A.D.M College for women (Autonomous) Nagapattinam

PG & Research Department of Mathematics

Course Material

ASTRONOMY

Course Code: UMO

Class: III B.Sc Mathematics

Staff Name: Dr.P.Jamuna Devi, Assistant Professor of Mathematics

Unit I

Spherical Trigonometry

Definition:

The Sphere:

A sphere is a geometrical object in three-dimensional space that is the surface of a ball. Like a circle in a two-dimensional space, a sphere is defined mathematically as the set of points that are all at the same distance r from a given point in a three-dimensional space.

Theorem:

A section of a sphere by a plane is a circle:

Proof:

The centre M of the circle is the point of intersection of the plane and line CM which passes through C and is perpendicular to the given plane. Centre : The foot of the perpendicular from the centre of the sphere to the plane is the centre of the circle.

Great Circle and Small Circle:

All the Great Circles divide the Earth into two equal halves. All meridians of longitude are Great Circles. The Small Circles are the circles whose center point is not the center of the Earth. The radii of the Small Circles are smaller than the radius of the Earth. All latitudes except the Equator are Small Circles.

Axis and Poles of a circle:

The latitude of the circle is approximately the angle between the Equator and the circle, with the angle's vertex at Earth's centre. The Equator is at 0°, and the North Pole and South Pole are at 90° north and 90° south, respectively.

Distance between two points on a sphere:

Angular distance is the angle between the two sightlines, or between two point objects as viewed from an observer.

Spherical angle:

A spherical angle is a particular dihedral angle; it is the angle between two intersecting arcs of great circles on a sphere. It is measured by the angle between the planes containing the arcs.

Secondaries:

Any Great circle passing through the poles of a circle is called secondary to that circle.

Theorem:

The point of intersection of two great circle are the poles of the great circle joining their poles

Proof:

Let O be the centre of the sphere. Let P be a point of intersection of two great circles PL and PM. Let X be a pole of the great circle PL and Y be a pole of PM. Arc PX = 90, Angle POX = 90, OX perpendicular to OP. Similarly OY Perpendicular to OP.

OP is perpendicular to the plane of the great circle AB. Since O is the centre, it follows that OP is the axis and so P is a pole of the great Circle AB.

Spherical Triangle: A spherical triangle is a figure formed on the surface of a sphere by three great circular arcs intersecting pairwise in three vertices. The spherical triangle is the spherical analog of the planar triangle, and is sometimes called an Euler triangle

Polar Triangle:

A spherical triangle formed by the arcs of three great circles each of whose poles is the vertex of a given spherical triangle

Theorem:

If A’B’C’ is the polar triangle of ABC, then ABC is the polar triangle of A’B’C’

Proof:

A’B’C’ is the polar triangle of ABC, therefore A’, B’, C’ lie on the same side of BC, CA and AB as the opposite vertices A, B, C and AA’, BB’, C C’ are each less than 90.

B’ is the pole of AC and so AB’= 90

C’ is the pole of AB and so AC’ = 90

Therefore A is the pole of B’C’

Some properties of spherical triangle:

The three angles of a spherical triangle must together be more than 180° and less than 540° . The greater side is opposite the greater angle , if two sides are equal their opposite angles are equal . , and if one side of the triangle 90° it is called a quadrantal triangle .

Cosine Formula:

The Spherical Law of Cosines Suppose that a spherical triangle on the unit sphere has side lengths a, b and c, and let C denote the angle adjacent to sides a and b. Then (using radian measure): cos(c) = cos(a) cos(b) + sin(a) sin(b) cos(C).

Sine Formula:

Sin a/ sin A = sin b/ sin B = sin c/ sin C Co tangent Formula:

Cos b. Cos C = Sin b. Cot a – Sin C. Cot A

Five parts formula:

In the spherical triangle ABC, prove that

Sin c. cos B = sin a. cos b – cos a. sinb. Cos C

Napiers Rules:

Napier's rules for right spherical triangles

Then Napier has two rules: The sine of a part is equal to the product of the tangents of the two adjacent parts. The sine of a part is equal to the product of the cosines of the two opposite parts. sine of the middle part = the product of the tangents of the adjacent parts. sine of the middle part = the product of the cosines of the opposite parts.

