Basic concepts:

Frame of reference:

A frame of reference (or reference frame) may refer to a coordinate system used to represent and measure properties of objects, such as their position and orientation, at different moments of time. It may also refer to a set of axes used for such representation. In a weaker sense, a reference frame does not specify coordinates, but only defines the same 3-dimensional space for all moments of time such that the frame can distinguish objects at rest from those that are moving.

COORDINATE SYSTEMS

A system of coordinates must be established to uniquely describe a position in space or to describe the magnitude and direction of target velocity with respect to a specified reference frame. For the location to have meaning we must designate a starting point or a reference point and a system of measure, such as length, time, weight, etc. In order to give the measuring system a zero point and to describe more easily the coordinate system, a reference frame will be established, as previously mentioned, consisting of three mutually perpendicular lines. The point from which these lines originate is the starting point, a zero position. In actual weapon systems, this zero position is generally the pivotal axes of the sensor and tracking subsystem since all initial target data is generated there. A coordinate, then, is one of a set of values used to locate a point relative to a reference system of lines or surfaces.

Any weapon system can be subdivided by function into three general categories: data measurement, computation, and data utilization. By the nature of the hardware design constraints, each functional category employs a coordinate system that allows the simplest expression of the problem and the easiest method of actual implementation.

Spherical Coordinate System

A spherical coordinate system is ideal for use in the process of actually measuring target parameters. This is due to the fact that sensor and tracking systems rotate about their coordinate axes and measure distance along a straight line to the target. A generalized spherical coordinate system consists of two angles and a linear distance. This system can therefore be represented by an ordered triple of characters, which describe any point in the reference frame.

Cylindrical Coordinate System

Most weapon control computation is accomplished using a rectangular coordinate system, although some older gun fire control systems remain that uses a cylindrical coordinate system. A cylindrical coordinate system employs an angle of azimuth, a vertical linear distance, and a horizontal linear distance.

Rectangular Coordinate System

The rectangular or Cartesian coordinate system employs the target’s range in the true north/south direction as the Y-axis distance, the target’s range in the east/west direction as the X-axis distance, and the vertical height as the Z-axis distance. The rectangular coordinate system is illustrated in figure 19-7. Calculation of target velocity and prediction of future target position is made easier in Cartesian coordinates, thus reducing the time required for computer solution of the fire control problem. For this reason, and the ease with which data can be shared by widely separated forces, the Cartesian coordinate system is used by most modern fire control systems and command and control facilities.

Celestial sphere:

The celestial sphere is an imaginary sphere of arbitrarily large radius, concentric with Earth. All objects in the observer’s sky can be thought of as projected upon the inside surface of the celestial sphere, as if it were the underside of a dome or a hemispherical screen. The celestial sphere is a practical tool for spherical astronomy, allowing observers to plot positions of objects in the sky when their distances are unknown or unimportant.

Celestial coordinate systems:

These concepts are important for understanding celestial reference systems, the methods in which the positions of objects in the sky are measured. Certain reference lines and planes on Earth, when projected onto the celestial sphere, form the bases of the reference systems. These include the Earth’s equator, axis, and the Earth’s orbit. At their intersections with the celestial sphere, these form the celestial equator, the north & south celestial poles and the ecliptic, respectively. As the celestial sphere is considered infinite in radius, all observers see the celestial equator, celestial poles and ecliptic at the same place against the background stars.

Directions toward objects in the sky can be quantified by constructing, from these bases, celestial coordinate systems. Similar to the terrestrial longitude and latitude, the equatorial system of right ascension and declination specifies positions relative to the celestial equator and celestial poles. The ecliptic system of celestial longitude and celestial latitude specifies positions relative to the Earth’s orbit. There are more coordinate systems besides the equatorial right ascension/declination system and the ecliptic system.

