The astrophysics for 4th class Department of Physics/ College of Science/ University of Kirkuk

2019-2020 Prof. Dr. Wafaa H. A. Zaki

Chapter one/ 4 weeks The Celestial : Coordinate Systems

1. Spherical

Spherical Astronomy is a science studying astronomical coordinate frames, directions and apparent motions of celestial objects, determination of position from astronomical observations, observational errors, etc.

For simplicity we will assume that the observer is always on the , we will also use degrees to express all unless otherwise mentioned.

The is an imaginary sphere of arbitrarily large , concentric with .

The is an imaginary directly "above" a particular location, on the imaginary celestial sphere. The opposite direction is toward the nadir. The zenith is the "highest" point on the celestial sphere. The is the great passing through the celestial poles, the zenith, and the nadir of an observer's location. Consequently, it contains also the horizon's north and south points, and it is perpendicular to the celestial and horizon.

The Celestial Meridian is coplanar with the analogous terrestrial meridian projected onto the celestial sphere. Hence, the number of astronomical meridians is infinite.

The celestial sphere. Earth is depicted in the center of the celestial sphere. 2. Celestial Coordinate System

The Celestial Coordinate System is a system for specifying positions of celestial objects: satellites, , , , and so on. Coordinate systems can specify a position in 3-dimensional space, or merely the direction of the object on the celestial sphere, Since the of the stars are ignored, we need only two coordinates to specify their directions. Each coordinate frame has some fixed reference passing through the center of the celestial sphere and dividing the sphere into two hemispheres along a . One of the coordinates indicates the angular from this reference plane. There is exactly one great circle going through the object and intersecting this plane perpendicularly; the second coordinate gives the between that point of intersection and some fixed direction.

Each coordinate system is named for its choice of fundamental plane.

The meridian on the celestial sphere.

A. Horizontal coordinate system

The horizontal coordinate system is a celestial coordinate system that uses the observer's local horizon as the fundamental plane. It is expressed in terms of altitude (or elevation) angle and .

This coordinate system divides the into the upper hemisphere where objects are visible, and the lower hemisphere where objects cannot be seen since the Earth obstructs vision. The great circle separating the hemispheres is called the celestial horizon. The Celestial Horizon is a great circle on the celestial sphere having a plane that passes through the center of the Earth and is to an observer’s horizon. The pole of the upper hemisphere is called the zenith. The pole of the lower hemisphere is called the nadir. There are two independent horizontal angular coordinates: Altitude (E), or elevation, is the angle between the object and the observer's local horizon. The altitude lies in the range [−90◦,+90◦]; it is positive for objects above the horizon and negative for the objects below the horizon. The zenith distance, or the angle between the object and the zenith, is obviously z = 90◦- E…………….(1)

The horizontal coordinate system

Azimuth (A), is the angle of the object around the horizon, usually measured from the north or south. Azimuth is measured from the north point (sometimes from the south point) of the horizon around to the east (0 -360 ).

The main disadvantage of this system:

 The horizontal coordinate system is fixed to the Earth, not the stars. Therefore, the altitude and azimuth of an object in the sky changes with , as the object appears to drift across the sky with the of the Earth.  Because the horizontal system is defined by the observer's local horizon, the same object viewed from different locations on Earth at the same time will have different values of altitude and azimuth. Horizontal coordinates are very useful for determining the rise and set of an object in the sky. When an object's altitude is 0°, it is on the horizon. If at that moment its altitude is increasing, it is rising, but if its altitude is decreasing, it is setting. One can determine whether altitude is increasing or decreasing by instead considering the azimuth of the celestial object:

 if the azimuth is between 0° and 180° (north–east–south), it is rising.  if the azimuth is between 180° and 360° (south–west–north), it is setting.

B. Equatorial Coordinate System

The equatorial coordinate system is a celestial coordinate system widely used to specify the positions of celestial objects. It may be applied in spherical or rectangular coordinates, both defined by an origin at the center of Earth, a fundamental plane consisting of the projection of Earth's equator onto the celestial sphere (forming the ), a primary direction towards the vernal , and a right-handed convention. The origin at the center of the Earth means the coordinates are geocentric. The fundamental plane and the primary direction mean that the coordinate system, while aligned with the Earth's equator and pole, does not rotate with the Earth, but remains relatively fixed against the background stars. A right-handed convention means that coordinates are positive toward the north and toward the east in the fundamental plane. A 's spherical coordinates are often expressed as a pair, rightascension and , without a distance coordinate.