CELESTIAL SPHERE DIURNAL MOTION Astronomy: Astronomy is the study of everything in the universe beyond Earth's atmosphere. That includes objects we can see with our naked eyes, like the Sun , the Moon , the planets, and the stars . It also includes objects we can only see with telescopes or other instruments, like faraway galaxies and tiny particles. Celestial Spheres: In astronomy and navigation, the celestial sphere is an abstract sphere that has an arbitrarily large radius and is concentric to Earth. All objects in the sky can be conceived as being projected upon the inner surface of the celestial sphere, which may be centered on Earth or the observer. Diurnal Motion: Diurnal motion is the daily motion of stars and other celestial bodies across the sky. This motion is due to the Earth's rotation from west to east, which causes celestial bodies to have an apparent motion from east to west. Celestial axis: The line joining the north and south celestial poles and passing through the center of the celestial sphere; the extension of the Earth's axis to the celestial sphere. Equator: An equator is an imaginary line around the middle of a planet or other celestial body. It is halfway between the North Pole and the South Pole, at 0 degrees latitude. An equator divides the planet into a Northern Hemisphere and a Southern Hemisphere. The Earth is widest at its Equator. Celestial Horizon: The celestial horizon is a plane passing through the earth's center perpendicular to the zenith- nadir axis. The visual horizon approximates this plane at the earth's surface. Zenith: The zenith is an imaginary point directly "above" a particular location, on the imaginary celestial sphere. "Above" means in the vertical direction opposite to the gravity direction at that location. The zenith is the "highest" point on the celestial sphere. Nadir: The nadir, is the direction pointing directly below a particular location; that is, it is one of two vertical directions at a specified location, orthogonal to a horizontal flat surface there. Since the concept of being below is itself somewhat vague, scientists define the nadir in more rigorous terms. Celestial Meridian: The celestial meridian is the line on the celestial sphere joining the observer's zenith (i.e. the point directly overhead) with the north and south celestial poles. Cardinal Points: The four cardinal directions, or cardinal points, are the directions north, east, south, and west, commonly denoted by their initials N, E, S, and W. Hemispheres: Any circle drawn around the Earth divides it into two equal halves called hemispheres. There are generally considered to be four hemispheres: Northern, Southern, Eastern, and Western. The Equator, or line of 0 degrees latitude, divides the Earth into the Northern and Southern hemispheres. The Northern Hemisphere is the half of the Earth that is north of the Equator. For other planets in the Solar System, north is defined as being in the same celestial hemisphere relative to the invariable plane of the solar system as Earth's North Pole. Declination Circle: A setting circle on the declination axis that enables an equatorially mounted telescope to be set at the declination of the object to be observed. Its scales are graduated from 0°, when the telescope is aligned with the celestial equator, to ±90°, when it is aligned with the north or south poles. Parallactic angle: In spherical astronomy, the parallactic angle is the angle between the great circle through a celestial object and the zenith, and the hour circle of the object. Annual motion of the sun: Over the course of a year, the Sun appears to move a little towards the East each day as seen with respect to the background stars. This daily eastward drift is <1° per day (there are 365 days in a year, but only 360° in a circle). This apparent motion is a reflection of the Earth's annual orbit around the Sun. Ecliptic, Obliquity: Obliquity of the ecliptic is the term used by astronomers for the inclination of Earth's equator with respect to the ecliptic, or of Earth's rotation axis to a perpendicular to the ecliptic. It is about 23.4° and is currently decreasing 0.013 degrees (47 arcseconds) per hundred years because of planetary perturbations. First point of Aries: The First Point of Aries, also known as the Cusp of Aries, is the location of the vernal equinox, used as a reference point in celestial coordinate systems. ... The First Point of Aries is considered to be the celestial "prime meridian" from which right ascension is calculated. First point of Libra: The point of intersection of the ecliptic and the celestial equator (equinoctial) when the sun is moving from the north to the south direction. It is denoted by the symbol λ. Also called autumnal equinox. Equinoxes and Solstices: The biggest difference between the equinox and the solstice is that a solstice is the point during the Earth's orbit around the sun at which the sun is at its greatest distance from the equator, while during an equinox, it's at the closest distance from the equator. When the sun is furthest north or south from the equator, it's a solstice. When neither hemisphere is tilted toward or away from the sun, it's an equinox. They are related to the seasons because it makes the days longer or shorter, warmer or colder. Celestial coordinates:

Horizontal system The horizontal, or altitude-azimuth, system is based on the position of the observer on Earth, which revolves around its own axis once per sidereal day (23 hours, 56 minutes and 4.091 seconds) in relation to the star background. The positioning of a celestial object by the horizontal system varies with time, but is a useful coordinate system for locating and tracking objects for observers on Earth. It is based on the position of stars relative to an observer's ideal horizon.

Equatorial system

The equatorial coordinate system is centered at Earth's center, but fixed relative to the celestial poles and the March equinox. The coordinates are based on the location of stars relative to Earth's equator if it were projected out to an infinite distance. The equatorial describes the sky as seen from the Solar System, and modern star maps almost exclusively use equatorial coordinates.