Ecliptic:

The ecliptic is the apparent path of the Sun on the celestial sphere, and is the basis for the ecliptic coordinate system. It also refers to the plane of this path, which is coplanar with both the orbit of the Earth around the Sun and the apparent orbit of the Sun around the Earth. The path of the Sun is not normally noticeable from the Earth’s surface because the Earth rotates, carrying the observer through the cycle of sunrise and sunset, obscuring the apparent motion of the Sun with respect to the stars.

Motion of Vernal Equinox:

It is either of two points on the celestial sphere where the ecliptic and the celestial equator intersect. The vernal equinox, also known as “the first point of Aries,” is the point at which the sun appears to cross the celestial equator from south to north. This occurs about Mar. 21, marking the beginning of spring in the Northern Hemisphere. At the autumnal equinox, about Sept. 23, the sun again appears to cross the celestial equator, this time from north to south; this marks the beginning of autumn in the Northern Hemisphere. On the date of either equinox, night and day are of equal length (12 hr each) in all parts of the world; the word equinox is often used to refer to either of these dates. The equinoxes are not fixed points on the celestial sphere but move westward along the ecliptic, passing through all the constellations of the zodiac in 26,000 years. This motion is called the precession of the equinoxes. The vernal equinox is a reference point in the equatorial coordinate system.

Sidereal time:

Sidereal time is a time-keeping system astronomers use to keep track of the direction to point their telescopes to view a given star in the night sky. Briefly, sidereal time is a “time scale that is based on the Earth’s rate of rotation measured relative to the fixed stars.

From a given observation point, a star found at one location in the sky will be found at nearly the same location on another night at the same sidereal time. This is similar to how the time kept by a sundial can be used to find the location of the Sun. Just as the Sun and Moon appear to rise in the east and set in the west due to the rotation of the Earth, so do the stars. Both solar time and sidereal time make use of the regularity of the Earth’s rotation about its polar axis, solar time following the Sun while sidereal time roughly follows the stars. More exactly, sidereal time is measured from the vernal equinox (first moment of spring), which is not fixed because the Earth’s orbit is not perfectly symmetrical; precession and nutation shift the equinox slightly from one day to the next, so sidereal time is not an exact measure of the rotation of the Earth relative to inertial space. Common time on a typical clock measures a slightly longer cycle, accounting not only for the Earth’s axial rotation but also for the Earth’s annual revolution around the Sun of slightly less than 1 degree per day (in fact to the nearest arc-second, it takes 365.2422 days to revolve therefore 360 degrees/365.2422 days = 0.9856 degrees or 59 arc-minutes, 8 arc-seconds per day, i.e., slightly less than 1 degree per day).

A mean sidereal day is about 23 hours, 56 minutes, 4.0916 seconds (23.9344699 hours or 0.99726958 mean solar days), the time it takes the Earth to make one rotation relative to the vernal equinox.[clarification needed] (Due to nutation, an actual sidereal day is not quite so constant.) The vernal equinox itself precesses slowly westward relative to the fixed stars, completing one revolution in about 26,000 years, so the misnamed sidereal day (“sidereal” is derived from the Latin sidus meaning “star”) is some 0.0084 seconds shorter than the Earth’s period of rotation relative to the fixed stars.

Sidereal time and solar time:

Solar time is measured by the apparent diurnal motion of the sun, and local noon in apparent solar time is the moment when the sun is exactly due south or north (depending on the observer’s latitude and the season). A mean solar day (what we normally measure as a “day”) is the average time between local solar noons (“average” since this varies slightly over the year).

The Earth makes one rotation around its axis in a sidereal day; during that time it moves a short distance (about 1°) along its orbit around the sun. So after a sidereal day has passed the Earth still needs to rotate slightly more before the sun reaches local noon according to solar time. A mean solar day is, therefore, nearly 4 minutes longer than a sidereal day.

The stars are so far away that the Earth’s movement along its orbit makes nearly no difference to their apparent direction (see, however, parallax), and so they return to their highest point in a sidereal day.