The declination symbol δ, measures the of an object perpendicular to the celestial equator, positive to the north, negative to the south. For example, the north has a declination of +90°. Declination is analogous to terrestrial ,(+90°, -90°).

The symbol ,measures the angular distance of an object eastward along the celestial equator from the vernal equinox to

the circle passing through the object (the vernal equinoxⱱ point is one of the two where the intersects the celestial equator). Analogous to terrestrial , right ascension is usually measured in sidereal , minutes and seconds instead of degrees, a result of the method of measuring right ascensions by timing the passage of objects across the meridian as the Earth rotates. There are (360° / 24h) = 15° in one hour of right ascension, 24h of right ascension around the entire celestial equator.

The hour angleh, measures the angular distance of an object westward along the celestial equator from the observer's meridian to the passing through the object. Unlike right ascension, is always increasing with the rotation of the Earth. Hour angle may be considered a means of measuring the time since an object crossed the meridian. A star on the observer's celestial meridian is said to have a zero-hour angle.

Hour circle

Observers celestial meridian

Hour angle (h)

Equatorial Coordinates system.

The The sidereal time Θ (the hour angle of the vernal equinox) equals the hour angle plus right ascension of any object. Θ =h+ ……..(2)

Equatorial Coordinates system with ecliptic plane

C. Ecliptic Coordinate System

The ecliptic coordinate system is a celestial coordinate system commonly used for representing the positions and of objects. Because most planets (except ), and many small Solar System bodies have orbits with small inclinations to the ecliptic, it is convenient to use it as the fundamental plane (The fundamental plane is the plane of the Earth's , called the ecliptic plane). The system's origin can be either the center of the or the center of the Earth, its primary direction is towards the vernal equinox. The angle between equator plane and ecliptic plane is 23.5o, which is known as the obliquity of the ecliptic (ε ).

Ecliptic longitude or celestial longitude (symbols: λ) measures the angular distance of an object along the ecliptic (counterclockwise) from the primary direction (vernal equinox) (0° ecliptic longitude)(0 -360 ). Ecliptic latitude or celestial latitude (symbols: β), measures the angular distance of an object from the ecliptic towards the north (positive) or south (negative) ecliptic pole. For example, the north ecliptic pole has a celestial latitude of +90° (+90 ,-90 ). D. Galactic Coordinate System The galactic coordinate system is a celestial coordinate system in spherical coordinates, with the Sun as its center, the primary direction aligned with the approximate center of the , and the fundamental plane approximately in the galactic plane. It uses the right-handed convention, meaning that coordinates are positive toward the north and toward the east in the fundamental plane.

Galactic longitude Longitude (symbol l) measures the angular distance of an object eastward along the galactic equator from the . Analogous to terrestrial longitude, galactic longitude is usually measured in degrees (°)(0 -360 ).

Galactic latitude Latitude (symbol b) measures the angular distance of an object perpendicular to the galactic equator, positive to the north, negative to the south. For example, the north galactic pole has a latitude of +90°. Analogous to terrestrial latitude, galactic latitude is usually measured in degrees (°)(+90 ,-90 ).

galactic coordinate system

Fundamental Coordinates Coordinate Center point Primary direction plane Poles system (origin) (0° longitude) (0° latitude) Latitude Longitude

Zenith, Altitude (E) or North or south point Horizontal Observer Horizon Azimuth (A) nadir elevation of horizon

Right Celestial Celestial ascension (α) Equatorial Declination (δ) Center of equator poles or hour the Earth (geocentric), angle (h) Vernal equinox or Sun (heliocentric) Ecliptic Ecliptic Ecliptic Ecliptic Ecliptic poles latitude (β) longitude (λ)

Galactic Galactic Galactic Galactic Galactic Center of the Sun Galactic center plane poles latitude (b) longitude (l)

3. Converting Coordinates:

Conversions between the various coordinate systems are given.

