Caliphate, the political-religious state comprising the Muslim community and the lands and peoples under its dominion in the centuries following the death (632 ce) of the Prophet Muhammad. Ruled by a caliph (Arabic khalīfah, “successor”), who held temporal and sometimes a degree of spiritual authority, the empire of the Caliphate grew rapidly through conquest during its first two centuries to include most of Southwest Asia, North Africa, and Spain. Dynastic struggles later brought about the Caliphate’s decline, and it ceased to exist with the Mongol destruction of Baghdad in 1258.

The urgent need for a successor to Muhammad as political leader of the Muslim community was met by a group of Muslim elders in Medina who designated Abū Bakr, the Prophet’s father-in-law, as caliph. Several precedents were set in the selection of Abū Bakr, including that of choosing as caliph a member of the Quraysh tribe. The first four caliphs—Abū Bakr, ʿUmar I, ʿUthmān, and ʿAlī, whose reigns constituted what later generations of Muslims would often remember as a golden age of pure Islam—largely established the administrative and judicial organization of the Muslim community and forwarded the policy begun by Muhammad of expanding the Islamic religion into new territories. During the 630s, Syria, Jordan, Palestine, and Iraq were conquered; Egypt was taken from Byzantine control in 645; and frequent raids were launched into North Africa,Armenia, and Persia.

The assassination of ʿUthmān and the ineffectual caliphate of ʿAlī that followed sparked the first sectarian split in the Muslim community. By 661 ʿAlī’s rivalMuʿāwiyah I, a fellow member of ʿUthmān’s Umayyad clan, had wrested away the Caliphate, and his rule established the Umayyad Caliphate that lasted until 750. Despite the largely successful reign of Muʿāwiyah, tribal and sectarian disputes erupted after his death. There were three caliphs between 680 and 685, and only by nearly 20 years of military campaigning did the next one, ʿAbd al-Malik, succeed in reestablishing the authority of the Umayyad capital of Damascus. ʿAbd al-Malik is also remembered for building the Dome of the Rock in Jerusalem. Under his son al-Walīd (705–715), Muslim forces took permanent possession of North Africa, converted the native Berbers to Islam, and overran most of the Iberian Peninsula as the Visigothic kingdom there collapsed. Progress was also made in the east with settlement in the Indus River valley. Umayyad power had never been firmly seated, however, and the Caliphate disintegrated rapidly after the long reign of Hishām (724–743). A serious rebellion broke out against the Umayyads in 747, and in 750 the last Umayyad caliph, Marwān II, was defeated in the Battle of Great Zab by the followers of the ʿAbbāsid family.

The ʿAbbāsids, descendants of an uncle of Muhammad, owed the success of their revolt in large part to their appeal to various pietistic, extremist, or merely disgruntled groups and in particular to the aid of the Shīʿites, a major dissident party that held that the Caliphate belonged by right to the descendants of ʿAlī. That the ʿAbbāsids disappointed the expectations of the Shīʿites by taking the Caliphate for themselves left the Shīʿites to evolve into a sect, permanently hostile to the orthodox Sunni majority, that would periodically threaten the established government by revolt. The first ʿAbbāsid caliph, al-Saffāḥ (749–754), ordered the elimination of the entire Umayyad clan; the only Umayyad of note who escaped was ʿAbd al-Raḥman, who made his way to Spain and established anUmayyad dynasty that lasted until 1031.

The period 786–861, especially the caliphates of Hārūn (786–809) and al-Maʾmūn(813–833), is accounted the height of ʿAbbāsid rule. The eastward orientation of the dynasty was demonstrated by al-Manṣūr’s removal of the capital to Baghdad in 762–763 and by the later caliphs’ policy of marrying non-Arabs and recruitingTurks, Slavs, and other non-Arabs as palace guards. Under al-Maʾmūn, the intellectual and artistic heritage of Iran (Persia) was cultivated, and Persian administrators assumed important posts in the Caliphate’s administration. After 861, anarchy and rebellion shook the empire. Tunisia and eastern Iran came under the control of hereditary governors who made token acknowledgment of Baghdad’s suzerainty. Other provinces became less-reliable sources of revenue. Shīʿite and similar groups, including the Qarmaṭians in Syria and the Fāṭimids in North Africa, challenged ʿAbbāsid rule on religious as well as political grounds.