Another way to see this difference is to notice that, relative to the stars, the Sun appears to move around the Earth once per year. Therefore, there is one fewer solar day per year than there are sidereal days. This makes a sidereal day approximately 365.24⁄366.24 times the length of the 24-hour solar day, giving approximately 23 hours, 56 minutes, 4.1 seconds (86,164.1 seconds).

Solar time:

Solar time is a reckoning of the passage of time based on the Sun‘s position in the sky. The fundamental unit of solar time is the day. Two types of solar time are apparent solar time (sundial time) and mean solar time (clock time).

Apparent solar time:

Apparent solar time or true solar time is based on the apparent motion of the actual Sun. It is based on the apparent solar day, the interval between two successive returns of the Sun to the local meridian. Solar time can be crudely measured by a sundial.

Standard time:

Standard time is the result of synchronizing clocks in different geographical locations within a time zone to the same time rather than using the local meridian as in local mean time or solar time. Historically, this helped in the process of weather forecasting and train travel. The concept became established in the late 19th century. The time so set has come to be defined in terms of offsets from Universal Time. Where daylight saving time is used, the term standard time typically refers to the time without the offset for daylight saving time.

Earth Atmosphere:

The atmosphere of Earth is a layer of gases surrounding the planet Earth that is retained by Earth’s gravity. The atmosphere protects life on Earth by absorbing ultraviolet solar radiation, warming the surface through heat retention (greenhouse effect), and reducing temperature extremes between day and night (the diurnal temperature variation).

Troposphere

The troposphere begins at Earth’s surface and extends to the tropopause at an average height of about 12 km, although this altitude actually varies from about 9 km (30,000 ft) at the poles and 17 km (56,000 ft) at the equator, with some variation due to weather. The troposphere is mostly heated by transfer of energy from the surface, so on average the lowest part of the troposphere is warmest and temperature decreases with altitude.

Stratosphere

The stratosphere ranges from the tropopause at about 12 km (7.5 mi; 39,000 ft) above Earth to the stratopause at an altitude of 50 to 55 km (31 to 34 mi; 160,000 to 180,000 ft). The atmospheric pressure at the top of the stratosphere is roughly 1/1000 the pressure at sea level.

Mesosphere

The mesosphere extends from the stratopause at an altitude of about 50 km to the mesopause at 80–85 km above sea level. It is the layer where most meteors burn up upon entering the atmosphere. Temperature decreases with height in the mesosphere. The mesopause, the temperature minimum that marks the top of the mesosphere, is the coldest place on Earth and has an average temperature around −85 °C. At this altitude, temperatures may drop to −100°C. Due to the cold temperature of this layer, water vapor is frozen, occasionally forming polar-mesospheric noctilucent clouds which are the highest water-based aerosols in the atmosphere. A type of lightning referred to as either sprites or ELVES, occasionally form far above tropospheric thunderclouds.

Thermosphere

Temperature increases with height in the thermosphere from the mesopause at an altitude of about 80 km (50 mi; 260,000 ft) up to the thermopause, and then is constant with height entering the exosphere. Since the thermopause is at the bottom of the exosphere, it also called the exobase. Its height averages about 700 km above Earth but actually varies with solar activity and ranges from about 500–1,000 km (310–620 mi; 1,600,000 – 3,300,000 ft).[6] Unlike the stratosphere, where a temperature inversion is caused by absorption of radiation by ozone, the inversion in the thermosphere is a result of the extremely low density of molecules. The temperature of this layer can rise to 1,500 °C (2,700 °F), though the gas molecules are so far apart that temperature in the usual sense is not well defined.

Exosphere

The exosphere is the outermost layer of Earth’s atmosphere, ranging from the exobase at an altitude of about 700 km above sea level to about half way to the Moon. It is mainly composed of hydrogen, helium and some heavier molecules such as nitrogen, oxygen and carbon dioxide closer to the exobase. The atoms and molecules are so far apart that they can travel hundreds of kilometres without colliding with one another, so the atmosphere no longer behaves like a gas. These free-moving particles follow ballistic trajectories and may migrate in and out of the magnetosphere or the solar wind.

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