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Apparent retrograde motion

Apparent retrograde motion is the apparent motion of a planet in a direction opposite to that of other bodies within its system, as observed from a particular vantage point. Direct motion or prograde motion is motion in the same direction as other bodies.

While the terms direct and prograde are equivalent in this context, the former is the traditional term in astronomy. The earliest recorded use of prograde was in the early 18th century, although the term is now less common.

Etymology and history
The term retrograde is from the Latin word retrogradus – "backward-step", the affix retro- meaning "backwards" and gradus "step". Retrograde is most commonly an adjective used to describe the path of a planet as it travels through the night sky, with respect to the zodiac, stars, and other bodies of the celestial canopy. In this context, the term refers to planets, as they appear from Earth, stopping briefly and reversing direction at certain times, though in reality, of course, we now understand that they perpetually orbit in the same uniform direction.

Although planets can sometimes be mistaken for stars as one observes the night sky, the planets actually change position from night to night in relation to the stars. Retrograde (backward) and prograde (forward) are observed as though the stars revolve around the Earth. Ancient Greek astronomer Ptolemy in 150 AD believed that the Earth was the center of the Solar System and therefore used the terms retrograde and prograde to describe the movement of the planets in relation to the stars. Although it is known today that the planets revolve around the Sun, the same terms continue to be used in order to describe the movement of the planets in relation to the stars as they are observed from Earth. Like the Sun, the planets appear to rise in the East and set in the West. When a planet travels eastward in relation to the stars, it is called prograde. When the planet travels westward in relation to the stars (opposite path) it is called retrograde.

This apparent retrogradation puzzled ancient astronomers, and was one reason they named these bodies 'planets' in the first place: 'Planet' comes from the Greek word for 'wanderer'. In the geocentric model of the Solar System proposed by Apollonius in the third century BCE, retrograde motion was explained by having the planets travel in deferents and epicycles. It was not understood to be an illusion until the time of Copernicus, although the Greek astronomer Aristarchus in 240 BCE proposed a heliocentric model for the Solar System.

Galileo's drawings show that he first observed Neptune on December 28, 1612, and again on January 27, 1613. On both occasions, Galileo mistook Neptune for a fixed star when it appeared very close—in conjunction—to Jupiter in the night sky, hence, he is not credited with Neptune's discovery. During the period of his first observation in December 1612, Neptune was stationary in the sky because it had just turned retrograde that very day. Since Neptune was only beginning its yearly retrograde cycle, the motion of the planet was far too slight to be detected with Galileo's small telescope.

From Earth
When standing on the Earth looking up at the sky, it would appear that the Moon travels from east to west, just as the Sun and the stars do. Day after day however, the Moon appears to move to the east with respect to the stars. In fact, the Moon orbits the Earth from west to east, as do the vast majority of manmade satellites such as the International Space Station. The apparent westward motion of the Moon from the Earth's surface is actually an artifact of its being in a supersynchronous orbit. This means that the Earth completes one sidereal rotation before the Moon is able to complete one orbit. As a result, it looks like the Moon is travelling in the opposite direction, otherwise known as apparent retrograde motion. A person standing on Earth "catches up" to the Moon and passes it because the Earth completes one rotation before the Moon completes one orbit.

This phenomenon also occurs on Mars, which has two natural satellites, Phobos and  Deimos. Both moons orbit Mars in an eastward (prograde) direction; however, Deimos has an orbital period of 1.23 Martian sidereal days, making it supersynchronous, whereas Phobos has an orbital period of 0.31 Martian  sidereal days, making it subsynchronous. Consequently, although both moons are traveling in an eastward (prograde) direction, they appear to be traveling in opposite directions when viewed from the surface of Mars due to their orbital periods in relation to the rotational period of the planet.

All other planetary bodies in the Solar System also appear to periodically switch direction as they cross Earth's sky. Though all stars and planets appear to move from east to west on a nightly basis in response to the rotation of Earth, the outer planets generally drift slowly eastward relative to the stars. Asteroids and Kuiper Belt objects (including Pluto) exhibit apparent retrograde motion. This motion is normal for the planets, and so is considered direct motion. However, since Earth completes its orbit in a shorter period of time than the planets outside its orbit, it periodically overtakes them, like a faster car on a multi-lane highway. When this occurs, the planet being passed will first appear to stop its eastward drift, and then drift back toward the west. Then, as Earth swings past the planet in its orbit, it appears to resume its normal motion west to east. Inner planets Venus and Mercury appear to move in retrograde in a similar mechanism, but as they can never be in opposition to the Sun as seen from Earth, their retrograde cycles are tied to their inferior conjunctions with the Sun. They are unobservable in the Sun's glare and in their "new" phase, with mostly their dark sides toward Earth; they occur in the transition from evening star to morning star.

The more distant planets retrograde more frequently, as they do not move as much in their orbits while Earth completes an orbit itself. The retrograde motion of a hypothetical extremely distant (and nearly non-moving) planet would take place during a half-year, with the planet's apparent yearly motion being reduced to a parallax ellipse.

The center of the retrograde motion occurs at the planet's opposition which is when the planet is exactly opposite the Sun. This position is halfway, or 6 months, around the ecliptic from the Sun. The planet's height in the sky is opposite that of the Sun's height. The planet is at its highest at the winter solstice, and at its lowest at the summer solstice, on those (rare) occasions when it passes through the center of its retrograde motion near a solstice. Note particularly that the hemisphere the observer is in is critical to what they observe. The December Solstice will place the planet high in the northern hemisphere sky where it is winter and place it low in the southern hemisphere sky where it is summer. The opposite is true if this happens at the June Solstice.

Since the planet's opposition retrograde motion is when the Earth passes closest, the planet appears at its brightest for the year.

The period between the center of such retrogradations is the synodic period of the planet.

From Mercury
From any point on the daytime surface of Mercury when the planet is near perihelion (closest approach to the Sun), the Sun undergoes apparent retrograde motion. This occurs because, from approximately four Earth days before perihelion until approximately four Earth days after it, Mercury's angular orbital speed exceeds its angular rotational velocity. Mercury's elliptical orbit is farther from circular than that of any other planet in the Solar System, resulting in a substantially higher orbital speed near perihelion. As a result, at specific points on Mercury's surface an observer would be able to see the Sun rise part way, then reverse and set before rising again, all within the same Mercurian day.

title: Apparent retrograde motion


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Zenith

The zenith is an imaginary point directly "above" a particular location, on the celestial sphere. "Above" means in the vertical direction (plumb line) opposite to the gravity direction at that location (nadir). The zenith is the "highest" point on the celestial sphere.

Origin
The word zenith derives from an inaccurate reading of the Arabic expression سمت الرأس (samt al-raʾs), meaning "direction of the head" or "path above the head", by Medieval Latin scribes in the Middle Ages (during the 14th century), possibly through Old Spanish. It was reduced to samt ("direction") and miswritten as senit/cenit, the m being misread as ni. Through the Old French cenith, zenith first appeared in the 17th century.

Relevance and use

The term zenith sometimes means the highest point, way, or level reached by a celestial body on its daily apparent path around a given point of observation. This sense of the word is often used to describe the position of the Sun ("The sun reached its zenith..."), but to an astronomer, the Sun does not have its own zenith and is at the zenith only if it is directly overhead.

In a scientific context, the zenith is the direction of reference for measuring the zenith angle (or zenith angular distance), the angle between a direction of interest (e.g. a star) and the local zenith - that is, the complement of the altitude angle (or elevation angle).

The Sun reaches the observer's zenith when it is 90° above the horizon, and this only happens between the Tropic of Cancer and the Tropic of Capricorn. In Islamic astronomy, the passing of the Sun over the zenith of Mecca becomes the basis of the qibla observation by shadows twice a year on 27/28 May and 15/16 July.