The equatorial system is the normal coordinate system for most professional and many amateur astronomers having an equatorial mount that follows the movement of the sky during the night. Celestial objects are found by adjusting the telescope's or other instrument's scales so that they match the equatorial coordinates of the selected object to observe.

Ecliptic system

The fundamental plane is the plane of the Earth's orbit, called the ecliptic plane. There are two principal variants of the ecliptic coordinate system: geocentric ecliptic coordinates centered on the Earth and heliocentric ecliptic coordinates centered on the center of mass of the Solar System.

The geocentric ecliptic system was the principal coordinate system for ancient astronomy and is still useful for computing the apparent motions of the Sun, Moon, and planets.

The celestial equivalent of latitude is called declination and is measured in degrees North (positive numbers) or South (negative numbers) of the Celestial Equator. The celestial equivalent of longitude is called right ascension.

Equinoxes and Solstices The zero point for celestial longitude (that is, for right ascension) is the Vernal Equinox, which is that intersection of the ecliptic and the celestial equator near where the Sun is located in the Northern Hemisphere Spring. The other intersection of the Celestial Equator and the Ecliptic is termed the Autumnal Equinox. When the Sun is at one of the equinoxes the lengths of day and night are equivalent (equinox derives from a root meaning "equal night"). The time of the Vernal Equinox is typically about March 21 and of the Autumnal Equinox about September 22.

The point on the ecliptic where the Sun is most north of the celestial equator is termed the Summer Solstice and the point where it is most south of the celestial equator is termed the Winter Solstice. In the Northern Hemisphere the hours of daylight are longest when the Sun is near the Summer Solstice (around June 22) and shortest when the Sun is near the Winter Solstice (around December 22). The opposite is true in the Southern Hemisphere. The term solstice derives from a root that means to "stand still"; at the solstices the Sun reaches its most northern or most southern position in the sky and begins to move back toward the celestial equator. Thus, it "stands still" with respect to its apparent North-South drift on the celestial sphere at that time.

Traditionally, Northern Hemisphere Spring and Fall begin at the times of the corresponding equinoxes, while Northern Hemisphere Winter and Summer begin at the corresponding solstices. In the Southern Hemisphere, the seasons are reversed (e.g., Southern Hemisphere Spring begins at the time of the Autumnal Equinox). Coordinates on the Celestial Sphere

The right ascension (R.A.) and declination (dec) of an object on the celestial sphere specify its position uniquely, just as the latitude and longitude of an object on the Earth's surface define a unique location.

UNIT II Zones of Earth:

The five main latitude regions of Earth's surface comprise geographical zones, divided by the major circles of latitude. The differences between them relate to climate. They are as follows:

1. The North Frigid Zone, between the North Pole at 90° N and the Arctic Circle at 66° 33' N, covers 4.12% of Earth's surface. 2. The North Temperate Zone, between the Arctic Circle at 66° 33' N and the Tropic of Cancer at 23° 27' N, covers 25.99% of Earth's surface. 3. The Torrid Zone, between the Tropic of Cancer at 23° 27' N and the Tropic of Capricorn at 23° 27' S, covers 39.78% of Earth's surface. 4. The South Temperate Zone, between the Tropic of Capricorn at 23° 27' S and the Antarctic Circle at 66° 33' S, covers 25.99% of Earth's surface. 5. The South Frigid Zone, from the Antarctic Circle at 66° 33' S and the South Pole at 90° S, covers 4.12% of Earth's surface.

Trace the variations in the durations of Day and night during the year at different stations: Earth rotates on its axis; this causes us to experience day and night. Because of this tilt and Earth's movement around our Sun, there is a time when Earth's north pole is tilting 23.5 degrees toward our Sun. If the sun passes through the zenith its declination equals the latitude of the place. As  lies between – and +  ,  must lie between S and N .

Therefore the sun could be seen at the zenith in some part of the year only in places of latitudes between S and N .

Again the sun remains above (or below) like a circumpolar star if  > 90 -  which give  > 90 -  north or south. Thus we find that latitudes  and 90 -  north or south define some peculiarities in the course of sun’s motion . For these reasons, The Terrestrial sphere is divided into a number of regions called the Zones of Earth by small circles parallel to the equator. These circles are called latitude circles.

The small circle parallel to the equator at a distance  north of it is called the Tropic of Cancer and the corresponding small circle in the southern hemisphere is called Tropic of Capricorn. For a Place, On the equator latitude  = 0 In the North Torrid Zone 0 <  <  On the Tropic of Cancer  =  In the North Temperate Zone  <  < 90 -  On the Arctic circle  = 90 -  In the North Frigid Zone 90 -  <  < 90  At the North Pole  = 90 

Similarly we get the latitudes of different zones in the Southern hemisphere To find the duration of perpetual day in a place of latitude  > 90 - 

Here, the perpetual day commences on the date when the sun’s north declination is 90 -  and ends on the date when it is again equal to 90 - .