ʿAbbāsid power ended in 945, when the Būyids, a family of rough tribesmen from northwestern Iran, took Baghdad under their rule. They retained the ʿAbbāsid caliphs as figureheads. The Sāmānid dynasty that arose in Khorāsān and Transoxania and the Ghaznavids in Central Asia and the Ganges River basin similarly acknowledged the ʿAbbāsid caliphs as spiritual leaders of Sunni Islam. On the other hand, the Fāṭimids proclaimed a new caliphate in 920 in their capital of Al-Mahdiyyah in Tunisia and castigated the ʿAbbāsids as usurpers; the Umayyad ruler in Spain, ʿAbd al-Raḥmān III, adopted the title of caliph in 928 in opposition to both the ʿAbbāsids and the Fāṭimids. Nominal ʿAbbāsid authority was restored to Egypt by Saladin in 1171. By that time the ʿAbbāsids had begun to regain some semblance of their former power, as the Seljuq dynasty of sultans in Baghdad, which had replaced the Būyids in 1055, itself began to decay. The caliph al-Nāṣir (1180–1225) achieved a certain success in dealing diplomatically with various threats from the east, but al-Mustaʿṣim (1242–58) had no such success and was murdered in the Mongol sack of Baghdad that ended the ʿAbbāsid line in that city. A scion of the family was invited a few years later to establish a puppet caliphate in Cairo that lasted until 1517, but it exercised no power whatever. From the 13th century onward a variety of rulers outside of Cairo also included caliph among their titles, although their claims to universal leadership of the Muslim community seem to have been more notional than real.

The concept of the caliphate took on new significance in the 18th century as an instrument of statecraft in the declining Ottoman Empire. Facing the erosion of their military and political power and territorial losses inflicted in a series of wars with European rivals, the Ottoman sultans, who had occasionally styled themselves as caliphs since the 14th century, began to stress their claim to leadership of the Islamic community. This served both as means of retaining some degree of influence over Muslim populations in formerly Ottoman lands and as means of bolstering Ottoman legitimacy within the empire. The caliphate was abolished in 1924, following the dissolution of the Ottoman Empire and the rise of the Turkish Republic.

In the 20th century the reestablishment of the caliphate, although occasionally invoked by Islamists as a symbol of global Islamic unity, was of no practical interest for mainstream Islamist groups such as the Muslim Brotherhood in Egypt. It did, however, figure prominently in the rhetoric of violent extremist groups such as al-Qaeda. In June 2014 an insurgent group known as the Islamic State in Iraq and the Levant (ISIL; also known as the Islamic State in Iraq and Syria [ISIS] and the Islamic State [IS]), which had taken control of areas of eastern Syria and western Iraq, declared the establishment of a caliphate with the group’s leader Abu Bakral-Baghdadi as caliph. Outside of extremist circles, the group’s claim was widely rejected.