At a given location during the course of a day, the Sun reaches not only its zenith but also its nadir, at the antipode of that location 12 hours from solar noon.

In astronomy, the altitude in the horizontal coordinate system and the zenith angle are complementary angles, with the horizon perpendicular to the zenith. The astronomical meridian is also determined by the zenith, and is defined as a circle on the celestial sphere that passes through the zenith, nadir, and the celestial poles.

A zenith telescope is a type of telescope designed to point straight up at or near the zenith, and used for precision measurement of star positions, to simplify telescope construction, or both. The NASA Orbital Debris Observatory and the Large Zenith Telescope are both zenith telescopes, since the use of liquid mirrors meant these telescopes could only point straight up.

On the International Space Station, zenith and nadir are used instead of up and down, referring to directions within and around the station, relative to the earth.

Zenith star
Zenith stars (also "star on top", "overhead star", "latitude star") are stars which culminate (pass) through the zenith. Used in celestial navigation it allowed the navigator to determine their latitudal position, since at every latitude different stars pass one's zenith, or with other words lie on the great circle of the zenith ("zenith circle"). Zenith stars are not to be confused with "steering stars" of a sidereal compass rose of a sidereal compass.

license: Creative Commons - Attribution Share-Alike - https://creativecommons.org/licenses/by-sa/4.0/

title: Zenith


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Horizon

The horizon is the apparent curve that separates the surface of a celestial body from its sky when viewed from the perspective of an observer on or near the surface of the relevant body. This curve divides all viewing directions based on whether it intersects the relevant body's surface or not.

The true horizon is a theoretical line, which can only be observed to any degree of accuracy when it lies along a relatively smooth surface such as that of Earth's oceans. At many locations, this line is obscured by terrain, and on Earth it can also be obscured by life forms such as trees and/or human constructs such as buildings. The resulting intersection of such obstructions with the sky is called the visible horizon. On Earth, when looking at a sea from a shore, the part of the sea closest to the horizon is called the offing.

The true horizon surrounds the observer and it is typically assumed to be a circle, drawn on the surface of a perfectly spherical model of the relevant celestial body, i.e., a small circle of the local osculating sphere. With respect to Earth, the center of the true horizon is below the observer and below sea level. Its radius or horizontal distance from the observer varies slightly from day to day due to atmospheric refraction, which is greatly affected by weather conditions. Also, the higher the observer's eyes are from sea level, the farther away the horizon is from the observer. For instance, in standard atmospheric conditions, for an observer with eye level above sea level by 1.70 m, the horizon is at a distance of about 4.7 km. When observed from very high standpoints, such as a space station, the horizon is much farther away and it encompasses a much larger area of Earth's surface. In this case, the horizon would no longer be a perfect circle, not even a plane curve such as an ellipse, especially when the observer is above the equator, as the Earth's surface can be better modeled as an oblate ellipsoid than as a sphere.

Etymology
The word horizon derives from the Greek ὁρίζων κύκλος (horízōn kýklos) 'separating circle', where ὁρίζων is from the verb ὁρίζω (horízō) 'to divide, to separate', which in turn derives from ὅρος (hóros) 'boundary, landmark'.

Appearance and usage
Historically, the distance to the visible horizon has long been vital to survival and successful navigation, especially at sea, because it determined an observer's maximum range of vision and thus of communication, with all the obvious consequences for safety and the transmission of information that this range implied. This importance lessened with the development of the radio and the telegraph, but even today, when flying an aircraft under visual flight rules, a technique called attitude flying is used to control the aircraft, where the pilot uses the visual relationship between the aircraft's nose and the horizon to control the aircraft. Pilots can also retain their spatial orientation by referring to the horizon.

In many contexts, especially perspective drawing, the curvature of the Earth is disregarded and the horizon is considered the theoretical line to which points on any horizontal plane converge (when projected onto the picture plane) as their distance from the observer increases. For observers near sea level, the difference between this geometrical horizon (which assumes a perfectly flat, infinite ground plane) and the true horizon (which assumes a spherical Earth surface) is imperceptible to the unaided eye. However, for someone on a 1000 m hill looking out across the sea, the true horizon will be about a degree below a horizontal line.

In astronomy, the horizon is the horizontal plane through the eyes of the observer. It is the fundamental plane of the horizontal coordinate system, the locus of points that have an altitude of zero degrees. While similar in ways to the geometrical horizon, in this context a horizon may be considered to be a plane in space, rather than a line on a picture plane.

Distance to the horizon
Ignoring the effect of atmospheric refraction, distance to the true horizon from an observer close to the Earth's surface is about

 * $$d \approx \sqrt{2\,h\,R} \,,$$

where h is height above sea level and R is the Earth radius.

The expression can be simplified as:

 * $$d \approx k \sqrt{h} \,,$$

where the constant equals k=$3.57 km/m^{½}$=$1.22 mi/ft^{½}$. In this equation, Earth's surface is assumed to be perfectly spherical, with R equal to about 6371 km.

Examples
Assuming no atmospheric refraction and a spherical Earth with radius R=6371 km:
 * For an observer standing on the ground with h = 1.70 m, the horizon is at a distance of 4.7 km.
 * For an observer standing on the ground with h = 2 m, the horizon is at a distance of 5 km.
 * For an observer standing on a hill or tower 30 m above sea level, the horizon is at a distance of 19.6 km.
 * For an observer standing on a hill or tower 100 m above sea level, the horizon is at a distance of 36 km.
 * For an observer standing on the roof of the Burj Khalifa, 828 m from ground, and about 834 m above sea level, the horizon is at a distance of 103 km.
 * For an observer atop Mount Everest (8848 m in altitude), the horizon is at a distance of 336 km.
 * For an observer aboard a commercial passenger plane flying at a typical altitude of 35000 ft, the horizon is at a distance of 369 km.
 * For a U-2 pilot, whilst flying at its service ceiling 21000 m, the horizon is at a distance of 517 km.

Other planets
On terrestrial planets and other solid celestial bodies with negligible atmospheric effects, the distance to the horizon for a "standard observer" varies as the square root of the planet's radius. Thus, the horizon on Mercury is 62% as far away from the observer as it is on Earth, on Mars the figure is 73%, on the Moon the figure is 52%, on Mimas the figure is 18%, and so on.

Derivation

If the Earth is assumed to be a featureless sphere (rather than an oblate spheroid) with no atmospheric refraction, then the distance to the horizon can easily be calculated.

The tangent-secant theorem states that
 * $$\mathrm{OC}^2 = \mathrm{OA} \times \mathrm{OB} \,.$$

Make the following substitutions:
 * d = OC = distance to the horizon
 * D = AB = diameter of the Earth
 * h = OB = height of the observer above sea level
 * D+h = OA = diameter of the Earth plus height of the observer above sea level,

with d, D, and h all measured in the same units. The formula now becomes
 * $$d^2 = h(D+h)\,\!$$

or
 * $$d = \sqrt{h(D+h)} =\sqrt{h(2R+h)}\,,$$

where R is the radius of the Earth.

The same equation can also be derived using the Pythagorean theorem. At the horizon, the line of sight is a tangent to the Earth and is also perpendicular to Earth's radius. This sets up a right triangle, with the sum of the radius and the height as the hypotenuse. With

 * d = distance to the horizon
 * h = height of the observer above sea level
 * R = radius of the Earth

referring to the second figure at the right leads to the following:

 * $$(R+h)^2 = R^2 + d^2 \,\!$$
 * $$R^2 + 2Rh + h^2 = R^2 + d^2 \,\!$$
 * $$d = \sqrt{h(2R + h)} \,.$$

The exact formula above can be expanded as:

 * $$d = \sqrt{2Rh + h^2} \,,$$

where R is the radius of the Earth (R and h must be in the same units). For example, if a satellite is at a height of 2000 km, the distance to the horizon is 5430 km; neglecting the second term in parentheses would give a distance of 5048 km, a 7% error.

