount of angular momentum depending on the location of the reference point.

The same linear force counts a a different amount of torque depending on the location of the reference point.

The two effects match so that no matter where you choose to put the reference point, the rate of change in angular momentum will match the applied torque. Moving the reference point simply gives a different set of coordinates to describe the same physical reality.

## Practical Introduction to Frequency-Domain Analysis

This example shows how to perform and interpret basic frequency-domain signal analysis. The example discusses the advantages of using frequency-domain versus time-domain representations of a signal and illustrates basic concepts using simulated and real data. The example answers basic questions such as: what is the meaning of the magnitude and phase of an FFT? Is my signal periodic? How do I measure power? Is there one, or more than one signal in this band?

Frequency-domain analysis is a tool of utmost importance in signal processing applications. Frequency-domain analysis is widely used in such areas as communications, geology, remote sensing, and image processing. While time-domain analysis shows how a signal changes over time, frequency-domain analysis shows how the signal's energy is distributed over a range of frequencies. A frequency-domain representation also includes information on the phase shift that must be applied to each frequency component in order to recover the original time signal with a combination of all the individual frequency components.

A signal can be converted between the time and frequency domains with a pair of mathematical operators called a transform. An example is the Fourier transform, which decomposes a function into the sum of a (potentially infinite) number of sine wave frequency components. The 'spectrum' of frequency components is the frequency domain representation of the signal. The inverse Fourier transform converts the frequency domain function back to a time function. The fft and ifft functions in MATLAB allow you to compute the Discrete Fourier transform (DFT) of a signal and the inverse of this transform respectively.

### Magnitude and Phase Information of the FFT

The frequency-domain representation of a signal carries information about the signal's magnitude and phase at each frequency. This is why the output of the FFT computation is complex. A complex number, , has a real part, , and an imaginary part, , such that . The magnitude of is computed as , and the phase of is computed as . You can use MATLAB functions abs and angle to respectively get the magnitude and phase of any complex number.

Use an audio example to develop some insight on what information is carried by the magnitude and the phase of a signal. To do this, load an audio file containing 15 seconds of acoustic guitar music. The sample rate of the audio signal is 44.1 kHz.

Use fft to observe the frequency content of the signal.

The output of the FFT is a complex vector containing information about the frequency content of the signal. The magnitude tells you the strength of the frequency components relative to other components. The phase tells you how all the frequency components align in time.

Plot the magnitude and the phase components of the frequency spectrum of the signal. The magnitude is conveniently plotted in a logarithmic scale (dB). The phase is unwrapped using the unwrap function so that we can see a continuous function of frequency.

You can apply an inverse Fourier transform to the frequency domain vector, Y, to recover the time signal. The 'symmetric' flag tells ifft that you are dealing with a real-valued time signal so it will zero out the small imaginary components that appear on the inverse transform due to numerical inaccuracies in the computations. Notice that the original time signal, y, and the recovered signal, y1, are practically the same (the norm of their difference is on the order of 1e-14). The very small difference between the two is also due to the numerical inaccuracies mentioned above. Play and listen the un-transformed signal y1.

## Orbital Dynamics 101

Historically, it was the observed the orbital motions of double stars that helped to prove the validity of Newton’s description of gravitational attraction. As well as his impressive laws of motion. He applied these rules to everything in the heavens. Not just to the planets and periodic comets but equally to the far away celestial motions of the stars as they danced about in the darkness above.

The observation of these distant stars helped lay the foundation for theories of stellar structure and evolution.

## Photometric Keywords in SCI Extensions of ACS Images

Here we describe several header keywords present in ACS FITS files. These keywords can be used to obtain photometric calibration information for your data.

- PHOTMODE : Observation configuration for photometric calibration.
- PHOTFLAM : Inverse sensitivity (units: erg cm &minus2 Å &minus1 electron &minus1 ). This represents the scaling factor necessary to transform an instrumental flux in units of electrons per second to a physical flux density.
- PHOTZPT : STMag zeropoint.
- PHOTPLAM : Pivot wavelength (units: Å)

The PHOTFLAM and PHOTPLAM header keywords are used to derive the instrumental zeropoint magnitudes, which are defined to be the magnitude of an object that produces one count per second. The instrumental magnitudes are defined as follows:

(ZP_

(ZP_

In addition to being present in the image headers, the PHOTFLAM value for a given date can be calculated using the acszpt module. See the Examples below.