Notes on conversion

 λo – observer's longitude.  φo – observer's latitude.  ε – obliquity of the ecliptic 23.5 h m s . Angles in the hours ( ), minutes ( ), and seconds ( ) of time measure must be converted to decimal degrees or radians before calculations are performed. 1h = 15° 1m = 15' 1s = 15" . Angles greater than 360° (2π) or less than 0° may need to be reduced to the range 0° - 360° (0 - 2π) depending upon the particular calculating machine or program. . Azimuth (A) is referred her to the south point of the horizon, the common astronomical reckoning. An object on the meridian to the south of the observer has A = h = 0° with this usage. In and some other disciplines, azimuth is figured from the north.

. Equatorial ←→ horizontal Note that Azimuth (A) is measured from the South point, turning positive to the West. Zenith distance, the angular distance along the great circle from the zenith to a celestial object, is simply the complementary angle of the altitude: 90° − E.

………(3) E …………(4)

E ………..(5) E E ………..(6) The altitude of an object is greatest when it is on the south meridian (the great circle arc between the celestial poles containing the zenith). At that moment(called upper , or transit) its hour angle is0 h. At the lower culmination the hour angle is h = 12 h. When h = 0 h, from the equationE (4)

Thus the altitude at the upper culmination is

E

……(7) The altitude is positive for objects with δ>φ−90◦.Objects with less than φ−90◦can never be seen at the latitude φ. On the other hand, when h = 12 h, we have

E

and the altitude at the lower culmination is

E ………..(8) Stars with δ> 90◦−φ will never set. For example, in Helsinki (φ ≈ 60◦), all stars with a declination higher than 30◦ are such circumpolar stars. And stars with a declination less than 30◦ can never be observed there. Suppose we observe a at its upper and lower ◦ culmination. At the upper transit, its altitude is Emax =90 −φ+δ and at the ◦ lower transit, Emin = δ+φ−90 . Eliminating the latitude, we get

E E …….(9) Thus we get the same value for the declination, independent of the observer’s location.

E

E

The altitude of a circumpolar star at upper and lower culmination

 Rising and Setting Times From the equation (4), we find the hour angle h of an object at the moment its altitude is E: E

……..(10) This equation can be used for computing rising and setting times. Then E = 0 and the hour angles corresponding to rising and setting times are obtained from cos h =-tan δ tan φ…………………(11) If higher accuracy is needed, we have to correct for the refraction of light caused by the atmosphere of the Earth. In that case, we must use a small negative value for E in (10). This value, the horizontally refraction, is about −34′.

. Equatorial ←→ eclipc The classical equations, derived from spherical trigonometry,

…….(12) ……….(13)

……….(14)

…………….(15)

. Equatorial ←→ galacc

…….(16) ……..(17)

………(18)

…………(19)

Example 1 :Find the altitude and azimuth of the in Helsinki at midnight at the beginning of 1996. The right ascension is α = 2h 55m 7s = 2.9186 hand declination δ = 14◦ 42′= 14.70◦, the sidereal time is Θ = 6h 19m 26s = 6.3239 h and latitude φ = 60.16◦.

Example 2 :The coordinates of are α = 14h 15.7m, δ = 19◦ 11′. Find the sidereal time at the moment Arcturus rises or sets in Boston (φ = 42◦ 19′). neglecting refraction. Example 3 :The right ascension of the Sun on June 1, 1983, was 4h 35m and declination 22◦ 00′. Find the ecliptic longitude and latitude of the Sun and the Earth.

Example 4 :A star of azimuth (A)=221.22o from the south and altitude (E)= 22.076o is observed when its sidereal time is 8h16m42s .If the observer’s latitude () is 60o N, calculate the star’s ecliptic longitude () and ecliptic latitude () at the time of observation.

Ecliptic and The ecliptic is an imaginary on the sky that marks the annual path of the sun. It is the projection of Earth’s orbit onto the celestial sphere, and is the basis for the ecliptic coordinate system.

Zodiac: an imaginary band, centered on the ecliptic, across the celestial sphere and about 16°-18° wide, in which the Sun, Moon and the planets Mercury, , and are always located. The band is divided into 12 intervals of 30°, each named (the Signs of the Zodiac) after the of stars which it contains.

The zodiac signs come from the which lie along the ecliptic (red line).

The The Earth's seasons are not caused by the differences in the distance from the Sun throughout the year (these differences are extremely small). The seasons are the result of the tilt of the Earth's axis. The Earth's axis is tilted from perpendicular to the plane of the ecliptic by 23.45°. This tilting is what gives us the four seasons of the year - spring, summer, autumn (fall) and winter. Since the axis is tilted, different parts of the are oriented towards the Sun at different times of the year.