Approximation

If the observer is close to the surface of the Earth, then it is valid to disregard h in the term (2R + h), and the formula becomes-

 * $$d = \sqrt{2Rh} \,.$$

Using kilometres for d and R, and metres for h, and taking the radius of the Earth as 6371 km, the distance to the horizon is

 * $$d \approx \sqrt{2\cdot6371\cdot{h/1000}} \approx 3.570\sqrt{h} \,$$.

Using imperial units, with d and R in statute miles (as commonly used on land), and h in feet, the distance to the horizon is

 * $$d \approx \sqrt{2\cdot3963\cdot{h/5280}} \approx \sqrt{1.5h} \approx 1.22 \sqrt{h} $$.

If d is in nautical miles, and h in feet, the constant factor is about 1.06, which is close enough to 1 that it is often ignored, giving:

 * $$d \approx \sqrt h $$

These formulas may be used when h is much smaller than the radius of the Earth (6371 km or 3959 mi), including all views from any mountaintops, airplanes, or high-altitude balloons. With the constants as given, both the metric and imperial formulas are precise to within 1% (see the next section for how to obtain greater precision). If h is significant with respect to R, as with most satellites, then the approximation is no longer valid, and the exact formula is required.

Arc distance
Another relationship involves the great-circle distance s along the arc over the curved surface of the Earth to the horizon; this is more directly comparable to the geographical distance on a map.

It can be formulated in terms of γ in radians,

 * $$s = R \gamma \,;$$

then

 * $$\cos \gamma = \cos\frac{s}{R}=\frac{R}{R+h}\,.$$

Solving for s gives

 * $$s=R\cos^{-1}\frac{R}{R+h} \,.$$

The distance s can also be expressed in terms of the line-of-sight distance d; from the second figure at the right,

 * $$\tan \gamma = \frac {d} {R} \,;$$

substituting for γ and rearranging gives

 * $$s=R\tan^{-1}\frac{d}{R} \,.$$

The distances d and s are nearly the same when the height of the object is negligible compared to the radius (that is, h ≪ R).

Zenith angle
When the observer is elevated, the horizon zenith angle can be greater than 90°. The maximum visible zenith angle occurs when the ray is tangent to Earth's surface; from triangle OCG in the figure at right,

 * $$\cos \gamma =\frac{R}{R+h}$$

where $$h$$ is the observer's height above the surface and $$\gamma$$ is the angular dip of the horizon. It is related to the horizon zenith angle $$z$$ by:

 * $$z = \gamma +90{}^\circ $$

For a non-negative height $$h$$, the angle $$z$$ is always ≥ 90°.

Objects above the horizon

To compute the greatest distance DBL at which an observer B can see the top of an object L above the horizon, simply add the distances to the horizon from each of the two points:
 * DBL = DB + DL

For example, for an observer B with a height of hB=1.70 m standing on the ground, the horizon is DB=4.65 km away. For a tower with a height of hL=100 m, the horizon distance is DL=35.7 km. Thus an observer on a beach can see the top of the tower as long as it is not more than DBL=40.35 km away. Conversely, if an observer on a boat (hB=1.7m) can just see the tops of trees on a nearby shore (hL=10m), the trees are probably about DBL=16 km away.

Referring to the figure at the right, and using the approximation above, the top of the lighthouse will be visible to a lookout in a crow's nest at the top of a mast of the boat if
 * $$D_\mathrm{BL} < 3.57\,(\sqrt{h_\mathrm{B}} + \sqrt{h_\mathrm{L}}) \,,$$

where DBL is in kilometres and hB and hL are in metres.

As another example, suppose an observer, whose eyes are two metres above the level ground, uses binoculars to look at a distant building which he knows to consist of thirty storeys, each 3.5 metres high. He counts the storeys he can see, and finds there are only ten. So twenty storeys or 70 metres of the building are hidden from him by the curvature of the Earth. From this, he can calculate his distance from the building:

 * $$D \approx 3.57(\sqrt{2}+\sqrt{70})$$

which comes to about 35 kilometres.

It is similarly possible to calculate how much of a distant object is visible above the horizon. Suppose an observer's eye is 10 metres above sea level, and he is watching a ship that is 20 km away. His horizon is:

 * $$ 3.57 \sqrt{10} $$

kilometres from him, which comes to about 11.3 kilometres away. The ship is a further 8.7 km away. The height of a point on the ship that is just visible to the observer is given by:

 * $$h\approx\left(\frac{8.7}{3.57}\right)^2$$

which comes to almost exactly six metres. The observer can therefore see that part of the ship that is more than six metres above the level of the water. The part of the ship that is below this height is hidden from him by the curvature of the Earth. In this situation, the ship is said to be hull-down.

Effect of atmospheric refraction
Due to atmospheric refraction the distance to the visible horizon is further than the distance based on a simple geometric calculation. If the ground (or water) surface is colder than the air above it, a cold, dense layer of air forms close to the surface, causing light to be refracted downward as it travels, and therefore, to some extent, to go around the curvature of the Earth. The reverse happens if the ground is hotter than the air above it, as often happens in deserts, producing mirages. As an approximate compensation for refraction, surveyors measuring distances longer than 100 meters subtract 14% from the calculated curvature error and ensure lines of sight are at least 1.5 metres from the ground, to reduce random errors created by refraction.

If the Earth were an airless world like the Moon, the above calculations would be accurate. However, Earth has an atmosphere of air, whose density and refractive index vary considerably depending on the temperature and pressure. This makes the air refract light to varying extents, affecting the appearance of the horizon. Usually, the density of the air just above the surface of the Earth is greater than its density at greater altitudes. This makes its refractive index greater near the surface than at higher altitudes, which causes light that is travelling roughly horizontally to be refracted downward. This makes the actual distance to the horizon greater than the distance calculated with geometrical formulas. With standard atmospheric conditions, the difference is about 8%. This changes the factor of 3.57, in the metric formulas used above, to about 3.86. For instance, if an observer is standing on seashore, with eyes 1.70 m above sea level, according to the simple geometrical formulas given above the horizon should be 4.7 km away. Actually, atmospheric refraction allows the observer to see 300 metres farther, moving the true horizon 5 km away from the observer.

This correction can be, and often is, applied as a fairly good approximation when atmospheric conditions are close to standard. When conditions are unusual, this approximation fails. Refraction is strongly affected by temperature gradients, which can vary considerably from day to day, especially over water. In extreme cases, usually in springtime, when warm air overlies cold water, refraction can allow light to follow the Earth's surface for hundreds of kilometres. Opposite conditions occur, for example, in deserts, where the surface is very hot, so hot, low-density air is below cooler air. This causes light to be refracted upward, causing mirage effects that make the concept of the horizon somewhat meaningless. Calculated values for the effects of refraction under unusual conditions are therefore only approximate. Nevertheless, attempts have been made to calculate them more accurately than the simple approximation described above.

Outside the visual wavelength range, refraction will be different. For radar (e.g. for wavelengths 300 to 3 mm i.e. frequencies between 1 and 100 GHz) the radius of the Earth may be multiplied by 4/3 to obtain an effective radius giving a factor of 4.12 in the metric formula i.e. the radar horizon will be 15% beyond the geometrical horizon or 7% beyond the visual. The 4/3 factor is not exact, as in the visual case the refraction depends on atmospheric conditions.