**WARNING:** The ACS absolute flux calibration represented by the PHOTFLAM keyword is applicable to the distortion corrected pipeline products (*_drz.fits or *_drc.fits) produced by AstroDrizzle. In order to extract photometry from non-geometrically-corrected pipeline products (*_flt.fits or *_flc.fits), the appropriate pixel area maps must be applied to the images first. See the ACS Data Handbook section 5.1.3 for more information.

## Astronomical Coordinate Systems

The coordinate systems considered here are all based at one reference point in space with respect to which the positions are measured, the *origin* of the reference frame (typically, the location of the observer, or the center of Earth, the Sun, or the Milky Way Galaxy). Any location in space is then described by the "radius vector" or "arrow" between the origin and the location, namely by the *distance* (length of the vector) and its *direction*. The direction is given by the straight half line from the origin through the location (to infinity). In the *spherical* coordinate systems used here, the direction is fixed by two angles, which are given as follows:

A reference plane containing the origin is fixed, or equivalently the axis through the origin and perpendicular to it (typically, an "equatorial" plane and a "polar" axis) elementarily, each of these uniquely determines the other. One can assign an orientation to the polar axis from "negative" to "positive", or "south" to "north", and simultaneously to the equatorial plane by assigning a positive sense of rotation to the equatorial plane these orientations are, by convention, usually combined by the right hand rule: If the thumb of the right hand point to the positive (north) polar axis, the fingers show in the positive direction of rotation (and vice versa, so that a physical rotation defines a north direction).

The reference plane or the reference axis define the set of planes which contain the origin and are perpendicular to the "equatorial" reference plane (or equivalently, contain the "polar" reference axis) each direction in space then lies precisely in one of these "meridional" planes (or half planes, if the reference axis is taken to divide each plane into halfs), with the exception of the (positive and negative) polar axis which lies in all of them by definition.

The first angle used to characterize a direction, typically the "latitude", is taken between the direction and the reference plane, within the "meridional" plane. For the second angle, it is required to select and fix one of the "meridional" half planes as zero, from which the angle (of "longitude") is measured to the "meridional" half plane containing our direction.

Note that this selection of angles to characterize a direction in a given reference frame is chosen by convention, which is especially common in astronomy and geography, and which is used in the following here, as well as in most astronomical databases. Other, equivalent, conventions are possible, e.g. physicists often use instead of the "latitude" angle to the reference plane, the angle between the direction and the "positive" or "north" polar axis (called "co-latitude" co-latitde = 90 deg - latitude). It depends on taste at last what the reader likes to use, but here we will stay as close to standard astronomical convention as possible. In order to minimize the requirement of case-to-case enumeration of conventions, we also recommend the reader to do the same.

### Positions on Earth

The natural reference plane here is that of the Earth's equator, and the natural reference axis is the rotational, polar axis which cuts the Earth's surface at the planet's North and South pole. The circles along Earth's surface which are parallel to the equator are the *latitude circles*, where the angle at the planet's center is constant for all points on these circles. Half circles from pole to pole, which are all perpendicular to the equatorialplane, are called *meridians*. One of the meridians, in practice that through the Greenwich Observatory near London, England, is taken as reference meridian, or *Null meridian*. *Geographical Longitude* is measured as the angle between this and the meridian under consideration (or more precisely, between the half planes containing them) it is of course the same for all points of the meridian.

*Geocentric Latitude*, measured as angle at the Earth's center, between the equatorial plane and the direction to the surface point under consideration, and*Geographical Latitude*, measured on the surface between the parallel plane to the equatorial plane and the line orthogonal to the surface, the*local vertical*or plumb line, which may be measured by the direction of gravitional force (e.g., plumb).

- Equatorial radius: a = 6378.140 km
- Polar radius: b = 6356.755 km
- Flattening/Oblateness: f = 1/298.253

In the following, we always deal with geographic latitude unless otherwise mentioned.

### The Celestial Sphere

Thus each observer can look at the skies as being manifested on the interior of a big sphere, the so-called *celestial sphere*. Then each direction away from the observer will intersect the celestial sphere in one unique point, and positions of stars and other celestial objects can be measured in angular coordinates (similar to longitude and latitude on Earth) on this virtual sphere. This can be done without knowing the actual distances of the stars. Moreover, any plane through the origin cuts the sphere in a great circle. Examples for celestial coordinate systems are treated below.