Let S1 and S2 be the positions of the sun at the beginning and the end of perpetual day.

The duration of perpetual day is the time taken by the sun to move from S1 to S2.

Let L be the midpoint of Ecliptic  . Clearly L is also the midpoint of S1S2.

Let S1L = x = LS2 so that S1S2= 2x.

Draw S1 D1 perpendicular to the equator.

From the spherical triangle S1 D1,

sin S sin S D 1  1 1 sin S1D1 sin S1D1 To describe 360 of longitude the sun takes a year of 365¼ days.

To describe arc S1S2 time taken

365¼ xarc S S days 360 1 2

365¼  2 xdays 360

365¼  coscoscos1   ecdays 180

i.e, Duration of perpetual day

365¼  coscoscos1   ecdays 180

Duration of perpetual night

If the number of days in the year be taken as 365 then the duration of perpetual day

73  coscoscos1   ecdays 36

Dip of Horizon If an observer is on the surface of the earth, he cannot see the objects below the horizon, which is the tangent plane to the earth at his position. But, if he is at an elevated position he can see objects below the ordinary horizon, the new zone of visibility being defined by the tangents from the observer to the surface of the earth. The curve passing through the points of contact to these tangents is called the visible horizon or the offing. The angle between the directions of ordinary horizon and the visible horizon is called the Dip of Horizon. To find an expression for Dip

Let E be the centre of earth and a its radius.

Let A be the position of the observer at the surface of earth and B his position at a height h above the surface of earth. BT be a tangent from B to the surface of earth and BX the direction of ordinary horizon so that Dip D= XBT

D= BET

BT2 = BE2 – ET2

= (a+h) 2-a2 = 2ah+h2 = 2ah.

Therefore, BT = 2ah (since h is small compared with a)

BT is called the distance of visible horizon from the observer.

From the triangle BET,

BT22 ah h tan D =  ET a a tan D = D (radians) , since D is very small

Therefore, tan D= (2h/a) radians.

To find the distance between two mountains whose tops are just visible from each other

Let A1B1 and A2B2 be the mountains of heights h1 and h2 whose tops B1 and B2 are just visible from each other

Therefore, B1B2 must be a tangent to the earth.

Let it touch the surface of earth at T.

Distance between Mountains

= arc A1 A2

= arc A1 T+ arc A2T = B1 T+ B2T(nearly)

= 2ah1 + 2ah2

Effects of Dip Due to dip the zone of visibility is enlarged. The positions of objects are apparently shifted towards the zenith through a distance D along the verticals. The azimuth of the body is not affected while its altitude is increased. Rising of a celestial body is accelerated and setting retarded. Duration of day time is increased and night decreased. The effect of dip is the same for all bodies as it depends only on the height of observer and the radius of earth. Acceleration in the time of rising of a star = D/{15(cos2 - sin2) } seconds

Twilight

As the sun goes below the horizon darkness does not fall-in instantaneously. This is because even after the sun set its rays fall on the atmosphere above the earth and of the light thus received a considerable portion is reflected or scattered in various directions.

Therefore, there is some diffused light lasting for some time. The intensity of this light gradually diminishes and finally gives way to complete darkness.

This subdued light which we get between sunset and complete darkness is called evening twilight or dusk.

Similarly, for an equal interval in the morning before sunrise there is twilight. It is called morning twilight or dawn.

It is found that twilight lasts as long as the sun is within 18 below horizon

To find the duration of the Twilight

Let  be the latitude of the place and  be the declination of the sun on the date of observation. Let S1 be the position of the sun 18 below the eastern horizon and S2 the position on the horizon.

Clearly S1 marks the beginning and S2 the end of morning twilight.

Let H and h be the hour angles of the sun at S1 and S2 respectively.

Duration of twilight is the time taken by the sun to move from S1 to S2 . That is time required to describe hour angle H-h.

Therefore, Duration of twilight is t = (H-h)/15 hours.

From the spherical triangle PS1Z,

cos ZS1 = cos PZ cos PS1 + sin PZ sin PS1 cos ZPS1

cos 108 = cos(90- ) cos(90-) + sin(90-) sin(90-) cos H

- sin18 = sin  sin + cos  cos cos H

1  sin 18  sin  sin   H  cos    cos cos 

Again from spherical triangle PS2Z,

cos ZS2 = cos PZ cos PS2 + sin PZ sin PS2 cos ZPS2

cos 90 = cos(90- ) cos(90-) + sin(90-) sin(90-) cos h

cos  cos cos h = - sin  sin 

h = cos -1[-tan  tan ]

Duration of twilight is t= (H-h)/15 hours, where H and h are given as above

Duration of evening twilight is t= (H-h)/15 hours.