If the density profile of the atmosphere is known, the distance d to the horizon is given by
 * Integration method—Sweer

 * $$d={{R}_{\text{E}}}\left( \psi +\delta \right) \,,$$

where RE is the radius of the Earth, ψ is the dip of the horizon and δ is the refraction of the horizon. The dip is determined fairly simply from

 * $$\cos \psi = \frac{{R}_{\text{E}}{\mu}_{0}}{\left( {{R}_{\text{E}}}+h \right)\mu } \,,$$

where h is the observer's height above the Earth, μ is the index of refraction of air at the observer's height, and μ0 is the index of refraction of air at Earth's surface.

The refraction must be found by integration of

 * $$\delta =-\int_{0}^{h}{\tan \phi \frac{\text{d}\mu }{\mu }} \,,$$

where $$\phi\,\!$$ is the angle between the ray and a line through the center of the Earth. The angles ψ and $$\phi\,\!$$ are related by

 * $$\phi =90{}^\circ -\psi \,.$$

A much simpler approach, which produces essentially the same results as the first-order approximation described above, uses the geometrical model but uses a radius R′ = 7/6 RE. The distance to the horizon is then
 * Simple method—Young

 * $$d=\sqrt{2 R^\prime h} \,.$$

Taking the radius of the Earth as 6371 km, with d in km and h in m,

 * $$d \approx 3.86 \sqrt{h} \,;$$

with d in mi and h in ft,

 * $$d \approx 1.32 \sqrt{h} \,.$$

In the case of radar one typically has R′ = 4/3 RE resulting (with d in km and h in m) in

 * $$d \approx 4.12 \sqrt{h} \,;$$

Results from Young's method are quite close to those from Sweer's method, and are sufficiently accurate for many purposes.

Vanishing points

The horizon is a key feature of the picture plane in the science of graphical perspective. Assuming the picture plane stands vertical to ground, and P is the perpendicular projection of the eye point O on the picture plane, the horizon is defined as the horizontal line through P. The point P is the vanishing point of lines perpendicular to the picture. If S is another point on the horizon, then it is the vanishing point for all lines parallel to OS. But Brook Taylor (1719) indicated that the horizon plane determined by O and the horizon was like any other plane:
 * The term of Horizontal Line, for instance, is apt to confine the Notions of a Learner to the Plane of the Horizon, and to make him imagine, that that Plane enjoys some particular Privileges, which make the Figures in it more easy and more convenient to be described, by the means of that Horizontal Line, than the Figures in any other plane;…But in this Book I make no difference between the Plane of the Horizon, and any other Plane whatsoever...

The peculiar geometry of perspective where parallel lines converge in the distance, stimulated the development of projective geometry which posits a point at infinity where parallel lines meet. In her book Geometry of an Art (2007), Kirsti Andersen described the evolution of perspective drawing and science up to 1800, noting that vanishing points need not be on the horizon. In a chapter titled "Horizon", John Stillwell recounted how projective geometry has led to incidence geometry, the modern abstract study of line intersection. Stillwell also ventured into foundations of mathematics in a section titled "What are the Laws of Algebra ?" The "algebra of points", originally given by Karl von Staudt deriving the axioms of a field was deconstructed in the twentieth century, yielding a wide variety of mathematical possibilities. Stillwell states
 * This discovery from 100 years ago seems capable of turning mathematics upside down, though it has not yet been fully absorbed by the mathematical community. Not only does it defy the trend of turning geometry into algebra, it suggests that both geometry and algebra have a simpler foundation than previously thought.

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Prince regent

A prince regent or princess regent is a prince or princess who, due to their position in the line of succession, rules a monarchy as regent in the stead of a monarch, e.g., as a result of the sovereign's incapacity (minority or illness) or absence (e.g., by remoteness, such as exile or long voyage, or the absence of an incumbent).

While the term itself can have the generic meaning and refer to any prince or princess who fills the role of regent, historically it has mainly been used to describe a small number of individual princes and princesses who were regents of non-principalities.

Prince Regent of the United Kingdom
In the English language the title Prince Regent is most commonly associated with George IV, who held the style His Royal Highness The Prince Regent during the Regency era due to the incapacity (by dint of mental illness) of his father, George III (see Regent for other regents). Regent's Park, Regent Street and Regent's Canal (which he commissioned) in London, were all named in honour of him. The architect John Nash, under the patronage of HRH The Prince Regent, planned a palatial summer residence for the prince, 50 detached villas in a parkland setting and elegant terraces around the exterior of the park. This was all part of an ambitious plan, to develop The Regent's Park and lay out an elegant new street, Regent's Street, to link it to St James's Park and the prince's London residence, Carlton House. Regent Terrace in Edinburgh is also named after the Prince Regent, who visited the area in 1822.

This period is known as the British Regency, or just the Regency.

The title was conferred by the Regency Act on 5 February 1811. Subject to certain limitations for a period, the prince regent was able to exercise the full powers of the King. The precedent of the Regency Crisis of 1788 (from which George III recovered before it was necessary to appoint a regent) was followed. The Prince of Wales continued as regent until his father's death in 1820, when he became George IV.

Prince regent in Germany
In Germany, the title Prinzregent (literally prince regent) is most commonly associated with Prince Luitpold of Bavaria, who served as regent for two of his nephews, King Ludwig II of Bavaria, who was declared mentally incompetent in 1886, and King Otto of Bavaria (who had been declared insane in 1875) from 1886 until 1912.

The years of Luitpold's regency were marked by tremendous artistic and cultural activity in Bavaria, where they are known after the regencies as the Prinzregentenjahre or the Prinzregentenzeit. Numerous streets in Bavarian cities and towns are called Prinzregentenstraße. Many institutions are named in Luitpold's honour, e.g., the Prinzregententheater in Munich. Prinzregententorte is a multi-layered cake with chocolate butter cream named in Luitpold's honour.

At Luitpold's death in 1912, his son Prince Ludwig succeeded as prince regent. Ludwig held the title for less than a year, since the Bavarian Legislature decided to set aside his cousin King Otto I of Bavaria and recognize him as king.

Prince regent in Belgium

 * The first head of state of Belgium after it seceded from the Dutch monarchy in 1831 was a regent (but not a prince in his own right), baron Erasme Louis Surlet de Chokier, before the new nation, which had chosen to become a parliamentary monarchy, had its first king sworn into the constitution.
 * Prince Charles of Belgium served as prince regent of Belgium from 1944 to 1950 during the German captivity and exile to Switzerland of his elder brother, King Leopold III of Belgium.

Prince regent in Bulgaria
Kiril, Prince of Preslav was appointed head of a regency council by the Bulgarian parliament following the death of his brother, Tsar Boris III on 28 August 1943, to act as Head of State until the late Tsar's son and successor, Tsar Simeon II, reached the age of 18 years. On 5 September 1944 the Soviet Union declared war on the Kingdom of Bulgaria and on 8 September Soviet armies crossed the Romanian border and occupied the country. On 1 February 1945 the prince regent Kyril, and the two other former regents – Professor Bogdan Filov and General Nikola Mikhov – as well as a range of former cabinet ministers, royal advisors and 67 MPs, were executed.

Prince Lieutenant in Luxembourg
The heir-apparent or heir-presumptive to the grand duke of Luxembourg may be titled prince-lieutenant ('prince deputy') during a period in which the incumbent remains formally on the grand ducal throne, but (progressively, most) functions of the crown are performed by the 'monarch apprentice', as prince Jean did 4 May 1961 – 12 November 1964 in the last years of his mother Charlotte's reign until she abdicated and he succeeded to the grand ducal throne (she lived until 1985), and Jean's own son prince Henri 3 March 1998 – 7 October 2000 until his father abdicated and he succeeded (Jean lived until 2019).