Resource Lesson A Derivation of the Formulas for Centripetal Acceleration An object is said to be moving in uniform circular motion when it maintains a constant speed while traveling in a circle. Remember that since acceleration is a vector quantity comprised of both magnitude and direction, objects can accelerate in any of these three ways:        1. constant direction, changing speed (linear acceleration);        2. constant speed, changing direction (centripetal acceleration);        3. change in both speed and direction (angular acceleration). In this lesson, we will be investigating centripetal acceleration and uniform circular motion - that is, objects moving in circular paths at constant speeds. While moving in a circular path, an object is constantly being pulled "towards the center" of the circle away from its tangential path. Envision a stopper on the end of a string being twirled over your head in a horizontal circle. If the string were to break, the stopper would "fly off at a tangent." The tension in the string is forcing the stopper to constantly be pulled back towards the center to follow a circular, instead of a linear, path. As shown in the diagram above, in a certain amount of time, Δt, an object traveling in a circular path would move from position A at time t1 where its velocity is labeled vo to position B at time t2 where its velocity is labeled vf. Note that the magnitude of vf equals that of vo since we are only changing the direction of the velocity, not the object's speed. Remember that acceleration equals Δv/Δt. To diagram this acceleration, we must be able to diagram the resultant change in velocity, or Δv. Thus we must recognize the orientation of the vector -vo. Since the vector vo points to the right, the vector -vo would have the exact same magnitude but point in the opposite direction.   The direction of the acceleration that an object experiences during an interval of time, Δt, is illustrated in the next diagram by showing the direction of vf - vo. To diagram the vector resultant vf - vo, we will use the head-to-tail method of vector addition where Δv = vf - vo = vf + (- vo) Notice that the resultant velocity, Δv, starts at the beginning of the vector vf and terminates at the end of the vector -vo. This relation can also be seen in the following diagram when we merely rearranged the vector equation Δv = vf - vo to read vo + Δv = vf . Notice that vf  is now the resultant vector since vf starts at the beginning of the vector vo and terminates at the end of the vector Δv. Note that in both cases, Δv points to the center of the circle reflecting that the acceleration is also directed towards the center of the circle.    We begin out derivation of the magnitude of this centripetal acceleration by comparing two similar triangles. The first triangle illustrates the relation s = rθ. In the time interval Δt, the object traveled from point A to point B, traversing through an angle of magnitude θ and along an arc of length s as shown in the following diagram. Note that as Δt → 0, the length of the arc, s, would approach the length of the chord, c**. Now let's look at the angles formed by vf and vo. Since vo and vf are both tangential velocities, they are perpendicular to their respective radii. Since OACB is a quadrilateral, the sum of its interior angles must equal 360º. Note the following relationships:     m∠A =  m∠B = 90º     m∠ACB + θ = 180º     m∠ACB + m∠BCD = 180     (supplementary angles) Therefore m∠BCD = θ. Consequently, in our second triangle formed by Δv = vf - vo, vf  and - vo will also meet each other at an angle θ.    A comparison of corresponding parts of these two similar isosceles triangles yields velocity triangle displacement triangle where |-vo| = |vf| = v In a small time interval Δt, the arc length s → c. And since we also know that distance = rate * time, we can replace the length of the chord, c, with the expression s = vΔt which results in the next ratio that algebraically simplifies to Since a = Δv / Δt, where we have added the subscript c to represent the term centripetal since this formula only applies to objects moving in uniform circular motion. For an object traveling with a constant speed we may use the relationship d = rt. For a circular path, d equals the circumference, C = 2πr and t equals the time for one revolution, or the period, T Substituting this expression for c into the equation for centripetal acceleration, yields Uniform circular motion requires that the object MUST move at a constant speed; therefore it can only move in a horizontal circle - that is, one in which gravity is always perpendicular to the object's tangential velocity. When moving in vertical circles, the object’s speed is always changing and the object is not considered to be moving in uniform circular motion. All units in these formulas are standard SI units:  m, m/sec, m/sec2, and seconds.

Circular Motion Principles for Satellites

A satellite is any object that is orbiting the earth, sun or other massive body. Satellites can be categorized as natural satellites or man-made satellites. The moon, the planets and comets are examples of natural satellites. Accompanying the orbit of natural satellites are a host of satellites launched from earth for purposes of communication, scientific research, weather forecasting, intelligence, etc. Whether a moon, a planet, or some man-made satellite, every satellite's motion is governed by the same physics principles and described by the same mathematical equations.
 

A Satellite is a Projectile

The fundamental principle to be understood concerning satellites is that a satellite is a projectile. That is to say, a satellite is an object upon which the only force is gravity. Once launched into orbit, the only force governing the motion of a satellite is the force of gravity. Newton was the first to theorize that a projectile launched with sufficient speed would actually orbit the earth. Consider a projectile launched horizontally from the top of the legendary Newton's Mountain - at a location high above the influence of air drag. As the projectile moves horizontally in a direction tangent to the earth, the force of gravity would pull it downward. If the launch speed was too small, it would eventually fall to earth. The diagram at the right resembles that found in Newton's original writings. Paths A and B illustrate the path of a projectile with insufficient launch speed for orbital motion. But if launched with sufficient speed, the projectile would fall towards the earth at the same rate that the earth curves. This would cause the projectile to stay the same height above the earth and to orbit in a circular path (such as path C). And at even greater launch speeds, a cannonball would once more orbit the earth, but now in an elliptical path (as in path D). At every point along its trajectory, a satellite is falling toward the earth. Yet because the earth curves, it never reaches the earth.