Duration of twilight on the date = 2  (H-h)/15 hours.

To find the condition that twilight may last throughout night Twilight may last throughout night if the sun must be within 18 below the horizon throughout night.

The sun is deepest below horizon at midnight, when it is at lower transit.

Therefore, if the distance of the sun below horizon at lower transit be less than 18 there will be twilight throughout night.

Let A be the position of the sun at the lower transit on any date and  be its declination on the date. Let  be the latitude of the place.

For twilight to last throughout night, nA ≤ 18. i.e, PA-Pn ≤ 18 i.e, PR-AR-Pn ≤ 18 i.e, 90- -  ≤ 18 i.e, 72-  ≤ 18 or  ≥ 72- 

Taking maximum value of = ,  ≥ 72- 

Therefore, twilight lasts throughout night only in places of latitude

 ≥ 72-  on the dates when the sun’s declination is  ≥ 72- .

To find the number of consecutive nights having twilight throughout night

For places of latitude  ≥ 72-  there will be twilight throughout night on the dates when the sun’s declination is  ≥ 72- .

Therefore, from the date when north declination of the sun is 72-  to the date when it is again equal to 72-  there will be twilight throughout night.

Let S1 ,S2 be the positions of the sun when its north declination is 72- . The number of nights having twilight throughout night is the number of nights in the period which the sun takes to move from S1 to S2.

Let L be the mid point of the ecliptic . L is also the midpoint of S1 S2.

Let S1 L= x so that  S1 = 90-x and S1 S2 = 2x.

From the spherical triangle S1 D1,

xeccossin(72)cos1  

Assuming that the motion of the sun along the ecliptic is uniform and the number of days in a year to be 365, the sun describes 360 in 365 days.

Therefore, time taken to describe S1 S2

= (365/360) S1 S2 days.

= (73/36) x days = (73/36) cos-1[sin(72- ) cosec ] days

Therefore, the number of nights having twilight throughout night is the integral part of (73/36) cos-1[sin(72- ) cosec ] or the next higher integer.

To find the duration of twilight when it is shortest

Let S1 be the position of the sun at the beginning of twilight(18 below the horizon) and S2 its position on the horizon at the end twilight.

Duration of twilight is the time taken by the sun to describe S1PS2 .

Rotate the celestial sphere westward, till S1 is brought into coincidence with S2.

Now, the angle through which the celestial sphere is rotated is equal to S1PS2 .

As the celestial sphere is rotated the original position of zenith when S1 is brought to S2.

Now, the angle described by the zenith at the pole is ZPZ1 = S1PS2.

Therefore, duration of twilight is shortest when ZPZ1 is minimum. Clearly, ZPZ1 is minimum when arc ZZ1 in minimum. In the spherical triangle ZZ1S1. arc ZZ1 + arc ZS1 ≥ arc Z1 S1

arc ZZ1 + 90 ≥ 108

arc ZZ1 ≥ 18

Therefore, minimum value of arc ZZ1 = 18.

When arc ZZ1 = 18, Z lies of Z1S1.

Therefore, for twilight to be shortest Z must lie on Z1S1.

Draw PD perpendicular to arc ZZ1 , PD bisects ZPZ1 and arc ZZ1.

Therefore, DZ = DZ1 = 9.

Let  be the latitude of the place and , the declination of sun on that day.

From spherical triangle PDZ,

cos PZ = cos PD cos DZ + sin PD sin DZ cos 90

cos(90- ) = cos PD cos 9  cos PD = sin / cos 9

From spherical triangle PDS, cos PS1 = cos PD cos DS1 + sin PD sin DS1 cos 90 cos(90- ) = cos PD cos 99  cos PD = sin  / sin 9

Therefore, (sin / cos 9) = (sin  / sin 9)

sin  = - sin  tan 9

 = sin-1[- sin  tan 9]

Therefore, duration of twilight is shortest on the date when the sun’s declination  = sin-1[- sin  tan 9]. Let ZPZ1 = h so that shortest duration of twilight = h/15 hours.

PD bisects ZPZ1 . Therefore, DPZ1 = h/2.

From spherical triangle PDZ,

sinsinDPZPDZ  sinsinDZPZ

sin(h/ 2)sin 90  sin 9sin90    sin(h/ 2)sin 9sec  (/h 2)sin[sin 9sec]1 

h  2sin[sin1 9sec] 

Civil, nautical and astronomical twilights

The time when the centre of the sun is 6 below the horizon is called civil twilight.