**Note**: In times up to Copernicus, people believed that there is actually a solid sphere to which the stars beyond the solar system are fixed: This idea was overcome when it was realized that stars are sunlike bodies, in the time of Newton and Halley. Today, the celestial sphere is only a virtual construct to make our understanding of positional astronomy easier.

### The Horizon System

Through any direction, or point on the celestial sphere, e.g. the position of a star, a unique [half] plane (or great [half] circle) perpendicular to the horizon can be found this is called *vertical circle* all vertical [half] circles contain (and intersect in) both the zenith and the nadir. Within the plane of its verticle circle, the position under consideration can be characterized by the angle to the horizon, called *altitude* `a`. Alternatively and equivalently, one could take the angle between the direction and the zenith, the *zenith distance* `z`, which is related to the altitude by the relation: `z = 90 deg - a`. All objects *above* the horizon have positive altitudes (or zenith distances smaller than 90 deg). The horizon itself can be defined, or recovered, as the set of all points for which `a = 0 deg` (or `z = 90 deg`).

In contrast to the apparent horizon which defines coordinates of objects as the observer perceives them, the *true horizon* is defined by the plane parallel to the apparent horizon, but through the center of Earth. The angle between the position of an object and the true horizon is referred to as *true altitude*. For nearby objects such as the Moon, the measured position can vary notably between these two reference systems (up to 1 deg for the Moon). Also, the apparent altitudes are subject to the effect of refraction by Earth's atmosphere.

The second coordinate of a position in the horizon system is defined by the point where the verticle circle of the position cuts the horizon. It is called *azimuth* `A` and, in astronomy and on the Northern hemisphere (the present author does not know the southern standards for this thread), is the angle from the south point (or direction) taken to the west, north, and east to the foot point of the vertical circle on the horizon, thus running from 0 to 360 deg. In geodesy, the north direction is often taken as zero point (this angle is sometimes called *bearing* and is given by `A +/- 180 deg`). Note that these conventions are not always uniquely used so that it may be advisable to clear up which conventions are used (e.g., by saying `A` is taken to the West).

Taking the astronomical standard, the *south*, *west*, *north*, and *east* points on the horizon are defined by `A = 0 deg`, `90 deg`, `180 deg`, and `270 deg`, respectively. The vertical circle passing through the south and north point (as well as zenith and nadir) is called *local meridian* the one perpendicular to it through west point, zenith, east point and nadir is called *prime vertical*. The local meridian coincides with the projection of the geographical meridian of the observer's location to the sky (celestial sphere) from Earth's center.

The terms introduced here are helpful in understanding the effects of Earth's rotation.

### The Equatorial Coordinate System

In principle, the celestial coordinate system can be introduced in the simplest way by projecting Earth's geocentric coordinates to the sky at a certain moment of time (actually, each time when star time is O:00 at Greenwich or anywhere on the Zero meridian on Earth, which occurs once each siderial day) the reader will hopefully understand this statement after reading this section. These coordinates are then left fixed at the celestial sphere, while Earth will rotate away below them.

Practically, projecting Earth's equator and poles to the celestial sphere by imagining straight half lines from the Earth's center produces the *celestial equator* as well as the *north* and the *south celestial pole*. Great circles through the celestial poles are always perpendicular to the celestial equator and called *hour circles* for reasons explained below.

The first coordinate in the equatorial system, corresponding to the latitude, is called Declination (`Dec`), and is the angle between the position of an object and the celestial equator (measured along the hour circle). Alternatively, sometimes the *polar distance* (`PD`) is used, which is given by `PD = 90 deg - Dec` the most prominent reference known to the present author using `PD` instead of `Dec` is John Herschel's *General Catalogue of Non-stellar Objects (GC)* of 1864, but this (equivalent) alternative has come more and more out of use since, so that virtually all current astronomical databases use `Dec`.