Summer is warmer than winter (in each hemisphere) because the Sun's rays hit the Earth at a more direct angle during summer than during winter and also because the days are much longer than the during the summer.

The Seasons on Earth

During the winter, the Sun's rays hit the Earth at an extreme angle, and the days are very short. These effects are due to the tilt of the Earth's axis.

Solstices The are days when the Sun reaches its farthest northern and southern declinations. The winter occurs on December 21 or 22 and marks the beginning of winter in the northern hemisphere and the beginning of summer in the (this is the shortest of the year). The occurs on June 21 and marks the beginning of summer (this is the longest day of the year).

Equinoxes are days in which day and are of equal duration. The two yearly equinoxes occur when the plane of Earth's equator passes through the center of the Sun.

The Vernal Equinox: one of two points at which the ecliptic intersects the celestial equator. At it, the sun is moving along the ecliptic in a northeasterly direction and occurs in 20-21 March, in this point is meaning the beginning of spring in the Northern Hemisphere and the beginning of autumn in the Southern Hemisphere.

The autumnal equinox: occurs in 22-23 September and opposite the vernal equinox and the ecliptic intersects the celestial equator. In this point is meaning the beginning of autumn in the Northern Hemisphere and the beginning of spring in the Southern Hemisphere.

The vernal equinox

Earth's rotation is the rotation of the Earth around its own axis. The Earth rotates from the west towards east. As viewed from North Star or polestar , the Earth turns counter-clockwise. The Earth rotates once in about 24h with respect to the sun and once every 23h 56 m 4 s with respect to the stars. Earth's rotation is slowing slightly with time; thus, a day was shorter in the past. This is due to the tidal effects the Moon has on Earth's rotation. Atomic clocks show that a modern-day is longer by about 1.7 milliseconds than a century ago.

Precession

Precession is the slow wobble of Earth's rotation axis due to our planet's non spherical shape and its gravitational interaction with the Sun and the Moon.

The precession of the Earth's axis has a number of observable effects.

First, the positions of the south and north celestial poles appear to move in against the space-fixed backdrop of stars, completing one circuit in approximately 25800 years. Thus, while today the star Polaris lies approximately 1 ° at the north celestial pole, this will change over time, and other stars will become the "north star". In approximately 3200 years, the star in the Cepheus constellation will succeed Polaris for this position.

Secondly, the position of the Earth in its orbit around the Sun at the solstices, equinoxes slowly changes. For example, suppose that the Earth's orbital position is marked at the summer solstice, when the Earth's is pointing directly toward the Sun. One full orbit later, when the Sun has returned to the same apparent position relative to the background stars, the Earth's axial tilt is not now directly toward the Sun: because of the effects of precession, it is a little way "beyond" this. In other words, the solstice occurred a little earlier in the orbit, is about 20 minutes shorter than the sidereal year. At present, the rate of precession corresponds to a period of 25800 years.

Effects of precession and nutation on Earth’s axis of rotation.

The relationship between the seasons and precession of Earth's axis. Nutation : Is a small irregularity in the precession of the equinoxes. Nutation superimposes a small oscillation, with a period of 18.6 years and an amplitude of 9.2 seconds of arc, upon this great slow movement. The cause of nutation lies chiefly in the fact that the plane of the Moon’s orbit around the Earth is tilted by about 5° from the plane of the Earth’s orbit around the Sun. Obliquity: The obliquity is the angle between the plane of the Earth's equator and the plane of the Earth's orbit around the Sun. it can change due to the gravitational effects of other bodies (nutation), which in turn can affect the earth’s climate. it is approximately 23°27' but it is not fixed. Instead, it varies slowly because both the Earth's axis of rotation and its orbital motion are affected by the gravitational attractions of the Moon and planets. it oscillates between 22.1 and 24.5 degrees on a 41,000-year cycle. If the obliquity was equal to zero, the Sun would rise at 6 a.m. and set at 6 p.m. every day of the year, everywhere in the world. There would be no long summer days and long winter nights. The Sun wouldn't be high in the sky in summer and low in winter. It would take the same path across the sky every day of the year. Every day would be the same as every other. There would be no seasons. The whole world's weather would be completely different. Life would probably have evolved in a totally different way.