Queen regent
If a king is unable to perform his duties then his consort or the queen mother may act for him as a Queen Regent. Queen mothers have acted in this way in the Kingdom of Eswatini.

Other notable princes and princesses regent

 * More prince-regents (often without such specific title) are to be found in List of regents.

 * Duke of Zhou Dan served as prince regent for his nephew King Cheng of Zhou until the latter came of age.
 * Philippe II, Duke of Orléans was Regent of France from the death of Louis XIV in 1715 until Louis XV came of age in 1723; this is also used as a period label for many aspects of French history, as "Régence" in French.
 * Te Wherowhero Tawhiao, younger brother of Māori King Mahuta Tāwhiao, served as Whirinaki-a-te-Kiingi (Prince Regent) for his brother from 1903 to 1910. This was because Mahuta Tāwhiao spent this period of his reign as a member of the Legislative Council.
 * Crown Prince Frederick of Denmark served as regent from 1784 to 1808 for his father, King Christian VII of Denmark, who was insane.
 * Prince William of Prussia served as regent from 1858 to 1861 for his older brother King Frederick William IV of Prussia, who had become mentally unfit to rule.
 * Prince Dorgon of the early Qing dynasty served as regent for his nephew, the Shunzhi Emperor, from 1643 to 1650, because the latter was only six at the time of his ascension. Dorgon was instrumental in moving Manchu forces into Beijing in 1644, proclaiming the Qing dynasty to be the legitimate successor to the Ming dynasty. In Qing dynasty historical records, Dorgon was the first to be referred to as Shezhengwang (摄政王; "Prince regent").
 * Rameses the Great
 * Prince Chun of the late Qing dynasty served as regent from 1908 to 1911 for his son Puyi, the Xuantong Emperor. Apart from Dorgon, Zaifeng was the only person in Chinese history who was specifically referred to as prince regent.
 * Crown Prince Hirohito served as regent from 1921 to 1926 for his ailing father, Emperor Taishō.
 * Prince Paul of Yugoslavia from 1934 to 1941, known in Serbian as Његово Краљевско Височанство, Кнез Намесник (English: His Royal Highness The Prince Regent) served as regent for his first cousin once removed, King Peter II who was a minor.
 * John, Prince of Brazil (1767–1826) served as regent of Portugal for his mother Queen Maria I, who had become mentally unfit to rule, from 1799 to 1816. His regency was associated with the transfer of the Portuguese court to Brazil.
 * Princess Erelu Kuti of Lagos, a Yoruba chieftess of the eighteenth century who served as queen mother of a line of tribal kings. The successors to the noble title that now shares her name have all subsequently served as regent of Lagos following the death of a reigning oba. Princesses are traditionally called upon to serve as regents in this fashion in most of the other kingdoms of Yorubaland as well. In Akure, for example, the eldest daughter of a recently deceased king ruled in his stead until a substantive successor to the royal title was chosen by the college of noble kingmakers, a period that lasted for an unusually long six years due to a succession crisis in the state.
 * Chief Tshekedi Khama of the Bamangwato, a Tswana prince who served as regent during the reign of his famous nephew, Sir Seretse Khama. He opposed Sir Seretse's marriage to Lady Khama on the grounds that it would have an adverse effect on the chieftaincy, and attempted to claim the tribal throne in his stead thereafter because of it. He was ultimately unsuccessful.
 * Anne of France served as regent of France for her younger brother between 1483-1491.
 * Sophia Alekseyevna from 1682-1689 as regent of Russia on behalf of her younger brothers.
 * Aurora Pinedo served as Princess Regent of the Afro-Bolivian monarchy from 1954-1992.

title: Prince regent


text:
Caroline of Brunswick

Caroline of Brunswick-Wolfenbüttel (Caroline Amelia Elizabeth; 17 May 1768 – 7 August 1821) was Queen of the United Kingdom of Great Britain and Ireland and Queen of Hanover from 29 January 1820 until her death in 1821 as the estranged wife of King George IV. She was Princess of Wales from 1795 to 1820.

The daughter of Charles William Ferdinand, Duke of Brunswick, and Princess Augusta of Great Britain, Caroline was engaged in 1794 to her cousin George, Prince of Wales, whom she had never met. He was already illegally married to Maria Fitzherbert. George and Caroline married the following year but separated shortly after the birth of their only child, Princess Charlotte, in 1796. By 1806, rumours that Caroline had taken multiple lovers and had an illegitimate child led to an investigation into her private life. The dignitaries who led the investigation concluded that there was "no foundation" to the rumours, but Caroline's access to her daughter was still restricted. In 1814, Caroline moved to Italy, where she employed Bartolomeo Pergami as a servant. Pergami soon became Caroline's closest companion, and it was widely assumed that they were lovers. In 1817, Caroline was devastated when Charlotte died in childbirth. She heard the news from a passing courier as George had refused to write and tell her. He was determined to divorce Caroline and set up a second investigation to collect evidence of her adultery.

In January 1820, George became King of the United Kingdom and Hanover, and Caroline became nominal queen consort. George insisted on a divorce from Caroline, which she refused. A legal divorce was possible but difficult to obtain. Caroline returned to Britain to assert her position as queen. She was wildly popular with the British people, who sympathised with her and despised the new king for his immoral behaviour. On the basis of the loose evidence collected against her, George attempted to divorce Caroline by introducing the Pains and Penalties Bill 1820 to Parliament, but he and the bill were so unpopular and Caroline so popular with the masses that it was withdrawn by the Liverpool ministry. The King barred Caroline from his coronation in July 1821. She fell ill in London and died three weeks later. Her funeral procession passed through London on its way to her native Brunswick, where she was buried.

Early life
Caroline was born a princess of Braunschweig, known in English as Brunswick, with the courtesy title of Duchess of Brunswick-Wolfenbüttel, on 17 May 1768 at Braunschweig in Germany. She was the daughter of Charles William Ferdinand, Duke of Brunswick-Wolfenbüttel, and his wife Princess Augusta of Great Britain, eldest sister of King George III.

Caroline was brought up in a difficult family situation. Her mother resented her father's open adultery with Baroness Luise von Hertefeld, whom he had installed as his official mistress in 1777, and Caroline was later to confide to Lady Charlotte Campbell that she was often tired of being a "shuttlecock" between her parents, as whenever she was civil to one of them, she was scolded by the other.

She was educated by governesses, but the only subject in which she was given a higher education was music. From 1783 until 1791, Countess Eleonore von Münster was her governess, and won her affection, but never managed to teach her to spell correctly, as Caroline preferred to dictate to a secretary. Caroline could understand English and French, but her father admitted that she was lacking in education.

According to Caroline's mother, who was British, all German princesses learned English in the hope that they would be chosen to marry George, Prince of Wales, George III's eldest son and heir apparent and Caroline's first cousin.

John Stanley, later 1st Baron Stanley of Alderley, saw her in 1781, and noted that she was an attractive girl with curly, fair hair. In 1784, she was described as a beauty, and two years later, Honoré Gabriel Riqueti, comte de Mirabeau described her as "most amiable, lively, playful, witty and handsome".

Caroline was brought up with extremely limited contact with the opposite sex even by the standards of her own time. She was reportedly constantly supervised by her governess and elder ladies, restricted to her room when the family was entertaining and ordered to keep away from the windows. She was normally refused permission to attend balls and court functions, and when allowed, she was forbidden to dance. Abbé Baron commented during the winter of 1789–90: "She is supervised with the greatest severity, as they claim she is already aware of what she is missing. I doubt if the torches of Hymen will illuminate for her. Although always attired with style and elegance, she is never allowed to dance", and that as soon as the first dance began, she was forced to sit down at the whist table with three old ladies.