It remains to fix the zero point of the longitudinal coordinate, called Right Ascension (`RA`). For this, the intersection points of the equatorial plane with Earth's orbital plane, the *ecliptic*, are taken, more precisely the so-called *vernal equinox* or "First Point of Aries". During the year, as Earth moves around the Sun, the Sun *appears* to move through this point each year around March 21 when spring begins on the Northern hemisphere, and crosses the celestial equator from south to north (Southerners are asked to forgive a certain amount of "hemispherism" in the official nomenclature). The opposite point is called the "autumnal equinox", and the Sun passes it around September 23 when it returns to the Southern celestial hemisphere. As a longitudinal coordinate, RA can take values between 0 and 360 deg. However, this coordinate is more often given in time units hours (h), minutes (m), and seconds (s), where 24 hours correspond to 360 degrees (so that RA takes values between 0 and 24 h) the correspondence of units is as follows: So the vernal equinox, where the Sun appears to be when Northern spring begins around March 21, is at RA = 0 h = 0 deg, the summer solstice where the Sun is when Northern summer begins around June 21, is at RA = 6 h = 90 deg, the autumnal equinox is at RA = 12 h = 180 deg, and the winter solstice is at RA = 18 h = 270 deg. Thus RA is measured from west to east in the celestial sphere.

Because of small periodic and secular changes of the rotation axis of Earth, especially precession, the vernal equinox is not constant but varies slowly, so that the whole equatorial coordinate system is slowly changing with time. Therefore, it is necessary to give an epoch (a moment of time) for which the equatorial system is taken currently, most sources use epoch 2000.0, the beginning of the year 2000 AD.

To go over from equatorial coordinates fixed to the stars to the horizon system, the concept of the *hour angle* (`HA`) is useful. In principle, this means introducing a new, *second* equatorial coordinate system which co-rotates with Earth. This system has again the celestial equator and poles as reference quantities, and declination as latitudinal coordinate, but a co-rotating longitudinal coordinate called hour angle. In this system, a star or other celestial object moves contrary to Earth's rotation along a circle of constant declination during the course of the day various effects of this *diurnal motion* are discussed below. This rotation leaves the celestial poles in the same invariant position for all time: They always stay on the *local meridian* of the observer (which goes through south and north point also), and the altitude of the north celestial pole is equal to the geographic latitude of the observer (thus negative for southerners, who cannot see it for this reason, but the south celestial pole instead). This meridian always coincides with in hour circle for this reason. Thus, as may be suggestive, the local meridian is taken as the hour circle for HA=0.

Celestial objects are at constant RA, but change their hour angle as time proceeds. If measured in units of hours, minutes and seconds, HA will change for the same amount as the elapsed time interval is, as measured in *star time* (ST), which is defined so that a siderial rotation of Earth takes 24 hours star time, which corresponds to 23 h 56 m 4.091 s standard (mean solar) time see our article on Astronomical Time Keeping for more details. This is actually the reason why RA and HA are measured in time units. The standard convention is that HA is measured from east to west so that it increases with time, and this is opposite to the convention for RA !

Star time is `ST = 0 h` by definition whenever the vernal equinox, `RA = 0 h`, crosses the local meridian, `HA = 0`. As time proceeds, RA stays constant, and both HA and ST grow by the amount of time elapsed, thus star time is always equal to the hour angle of the vernal equinox. Moreover, objects with "later" RA come into the meridian `HA = 0`, more precisely with RA which is later by the amount of elapsed star time, so that also star time is equal to the current Right Ascension of the local meridian.

More generally, for any object in the sky, the following relation between right ascension, hour angle, and star time always holds: (here given to determine the current HA from known RA and ST).

#### Transformation of Horizontal to Equatorial Coordinates, and Vice Versa

#### Effects of Earth's Rotation

By doing so, stars will cross the local meridian (defined e.g. by zero hour angle `HA`) twice a day these events are called *transits* or *culminations*, i.e., the *upper* and the *lower* transit, or the *upper* and the *lower* culmination. These events also mark the maximal and minimal altitude `a` the objects can reach in the observer's sky, and may both take place above or below the horizon of the observer, depending on the declination `Dec` of the object and the geographic latitude *B* of the observer.