The name nautical twilight is applied to the time when the centre of the sun is 12 below the horizon.

Astronomical twilight is the name given to the time when the sun is at a depth of 18 below the horizon.

The Nautical Almanac gives the times of beginning of civil, nautical and astronomical twilights in the morning and of their evening for different latitudes.

UNIT III

KEPLER’S LAWS

Kepler's laws :

First Law: Planetary orbits are elliptical with the sun at a focus.

Second Law: The radius vector from the sun to a planet sweeps equal areas in equal times. Third Law: The ratio of the square of the period of revolution and the cube of the ellipse semimajor axis is the same for all planets.

Perigee:

The point in the orbit of the moon or a satellite at which it is nearest to the earth.

Apogee: The point in the orbit of an object (such as a satellite) orbiting the earth that is at the greatest distance from the center of the earth also : the point farthest from a planet or a satellite (such as the moon) reached by an object orbiting it compare perigee.

Newton’s Deduction from Kepler’s laws: Newton’s Law of Gravitation is states that in this universe attracts every other body with a force which is directly proportional to the product of their masses and is inversely proportional to the product of the squares of the distance between them.

Newton’s Law of Gravitation can be easily obtained from Kepler’s Laws of Planetary Motion.

Suppose a planet of mass m is revolving around the sun of mass M in a nearly circular orbit of radius r, with a constant angular velocity ω. Let, T be the time period of revolution of the planet around the sun.

True Anomaly: True anomaly is the angle, V, between lines drawn from the centre of mass (near the centre of the Sun, S), to a planet P, and to the perihelion point B, where the planet comes closest to the Sun. Mean Anomaly: If the mean anomaly is known at any given instant, it can be calculated at any later (or prior) instant by simply adding (or subtracting) n δt where δt represents the small time difference. Mean anomaly does not measure an angle between any physical objects.

Results: M= u – e sin u V= m + 2e sin m + 5/4 e sin 2m M = v- 2e sin v

Equation of Time: Apparent Solar time – E = Mean solar Time The equation of time describes the discrepancy between two kinds of solar time. The word equation is used in the medieval sense of "reconcile a difference". Mean solar time is the hour angle of the mean Sun plus 12 hours. This 12 hour offset comes from the decision to make each day start at midnight for civil purposes whereas the hour angle or the mean sun is measured from the local meridian.

Equation of Time vanishes Four Times a year.

The equation of time is determined by the following parameters:

The eccentricity of the orbit of the earth

The angle between the ecliptic and the equatorial planes

The last two parameters change gradually by nutation and precession. It is therefore interesting to examine the influence of each parameter. parameter: the eccentricity. If e = 0 a regular variation results that is caused by the inclination of the ecliptic plane. The deviations of the apparent solar time from the mean solar time increase with growing e in winter and autumn. Thus, the yearly variation becomes dominant. Since at the perihelion and aphelion the equation of time is only a function of the ecliptic inclination and the angle P, all plots have the same value at these two points. 2. parameter: the inclination of the ecliptic. ε = 0 yields a plot which is symmetric to the passage through the aphelion. The greater ε the more dominant the variation with a period of half a year. All plots have four common points at the beginning of each season, for the equation of time depends only on the two other parameters there (eccentricity and P). As the projection from the ecliptic plane onto the equatorial plane does not change the polar angle relative to the winter solstice, ε does not influence the value of the equation of time at the beginning of a season. 3. parameter: the time interval between the beginning of winter and the passage through the perihelion. If ∆t = 0 the two main variations vanish both at the beginning of winter and summer (because winter begins when the earth passes the perihelion; the aphelion is the summer solstice). Therefore, the resulting function is symmetric and the extreme values are in autumn and winter. If ∆t increases, the two components tend to compensate each other in winter whereas the negative value in summer begins to dominate.

Seasons:

A season is a division of the year based on changes in weather, ecology, and the number of daylight hours in a given region. On Earth, seasons are the result of Earth's orbit around the Sun and Earth's axial tilt relative to the ecliptic plane.[2][3] In temperate and polar regions, the seasons are marked by changes in the intensity of sunlight that reaches the Earth's surface, variations of which may cause animals to undergo hibernation or to migrate, and plants to be dormant. Various cultures define the number and nature of seasons based on regional variations, and as such there are a number of both modern and historical cultures whose number of seasons vary.