A rare occasion was the wedding of her elder brother Charles, when she was finally allowed to dance, though only with the groom and his new brother-in-law, the Prince of Orange – she was, however, still forbidden to dine alone with her brother.

Her secluded isolation tormented her, which was demonstrated when she was later again banned from attending a ball. She simulated an illness so severe that her parents left the ball to see her. When they arrived, she claimed to be in labour and forced them to send for a midwife. When the midwife arrived, she stopped her simulation and asked her mother: "Now, Madam, will you keep me another time from a ball?"

Her mother early favoured a match between one of her children and a member of her English family, and when her nephew Prince Frederick, Duke of York and Albany visited Brunswick in June 1781, she lamented the fact that Caroline, because of her age, could not be present very often. Caroline was given a number of proposals from 1782 onward. Marriage with the Prince of Orange, Prince George of Hesse-Darmstadt, Charles, Duke of Mecklenburg-Strelitz, and the second son of the Margrave of Baden were all suggested, while her mother and father supported an English and a Prussian Prince, respectively, but none came to fruition. Caroline was later to state that her father had forbidden her to marry a man she had fallen in love with because of his low status. The identity of this man is not clear, but contemporaries point out an officer who was referred to at the time as the "Handsome Irishman" who lived in Brunswick, and with whom Caroline was said to have been in love. There was also a rumour that Caroline had given birth at the age of fifteen.

Though she was not allowed to socialise with men, she was allowed to ride, and during riding, she visited the cottages of the peasantry. She had done this already as a child, during which she had met children to play with, and as an adult, one of these visits allegedly led to a pregnancy. There is no confirmation of this rumour, but it was well known during her life, and referred to as a reason why she married at an older age than was usual, despite being regarded as good-looking and having been given so many proposals.

Engagement
In 1794, Caroline and the Prince of Wales were engaged. They had never met—George had agreed to marry her because he was heavily in debt, and if he contracted a marriage with an eligible princess, Parliament would increase his allowance. Caroline seemed eminently suitable: she was a Protestant of royal birth, and the marriage would ally Brunswick and Britain. Although Brunswick was only a small country, Britain was at war with revolutionary France and so was eager to obtain allies on the European mainland. Brunswick was ruled by Caroline's father, the esteemed soldier Charles William Ferdinand, Duke of Brunswick, who himself had married Princess Augusta, the sister of George III. On 20 November 1794, Lord Malmesbury arrived at Brunswick to escort Caroline to her new life in Britain. In his diary, Malmesbury recorded his reservations about Caroline's suitability as a bride for the prince: she lacked judgement, decorum and tact, spoke her mind too readily, acted indiscreetly, and often neglected to wash, or change her dirty clothes. He went on to say that she had "some natural but no acquired morality, and no strong innate notions of its value and necessity". However, Malmesbury was impressed by her bravery; on the journey to England, the party heard cannon fire, as they were not far from the French lines. While Caroline's mother, who was accompanying them to the coast as chaperone, was concerned for their safety, Caroline was unfazed.

On 28 March 1795, Caroline and Malmesbury left Cuxhaven on the Jupiter. Delayed by poor weather, they landed a week later, on Easter Sunday, 5 April, at Greenwich. There, she met Frances Villiers, Countess of Jersey, George's mistress, who had been appointed Caroline's Lady of the Bedchamber. Smith concludes that:
 * She was chosen as the intended bride of George, Prince of Wales partly because her mother was a favourite sister of George III, partly through the favourable reports of her given by his brothers the Dukes of York and Clarence when they visited Germany, and partly for lack of a suitable alternative German Protestant princess.

On meeting his future wife for the first time, George called for a glass of brandy. He was evidently disappointed. Similarly, Caroline told Malmesbury, "[the Prince is] very fat and he's nothing like as handsome as his portrait." At dinner that evening, the Prince was appalled by Caroline's garrulous nature and her jibes at the expense of Lady Jersey. She was upset and disappointed by George's obvious partiality for Lady Jersey over her.

Troubled marriage
Caroline and George were married on 8 April 1795 at the Chapel Royal, St. James's Palace, in London. At the ceremony, George was drunk. He regarded Caroline as unattractive and unhygienic, and told Malmesbury that he suspected that she was not a virgin when they married. He had already secretly married Maria Fitzherbert, but his marriage to Fitzherbert violated the Royal Marriages Act 1772 and so was not legally valid. In a letter to a friend, the prince claimed that the couple only had sexual intercourse three times: twice the first night of the marriage, and once the second night. He wrote, "it required no small [effort] to conquer my aversion and overcome the disgust of her person." Caroline claimed George was so drunk that he "passed the greatest part of his bridal night under the grate, where he fell, and where I left him".

Nine months after the wedding, Caroline gave birth to Princess Charlotte, George's only legitimate child, at Carlton House on 7 January 1796. Charlotte was second in the line of succession to the British throne after her father. Just three days after Charlotte's birth, George made out a new will. He left all his property to "Maria Fitzherbert, my wife", while to Caroline he left one shilling.

Gossip about Caroline and George's troubled marriage was already circulating. The newspapers claimed that Lady Jersey opened, read and distributed the contents of Caroline's private letters. She despised Lady Jersey and could not visit or travel anywhere without George's permission. The press vilified George for his extravagance and luxury at a time of war and portrayed Caroline as a wronged wife. She was cheered in public and gained plaudits for her "winning familiarity" and easy, open nature. George was dismayed at her popularity and his own unpopularity, and felt trapped in a loveless marriage with a woman he loathed. He wanted a separation. In April 1796, George wrote to Caroline, "We have unfortunately been oblig'd to acknowledge to each other that we cannot find happiness in our union. ... Let me therefore beg you to make the best of a situation unfortunate for us both." In June, Lady Jersey resigned as Caroline's Lady of the Bedchamber. George and Caroline were already living separately, and in August 1797 Caroline moved to a private residence: The Vicarage or Old Rectory in Charlton, London. Later, she moved to Montagu House in Blackheath. No longer constrained by her husband, or, according to rumour, her marital vows, she entertained whomever she pleased. She flirted with Admiral Sir Sidney Smith and Captain Thomas Manby, and may have had a brief relationship with the politician George Canning.

Delicate Investigation
Charlotte was placed in the care of a governess, in a mansion near Montagu House in the summers, and Caroline visited her often. It seems that a single daughter was not sufficient to sate Caroline's maternal instincts, and she adopted eight or nine poor children who were fostered out to people in the district. In 1802, she adopted a three-month-old boy, William Austin, and took him into her home. By 1805, Caroline had fallen out with her near neighbours, Sir John and Lady Douglas, who claimed that Caroline had sent them obscene and harassing letters. Lady Douglas accused Caroline of infidelity, and alleged that William Austin was Caroline's illegitimate son.

In 1806, a secret commission was set up, known as the "Delicate Investigation", to examine Lady Douglas's claims. The commission comprised four of the most eminent men in the country: Prime Minister Lord Grenville, the Lord Chancellor Lord Erskine, the Lord Chief Justice of England and Wales Lord Ellenborough and the Home Secretary Lord Spencer. Lady Douglas testified that Caroline herself had admitted to her in 1802 that she was pregnant, and that Austin was her son. She further alleged that Caroline had been rude about the royal family, touched her in an inappropriately sexual way, and had admitted that any woman friendly with a man was sure to become his lover. In addition to Smith, Manby and Canning, artist Thomas Lawrence and Henry Hood, son of Lord Hood, were also mentioned as potential paramours. Caroline's servants could or would not confirm that these gentlemen were her lovers, nor that she had been pregnant, and said that the child had been brought to Caroline's house by his true mother, Sophia Austin. Sophia was summoned before the commissioners, and testified that the child was hers.