The altitudes for *upper transits* are as follows: where the transit takes place *north* of the zenith if `Dec > B` and south otherwise. If `|B - Dec| > 90 deg`, the upper transit will take place at negative altitude, i.e. below the horizon, so that the object will never come above the horizon and thus never be visible for the Northern hemisphere, this is true for all objects with and for the Southern hemisphere for The altitudes for the *lower transit* are given by For an observer on the Northern hemisphere, stars with `Dec > 90 deg - B (> 0)`, and for southern hemisphere observers, stars with `Dec < - 90 deg - B (< 0)` will have their lower transit at positive altitudes, i.e., above horizon, and will never set such stars are called *circumpolar*.

All stars which are neither circumpolar nor never visible will have their upper transit above and their lower transit below horizon, and thus rise and set during a siderial day. Disregarding refraction effects, the hour angle of the rise and set of a celestial object, the *semidiurnal arc* `H0`, is given by while the azimuth of the rising and setting points, the evening and morning *elongation* `A0` is where `A0 > 90 dec` if `Dec` and `B` have same sign (i.e., are on the same hemisphere). Rising and setting times differ from transit time by the amount of the diurnal arc `H0`, given in time units (hours), taken as hours of star time.

- If
`|Dec| < |B|`, the object transits the prime vertical,`A = +/- 90 deg`this occurs at altitude and hour angle given by - If
`|Dec| > |B|`, the object will stay within a certain region of azimuth around the visible celestial pole, where the extremal azimuth points are given by

### The Ecliptical Coordinate System

The *ecliptic latitude* (`be`) is defined as the angle between a position and the ecliptic and takes values between -90 and +90 deg, while the *ecliptic longitude* (`le`) is again starting from the vernal equinox and runs from 0 to 360 deg in the same eastward sense as Right Ascension.

The obliquity, or inclination of Earth's equator against the ecliptic, amounts `eps[ilon] = 23deg 26' 21.448" (2000.0)` and changes very slightly with time, due to gravitational perturbations of Earth's motion. Knowing this quantity, the transformation formulae from equatorial to ecliptical coordinates are quite simply given (mathematically, by a rotation around the "X" axis pointing to the vernal equinox by angle `eps`): and the reverse transformation:

Ecliptical coordinates are most frequently used for solar system calculations such as planetary and cometary orbits and appearances. For this purpose, two ecliptical systems are used: The heliocentric coordinate system with the Sun in its center, and the geocentric one with the Earth in its origin, which can be transferred into each other by a coordinate translation.

### Galactic Coordinates

Here, the galactic plane, or galactic equator, is used as reference plane. This is the great circle of the celestial sphere which best approximates the visible Milky Way. For historical reasons, the direction from us to the Galactic Center has been selected as zero point for *galactic longitude* `l`, and this was counted toward the direction of our Sun's rotational motion which is therefore at `l = 90 deg`. This sense of rotation, however, is opposite to the sense of rotation of our Galaxy, as can be easily checked ! Therefore, the *galactic north pole*, defined by the *galactic coordinate system*, coincides with the rotational south pole of our Galaxy, and vice versa.

*Galactic latitude* `b` is the angle between a position and the galactic equator and runs from -90 to +90 deg. Glalactic longitude runs of course from 0 to 360 deg.

The galactic north pole is at RA = 12:51.4, Dec = +27:07 (2000.0), the galactic center at RA = 17:45.6, Dec = -28:56 (2000.0). The inclination of the galactic equator to Earth's equator is thus 62.9 deg. The intersection, or node line of the two equators is at RA = 18:51.4, Dec = 0:00 (2000.0), and at l = 33 deg, b=0.

The transformation formulae for this frame get more complicated, as the transformation is consisted of (1.) a rotation around the celestial polar axis by 18:51.4 hours, so that the reference zero longitude matches the node, (2.) a rotation around the node by 62.9 deg, followed by (3.) a rotation around the galactic polar axis by 33 deg so that the zero longitude meridian matches the galactic center. This complicated transformation will not be given here formally.

Before 1959, the intersection line had been taken as zero galactic longitude, so that the old differred from the new latitude by 33.0 deg (the longitude of the node just discussed, but for the celestial equator of the epoch 1950.0): For a transition time, the old coordinate had been assigned a superscript "I", the new longitude a superscript "II", which can be found in some literature.

For some considerations, besides the geo- or heliocentric galactic coordinates described above, *galactocentric galactic coordinates* are useful, which have the galactic center in their origin these can be obtained from the helio/geocentric ones by a parallel translation.