Astronomical System of Units:

 The astronomical unit (AU) is a unit of , the average distance from Earth to the Sun. it is now defined as exactly 149597870700 meters (about 150 million kilometers, or 93 million miles). The astronomical unit is used primarily as a convenient yardstick for measuring distances within the Solar System or around other stars.  A light year ( L.Y) is a unit of length used informally to express astronomical distances. it is the distance that light travels in vacuum in one Julian year (365.25 days) is equal L.Y=9.45x1012 Km. it is most often used when expressing distances to stars and other distances on a galactic scale.  (P") is the apparent displacement of an object because of a change in the observer's point of view. This effect can be used to measure the distances to nearby stars. As the Earth orbits the sun, a nearby star will appear to move against the more distant background stars. can measure a star's position once, and then again 6 months later and calculate the apparent change in position. The star's apparent motion is called .Distance to the star can be computed by the formula:

d = 206265/p″

where the distance d is in unit of A.U., and the parallax p″ in second of arc.

Parallax angles of less than 0.01 arc sec are very difficult to measure from Earth because of the effects of the Earth's atmosphere. This limits Earth based telescopes to measuring the distances to stars about 1/0.01 or 100 away. Space based telescopes can get accuracy to 0.001, which has increased the number of stars whose distance could be measured with this method.

(Pc) The distance corresponding to a parallax of one second of arc. A unit of distance defined as the distance at which 1 astronomical unit (AU) subtends an angle of 1 sec of arc. It is a unit of length used to measure large distances to objects outside our Solar System.1 parsec = 206265 AU = 3.26 light years. . (. ) = " ( ) = " = . .

A distance of 1000 parsecs (3262 light-years) is commonly denoted by the kilo parsec (kpc). Astronomers typically use kilo parsecs to express distances between parts of a galaxy, or within groups of galaxies. A distance of one million parsecs is commonly denoted by the mega parsec (Mpc). Astronomers typically express the distances between neighboring galaxies and galaxy clusters in mega parsecs. One giga parsec (Gpc) is one billion parsecs — one of the largest units of length commonly used. One giga parsec is about 3.26 billion light-years. Astronomers typically use giga parsecs to express the sizes of large-scale structures such as the size of, and distance to, the CfA2 Great Wall; the distances between galaxy clusters; and the distance to .

For example:  The is about 0.78 Mpc (2.5 million light-years) from the Earth.  The nearest large galaxy cluster, the Cluster, is about 16.5 Mpc (54 million light-years) from the Earth.  The galaxy RXJ1242-11, observed to have a super massive black hole core similar to the Milky Way's, is about 200 Mpc (650 million light-years) from the Earth.  The –Corona Borealis Great Wall, currently the largest known structure in the universe, is about 3 Gpc (10 billion light-years) across.  The particle horizon (the boundary of the ) has a radius of about 14.0 Gpc (46 billion light-years)

The Laws Of Motion Of Celestial Bodies:

Kepler’s Laws of Planetary Motion:

Johannes Kepler (1571–1630) published his three laws of planetary motion:

1. The orbits of the planets are ellipses, with the Sun at one focus of the ellipse. 2. Planets move proportionally faster in their orbits when they are nearer the Sun (cover equal areas in equal time.

Kepler’s Laws of Planetary Motion

3. More distant planets take proportionally longer to orbit the Sun. The ratio of the squares of the orbital periods for two planets is equal to the ratio of the cubes of their semi-major axes. R3=P2 (R in AU; P in years) As an example, the "radius" of the orbit of (the length of the semi-major axis of the orbit) is: R=P2/3=(1.88)2/3=1.52 AU

Newton’s laws of motion

1. A body at rest remains at rest, and a body in uniform motion remains in uniform motion, unless acted on by an external force. 2. The acceleration of a body is equal to the net applied force, divided by the mass of the body: a = F/m. 3. For every force there is an equal and opposite reaction force. The third law is exemplified by a person attracted by gravity toward a planet. There is simultaneously an equal opposite force attracting the planet toward the person.

Newtonian gravity: The description of gravity due to Newton: The

attractive gravitational force of point mass m1 on point mass m2 separated by displacement r = where G is Newton’s gravitational constant G=6.67x10-11 N.M2/Kg2

CH.1 : 4 Weeks