The Northern Hemisphere experiences more direct sunlight during May, June, and July, as the hemisphere faces the Sun. The same is true of the Southern Hemisphere in November, December, and January. It is Earth's axial tilt that causes the Sun to be higher in the sky during the summer months, which increases the solar flux. However, due to seasonal lag, June, July, and August are the warmest months in the Northern Hemisphere while December, January, and February are the warmest months in the Southern Hemisphere. In temperate and sub-polar regions, four seasons based on the Gregorian calendar are generally recognized: spring, summer, autumn or fall, and winter. Ecologists often use a six-season model for temperate climate regions which are not tied to any fixed calendar dates: prevernal, vernal, estival, serotinal, autumnal, and hibernal. Many tropical regions have two seasons: the rainy, wet, or monsoon season and the dry season. Some have a third cool, mild, or harmattan season. "Seasons" can also be dictated by the timing of important ecological events such as hurricane season, tornado season, and wildfire season

Seasons often hold special significance for agrarian societies, whose lives revolve around planting and harvest times, and the change of seasons is often attended by ritual. The definition of seasons is also cultural. In India, from ancient times to the present day, six seasons or Ritu based on south Asian religious or cultural calendars are recognised and identified for purposes such as agriculture and trade.

 Winter – December, January and February.  Spring – March, April and May.  Summer – June, July and August.  Autumn – September, October and November. Spring, lasts 92 days, 19 hours; summer 93 days, 15 hours; autumn, 89 days, 20 hours; winter, 89 days, zero hours. One last postscript: Some people may wonder why we in the Northern Hemisphere have our spring and summer when our planet is far away from the sun.

Causes of Seasons:

As the earth spins on its axis, producing night and day, it also moves about the sun in an elliptical (elongated circle) orbit that requires about 365 1/4 days to complete. The earth's spin axis is tilted with respect to its orbital plane. This is what causes the seasons.

Seasons happen because Earth's axis is tilted at an angle of about 23.4 degrees and different parts of Earth receive more solar energy than others.

UNIT IV

PARALLAX

Geocentric Parallax:

The difference in the apparent direction or position of a celestial body as observed from the center of the earth and from a point on the surface of the earth.

Horizontal parallax

It is the geocentric parallx of a heavenly body when in the horizon, or the angle subtended at the body by the earth's radius. Optical parallax, the apparent displacement in position undergone by an object when viewed by either eye singly.

Parallax:

The apparent displacement or the difference in apparent direction of an object as seen from two different points not on a straight line with the object especially : the angular difference in direction of a celestial body as measured from two points on the earth's orbit.

Annual Parallax:

The maximum apparent difference in position of a star during the course of a year due to the changing position of the Earth in its orbit around the Sun; also known as heliocentric parallax.

Helio Centric Parallax:

The parallax of a celestial body measured with the earth's orbit around the sun as a baseline : the angle subtended at the celestial body by the radius of the earth's orbit. — called also annual parallax, stellar parallax.

Unit V

The Moon and Eclipse

The Moon is Earth's only proper natural satellite. At one-quarter the diameter of Earth, it is the largest natural satellite in the Solar System relative to the size of its planet, and the fifth largest satellite in the Solar System overall.

The Moon is Earth's only proper natural satellite. At one-quarter the diameter of Earth (comparable to the width of Australia), it is the largest natural satellite in the Solar System relative to the size of its planet, and the fifth largest satellite in the Solar System overall (larger than any dwarf planet). Orbiting Earth at an average lunar distance of 384,400 km (238,900 mi), or about 30 times Earth's diameter, its gravitational influence is the main driver of Earth's tides and slightly lengthens Earth's day. The Moon is classified as a planetary-mass object and a differentiated rocky body, and lacks any significant atmosphere, hydrosphere, or magnetic field. Its surface gravity is about one-sixth of Earth's (0.1654 g); Jupiter's moon Io is the only satellite in the Solar System known to have a higher surface gravity and density.

The Moon's orbit around Earth has a sidereal period of 27.3 days, and a synodic period of 29.5 days. The synodic period drives its lunar phases, which form the basis for the months of a lunar calendar. The Moon is tidally locked to Earth, which means that the length of a full rotation of the Moon on its own axis (a lunar day) is the same as the synodic period, resulting in its same side (the near side) always facing Earth. That said, 59% of the total lunar surface can be seen from Earth through shifts in perspective (its libration).

The near side of the Moon is marked by dark volcanic maria ("seas"), which fill the spaces between bright ancient crustal highlands and prominent impact craters. The lunar surface is relatively non-reflective, with a reflectance just slightly brighter than that of worn asphalt. However, because it reflects direct sunlight, is contrasted by the relatively dark sky, and has a large apparent size when viewed from Earth, the Moon is the brightest celestial object in Earth's sky after the Sun. The Moon's apparent size is nearly the same as that of the Sun, allowing it to cover the Sun almost completely during a total solar eclipse.