The commissioners decided that there was "no foundation" for the allegations, but despite being a supposedly secret investigation, it proved impossible to prevent gossip from spreading, and news of the investigation leaked to the press. Caroline's conduct with her gentlemen friends was considered improper, but there was no direct proof that she had been guilty of anything more than flirtation. Perhaps Caroline had told Lady Douglas that she was pregnant out of frustrated maternal desire, or as part of a foolish prank that, unfortunately for her, backfired. Later in the year, Caroline received further bad news as Brunswick was overrun by the French, and her father was killed in the battle of Jena-Auerstadt. Her mother and brother, Frederick William, Duke of Brunswick-Wolfenbüttel, fled to England. Caroline had wanted to return to Brunswick and leave Britain behind her, but with much of Europe controlled by the French she had no safe haven to run to.

During the Delicate Investigation, Caroline was not permitted to see her daughter, and afterwards her visits were essentially restricted to once a week and only in the presence of Caroline's own mother, the Dowager Duchess of Brunswick. Meetings took place at either Blackheath or an apartment in Kensington Palace designated for Caroline's use.

Social isolation
By the end of 1811, King George III had become permanently insane, and the Prince of Wales was appointed regent. He restricted Caroline's access to Princess Charlotte further, and Caroline became more socially isolated as members of high society chose to patronise George's extravagant parties rather than hers. She moved her London residence to Connaught House in Bayswater. Caroline needed a powerful ally to help her oppose George's increasing ability to prevent her from seeing her daughter. In league with Henry Brougham, an ambitious Whig politician who favoured reform, she began a propaganda campaign against George. George countered by leaking Lady Douglas's testimony from the "Delicate Investigation", which Brougham repudiated by leaking the testimonies of the servants and Mrs Austin. Charlotte favoured her mother's point of view, as did most of the public. Jane Austen wrote of Caroline: "Poor woman, I shall support her as long as I can, because she is a Woman and because I hate her Husband."

In 1814, after Napoleon's defeat, nobility from throughout Europe attended celebrations in London, but Caroline was excluded. George's relationship with his daughter was also deteriorating, as Charlotte sought greater freedom from her father's strictures. On 12 July, he informed Charlotte that she would henceforth be confined at Cranbourne Lodge, Windsor, that her household would be replaced, and that she could have no visitors except her grandmother Queen Charlotte once a week. Horrified, Princess Charlotte ran away to her mother's house in Bayswater. After an anxious night, Charlotte was eventually persuaded to return to her father by Brougham, since legally she could be placed in her father's care and there was a danger of public disorder against George, which might prejudice Charlotte's position if she continued to disobey him.

Caroline, unhappy at her situation and treatment in Britain, negotiated a deal with the Foreign Secretary, Lord Castlereagh. She agreed to leave the country in exchange for an annual allowance of £35,000. Both Brougham and Charlotte were dismayed by Caroline's decision, as they both realised that Caroline's absence would strengthen George's power and weaken theirs. On 8 August 1814, Caroline left Britain.

Exile
After a two-week visit to Brunswick, Caroline headed for Italy through Switzerland. Along the way, possibly in Milan, she hired Bartolomeo Pergami as a servant. Pergami soon rose to the head of Caroline's household, and managed to get his sister, Angelica, Countess of Oldi, appointed as Caroline's lady-in-waiting. In mid-1815, Caroline bought a house, Villa d'Este, on the shores of Lake Como, even though her finances were stretched.

From early 1816, she and Pergami went on a cruise around the Mediterranean, visiting Napoleon's former palace on Elba, and Sicily, where Pergami obtained the Order of Malta and a barony. By this time, Caroline and Pergami were openly eating together, and it was widely rumoured that they were lovers. They visited Tunis, Malta, Milos, Athens, Corinth, Constantinople, and Nazareth. Caroline entered Jerusalem riding on a donkey in a convoy of camels. Pergami was made a Knight of the Order of Saint Lazarus. Caroline instituted the Order of Saint Caroline, nominating Pergami its Grand Master. In August, they returned to Italy, stopping at Rome to visit Pope Pius VII.

By this time, gossip about Caroline was everywhere. Lord Byron wrote to his publisher that Caroline and Pergami were lovers, and Sir Francis Ronalds described and made a sketch of their sleeping arrangements in Karlsruhe. Baron Friedrich Ompteda, a Hanoverian spy, bribed one of Caroline's servants so that he could search her bedroom for proof of adultery. He found none. By August 1817, Caroline's debts were growing, so she sold Villa d'Este and moved to the smaller Villa Caprile near Pesaro. Pergami's mother, brother and daughter, but not his wife, joined Caroline's household.

The previous year, Caroline's daughter, Princess Charlotte, had married Prince Leopold of Saxe-Coburg-Saalfeld, and the future of the British monarchy looked bright. Then tragedy struck: in November 1817, Charlotte died after giving birth to her only child, a stillborn son. For the most part, Charlotte had been immensely popular with the public, and her death was a blow to the country. George refused to write to Caroline to inform her, leaving it for their son-in-law Leopold to do, but Leopold was deep in grief and delayed writing. George did, however, write to the Pope of the tragedy, and by chance the courier carrying the letter passed by Pesaro, and so it was that Caroline heard the devastating news. Caroline had lost her daughter, but she had also lost any chance of regaining position through the succession of her daughter to the throne.

George was determined to press ahead with a divorce and set up a commission chaired by the Vice-Chancellor John Leach to gather evidence of Caroline's adultery. Leach sent three commissioners to Milan to interrogate Caroline's former servants, including Theodore Majocchi and Caroline's maid, Louise Demont. In London, Brougham was still acting as Caroline's agent. Concerned that the "Milan commission" might threaten Caroline, he sent his brother James to Caroline's villa in the hope of establishing whether George had any grounds for divorce. James wrote back to his brother of Caroline and Pergami, "they are to all appearances man and wife, never was anything so obvious." The Milan commission was assembling more and more evidence, and by 1819 Caroline was worried. She informed James Brougham that she would agree to a divorce in exchange for money. However, at this time in England divorce by mutual consent was illegal; it was only possible to divorce if one of the partners admitted or was found guilty of adultery. Caroline said it was "impossible" for her to admit that, so the Broughams advised that only formal separation was possible. Both keen to avoid publicity, the Broughams and the Government discussed a deal where Caroline would be called by a lesser title, such as "Duchess of Cornwall" rather than "Princess of Wales". As the negotiations continued at the end of 1819, Caroline travelled to France, which gave rise to speculation that she was on her way back to England. In January 1820, however, she made plans to return to Italy, but then on 29 January 1820 George III died. Caroline's husband became king and, at least nominally, she was queen of the United Kingdom.

Queen consort
Instead of being treated like a queen, Caroline found that her estranged husband's accession paradoxically made her position worse. On visiting Rome, Pope Pius VII refused her an audience, and the Pope's minister Cardinal Consalvi insisted that she be greeted only as a duchess of Brunswick, and not as a queen. In an attempt to assert her rights, she made plans to return to Britain. The King demanded that his ministers get rid of her. He successfully persuaded them to remove her name from the liturgy of the Church of England, but they would not agree to a divorce because they feared the effect of a public trial. The government was weak and unpopular, and a trial detailing salacious details of both Caroline's and George's separate love lives was certain to destabilise it further. Rather than run the risk, the government entered into negotiations with Caroline, and offered her an increased annuity of £50,000 if she stayed abroad.