So what launch speed does a satellite need in order to orbit the earth? The answer emerges from a basic fact about the curvature of the earth. For every 8000 meters measured along the horizon of the earth, the earth's surface curves downward by approximately 5 meters. So if you were to look out horizontally along the horizon of the Earth for 8000 meters, you would observe that the Earth curves downwards below this straight-line path a distance of 5 meters. For a projectile to orbit the earth, it must travel horizontally a distance of 8000 meters for every 5 meters of vertical fall. It so happens that the vertical distance that a horizontally launched projectile would fall in its first second is approximately 5 meters (0.5*g*t²). For this reason, a projectile launched horizontally with a speed of about 8000 m/s will be capable of orbiting the earth in a circular path. This assumes that it is launched above the surface of the earth and encounters negligible atmospheric drag. As the projectile travels tangentially a distance of 8000 meters in 1 second, it will drop approximately 5 meters towards the earth. Yet, the projectile will remain the same distance above the earth due to the fact that the earth curves at the same rate that the projectile falls. If shot with a speed greater than 8000 m/s, it would orbit the earth in an elliptical path. 

Velocity, Acceleration and Force Vectors

The motion of an orbiting satellite can be described by the same motion characteristics as any object in circular motion. The velocity of the satellite would be directed tangent to the circle at every point along its path. The acceleration of the satellite would be directed towards the center of the circle - towards the central body that it is orbiting. And this acceleration is caused by a net force that is directed inwards in the same direction as the acceleration.

This centripetal force is supplied by gravity - the force that universally acts at a distance between any two objects that have mass. Were it not for this force, the satellite in motion would continue in motion at the same speed and in the same direction. It would follow its inertial, straight-line path. Like any projectile, gravity alone influences the satellite's trajectory such that it always falls below its straight-line, inertial path. This is depicted in the diagram below. Observe that the inward net force pushes (or pulls) the satellite (denoted by blue circle) inwards relative to its straight-line path tangent to the circle. As a result, after the first interval of time, the satellite is positioned at position 1 rather than position 1'. In the next interval of time, the same satellite would travel tangent to the circle in the absence of gravity and be at position 2'; but because of the inward force the satellite has moved to position 2 instead. In the next interval of time, the same satellite has moved inward to position 3 instead of tangentially to position 3'. This same reasoning can be repeated to explain how the inward force causes the satellite to fall towards the earth without actually falling into it.

Elliptical Orbits of Satellites

Occasionally satellites will orbit in paths that can be described as ellipses. In such cases, the central body is located at one of the foci of the ellipse. Similar motion characteristics apply for satellites moving in elliptical paths. The velocity of the satellite is directed tangent to the ellipse. The acceleration of the satellite is directed towards the focus of the ellipse. And in accord with Newton's second law of motion, the net force acting upon the satellite is directed in the same direction as the acceleration - towards the focus of the ellipse. Once more, this net force is supplied by the force of gravitational attraction between the central body and the orbiting satellite. In the case of elliptical paths, there is a component of force in the same direction as (or opposite direction as) the motion of the object.  Such a component of force can cause the satellite to either speed up or slow down in addition to changing directions. So unlike uniform circular motion, the elliptical motion of satellites is not characterized by a constant speed.

In summary, satellites are projectiles that orbit around a central massive body instead of falling into it. Being projectiles, they are acted upon by the force of gravity - a universal force that acts over even large distances between any two masses. The motion of satellites, like any projectile, is governed by Newton's laws of motion. For this reason, the mathematics of these satellites emerges from an application of Newton's universal law of gravitation to the mathematics of circular motion.

Law of gravitation

The idealized observation of Galileo that all bodies in free-fall accelerate equally implies that the gravitational force causing acceleration bears a constant relation to the inertial mass.

 According to Newton’s postulated law of gravitation, two bodies of mass m₁ and m₂, separated by a distance r, exert equal attractive forces on each other (the equal action and reaction of the third law of motion) of magnitude proportional to m₁m₂/r². The constant of proportionality, G, in the gravitational law:

 F = Gm₁m₂/r²,

 is thus to be regarded as a universal constant, applying to all bodies, whatever their constitution. The constancy of gravitational acceleration, g, at a given point on the Earth is a particular case of this general law.