The first manmade object to reach the Moon was the Soviet Union's Luna 2 uncrewed spacecraft in 1959; this was followed by the first successful soft landing by Luna 9 in 1966. The only human lunar missions to date have been those of the United States' NASA Apollo program, which conducted the first crewed lunar orbiting mission with Apollo 8 in 1968. Beginning with Apollo 11, six human landings took place between 1969 and 1972. These and later uncrewed missions returned lunar rocks which have been used to develop a detailed geological understanding of the Moon's origins, internal structure, and subsequent history; the most widely accepted origin explanation posits that the Moon formed about 4.51 billion years ago, not long after Earth, out of the debris from a giant impact between the planet and a hypothesized Mars-sized body called Theia.

Both the Moon's natural prominence in the earthly sky and its regular cycle of phases as seen from Earth have provided cultural references and influences for human societies and cultures throughout history. Such cultural influences can be found in language, calendar systems, art, and mythology.

Phases of Moon:

 new Moon.

 waxing crescent Moon.

 first quarter Moon.

 waxing gibbous Moon.

 full Moon.

 waning gibbous Moon.

 last quarter Moon.

 waning crescent Moon. New: We cannot see the Moon when it is a new moon. Waxing Crescent: In the Northern Hemisphere, we see the waxing crescent phase as a thin crescent of light on the right. First Quarter: We see the first quarter phase as a half moon. Waxing Gibbous: The waxing gibbous phase is between a half moon and full moon. Metonic Cycle: The Metonic cycle or is a period of approximately 19 years after which the phases of the moon recur at the same time of the year.

Age of Moon: Scientists looked to the moon's mineral composition to estimate that the moon is around 4.425 billion years old, or 85 million years younger than what previous studies had proven. Harvest moon: The term "harvest moon" refers to the full, bright Moon that occurs closest to the start of autumn. The name dates from the time before electricity, when farmers depended on the Moon's light to harvest their crops late into the night. Epact: The epact used to be described by medieval computists as the age of a phase of the Moon in days on 22 March; in the newer Gregorian calendar, however, the epact is reckoned as the age of the ecclesiastical moon on 1 January. Solar Eclipse: A solar eclipse occurs when a portion of the Earth is engulfed in a shadow cast by the Moon which fully or partially blocks sunlight. This occurs when the Sun, Moon and Earth are aligned. Such alignment coincides with a new moon (syzygy) indicating the Moon is closest to the ecliptic plane.In a total eclipse, the disk of the Sun is fully obscured by the Moon. In partial and annular eclipses, only part of the Sun is obscured. If the Moon were in a perfectly circular orbit, a little closer to the Earth, and in the same orbital plane, there would be total solar eclipses every new moon. However, since the Moon's orbit is tilted at more than 5 degrees to the Earth's orbit around the Sun, its shadow usually misses Earth. A solar eclipse can only occur when the Moon is close enough to the ecliptic plane during a new moon. Special conditions must occur for the two events to coincide because the Moon's orbit crosses the ecliptic at its orbital nodes twice every draconic month (27.212220 days) while a new moon occurs one every synodic month (29.530587981 days). Solar (and lunar) eclipses therefore happen only during eclipse seasons resulting in at least two, and up to five, solar eclipses each year; no more than two of which can be total eclipses.

Total eclipses are rare because the timing of the new moon within the eclipse season needs to be more exact for an alignment between the observer (on Earth) and the centers of the Sun and Moon. In addition, the elliptical orbit of the Moon often takes it far enough away from Earth that its apparent size is not large enough to block the Sun entirely. Total solar eclipses are rare at any particular location because totality exists only along a narrow path on the Earth's surface traced by the Moon's full shadow or umbra.

An eclipse is a natural phenomenon. However, in some ancient and modern cultures, solar eclipses were attributed to supernatural causes or regarded as bad omens. A total solar eclipse can be frightening to people who are unaware of its astronomical explanation, as the Sun seems to disappear during the day and the sky darkens in a matter of minutes.

Since looking directly at the Sun can lead to permanent eye damage or blindness, special eye protection or indirect viewing techniques are used when viewing a solar eclipse. It is safe to view only the total phase of a total solar eclipse with the unaided eye and without protection. This practice must be undertaken carefully, though the extreme fading of the solar brightness by a factor of over 100 times in the last minute before totality makes it obvious when totality has begun and it is for that extreme variation and the view of the solar corona that leads people to travel to the zone of totality (the partial phases span over two hours while the total phase can only last a maximum of 7.5 minutes for any one location and is usually less). People referred to as eclipse chasers or umbraphiles will travel even to remote locations to observe or witness predicted central solar eclipses.

Lunar eclipse: A lunar eclipse occurs when the Moon moves into the Earth's shadow. This can occur only when the Sun, Earth, and Moon are exactly or very closely aligned with Earth between the other two, and only on the night of a full moon.