By the beginning of June 1820, Caroline had travelled north from Italy, and was at St Omer near Calais. Acting on the advice of Alderman Matthew Wood and her lady-in-waiting Lady Anne Hamilton, she rejected the government's offer. She bid farewell to Pergami, and embarked for England. When she arrived on 5 June, riots broke out in support of her. Caroline was a figurehead for the growing Radical movement that demanded political reform and opposed the unpopular king. Nevertheless, the King still adamantly desired a divorce and, the following day, he submitted the evidence gathered by the Milan commission to Parliament in two green bags. On 15 June, the guards in the King's Mews mutinied. The mutiny was contained, but the government was fearful of further unrest. Examination of the bags of evidence was delayed as Parliament debated the form of the investigation, but eventually, on 27 June, they were opened and examined in secret by 15 peers. The peers considered the contents scandalous, and a week later, after their report to the House, the government introduced a bill in Parliament, the Pains and Penalties Bill 1820, to strip Caroline of the title of queen and dissolve her marriage. It was claimed that Caroline had committed adultery with a low-born man: Bartolomeo Pergami. Various witnesses, such as Theodore Majocchi, were called during the reading of the bill, which was effectively a public trial of the Queen. The trial caused a sensation, as details of Caroline's familiarity with Pergami were revealed. Witnesses said the couple had slept in the same room, kissed, and been seen together in a state of undress. The bill passed the House of Lords, but was not submitted to the House of Commons as there was little prospect that the Commons would pass it. To her friends, Caroline joked that she had indeed committed adultery once—with the husband of Mrs. Fitzherbert, the King.

Even during the trial, the Queen remained immensely popular, as witnessed by over 800 petitions and nearly a million signatures that favoured her cause. As a figurehead of the opposition movement demanding reform, many revolutionary pronouncements were made in Caroline's name. "All classes will ever find in me a sincere friend to their liberties, and a zealous advocate of their rights."

- Queen Caroline, September 1820, quoted in Robins, p. 240

"A government cannot stop the march of intellect any more than they can arrest the motion of the tides or the course of the planets."

- Queen Caroline quoted in The Times, 7 October 1820

But with the end of the trial, her alliance with the radicals came to an end. The government again extended the offer of £50,000 a year, this time without preconditions, and Caroline accepted.

On 5 May 1821, Napoleon died on St Helena. Sir Edmund Nagle informed George, "Sir, your bitterest enemy is dead". He replied, "Is she, by God!".

Despite the King's best attempts, Caroline retained a strong popularity among the masses, and pressed ahead with plans to attend the coronation service on 19 July 1821 as queen. Lord Liverpool told Caroline that she should not go to the service, but she turned up anyway. George had Caroline turned away from the coronation at the doors of Westminster Abbey. Refused entry at both the doors to the East Cloister and the doors to the West Cloister, Caroline attempted to enter via Westminster Hall, where many guests were gathered before the service began. A witness described how the Queen stood at the door fuming as bayonets were held under her chin until the deputy lord chamberlain had the doors slammed in her face. Caroline then proceeded back to an entrance near Poets' Corner, where she was met by Sir Robert Inglis, who held the office of "Gold Staff". Inglis persuaded her to return to her carriage, and she left. Caroline lost support through her exhibition at the coronation; the crowds jeered her as she rode away, and even Brougham recorded his distaste at her undignified behaviour.

Death
The night following Caroline's failed attempt to attend her husband's coronation, she fell ill and took a large dose of milk of magnesia and some drops of laudanum. Over the next three weeks, she suffered more and more pain as her condition deteriorated. She realised she was nearing death and put her affairs in order. Her papers, letters, memoirs, and notebooks were burned. She wrote a new will, and settled her funeral arrangements: she was to be buried in her native Brunswick in a tomb bearing the inscription "Here lies Caroline, the Injured Queen of England [sic]". She died at Brandenburg House in Hammersmith at 10:25 p.m. on 7 August 1821 at the age of 53. Her physicians thought she had an intestinal obstruction, but she may have had cancer, and there were rumours at the time that she had been poisoned.

Afraid that a procession of the funeral bier through London could spark public unrest, Lord Liverpool decided the Queen's cortège would avoid the city, passing to the north on the way to Harwich and Brunswick. The crowd accompanying the procession was incensed and blocked the intended route with barricades to force a new route through Westminster and London. The scene soon descended into chaos; the soldiers forming the honour guard opened fire and rode through the crowd with drawn sabres. People in the crowd threw cobblestones and bricks at the soldiers, and two members of the public—Richard Honey, a carpenter, and George Francis, a bricklayer —were killed. Eventually, Chief Metropolitan Magistrate Sir Robert Baker ordered that the official route be abandoned, and the cortège passed through the city. As a result, Baker was dismissed from office.

The final route (in heavy rain) took the following course: Hammersmith, Kensington (blocked), Kensington Gore (blocked), Hyde Park, Park Lane (blocked), return to Hyde Park where soldiers forced the gates open, Cumberland Gate (blocked), Edgware Road, Tottenham Court Road, Drury Lane, the Strand, and from there through the City of London, then by way of Romford and Chelmsford to Colchester. The coffin was kept overnight at St Peter's Church, Colchester, where Caroline's executors tried unsuccessfully to replace the official inscription plate with one including the phrase "Injured Queen of England". The next day, the coffin was taken to the seaport of Harwich, and placed on a ship bound for Germany. Brunswick Cathedral is Caroline's final resting place.

Legacy
The American historian Thomas W. Laqueur emphasises that the sordid royal squabble captivated all Britons:
 * During much of 1820 the "queen's business" captivated the nation. "It was the only question I have ever known," wrote the radical critic William Hazlitt, "that excited a thorough popular feeling. It struck its roots into the heart of the nation; it took possession of every house or cottage in the kingdom."

In 1822, the publishers of the newspaper John Bull were found guilty over a series of libels published in the Queen's lifetime, including one that alluded to her as "a shameless woman". It was alleged the libels had embittered the Queen and shortened her life.

The story of Caroline's marriage to George and her battle to be recognised as queen served as the basis for the 1996 BBC docudrama A Royal Scandal with Susan Lynch as Caroline and Richard E. Grant as George IV. The 2008 radio play The People's Princess, with Alex Jennings as George IV and Rebecca Saire as Caroline, drew parallels with the marriage and divorce of Charles, Prince of Wales (later Charles III), and Diana, Princess of Wales. Caroline was also the subject of Richard Condon's 1977 novel The Abandoned Woman.

Arms
The royal coat of arms of the United Kingdom are impaled with her father's arms as Duke of Brunswick. The arms were Quarterly of twelve, 1st, Or, a semé of hearts Gules, a lion rampant Azure (Lüneburg); 2nd, Gules, two lions passant guardant Or (Brunswick); 3rd, Azure, a lion rampant Argent crowned Or (Eberstein); 4th, Gules a lion rampant Or, within a border componé Argent and Azure (Homburg); 5th, Or, a lion rampant Gules crowned Azure (Diepholz); 6th, Gules, a lion rampant Or (Lauterberg); 7th, Per fess, in chief Or, two bears' paws erect Sable (Hoya), in the base a gyronny, Argent and Azure (Old Bruckhausen); 8th, Azure, an eagle displayed Argent, langued, beaked and membered Gules (Diepholz eagle); 9th, Chequy Argent and Gules (Hohnstein); 10th, Argent, a stag's attire in bend Gules (Regenstein); 11th, Argent, a stag trippant Sable (Klettenberg); 12th, Argent, a stag's attire in bend sinister Sable (Blankenburg).

As Princess of Wales she used the arms of her husband (the royal arms with a label of three points Argent) impaled with those of her father, the whole surmounted by a coronet of the heir apparent.

title: Caroline of Brunswick