An alternative metric to assess global warming

Posted on April 28, 2014

by Roger A. Pielke Sr., Richard T. McNider, and John Christy

Original link:  http://judithcurry.com/2014/04/28/an-alternative-metric-to-assess-global-warming/

The thing we’ve all forgotten is the heat storage of the ocean – it’s a thousand times greater than the atmosphere and the surface.  – James Lovelock

This aspect of the climate system is why it has been proposed to use the changes in the ocean heat content to diagnose the global radiative imbalance, as summarized in Pielke (2003, 2008). In this weblog post, we take advantage of this natural space and time integrator of global warming and cooling.

We present this alternate tool to assess the magnitude of global warming based on assessing the magnitudes of the annual global average radiative imbalance, and the annual global average radiative forcing and feedbacks. Among our findings is the difficulty of reconciling the three terms.

Introduction

As summarized in NRC (2005) “the concept of radiative forcing is based on the hypothesis that the change in global annual mean surface temperature is proportional to the imposed global annual mean forcing, independent of the nature of the applied forcing. The fundamental assumption underlying the radiative forcing concept is that the surface and the troposphere are strongly coupled by convective heat transfer processes; that is, the earth-troposphere system is in a state of radiative-convective equilibrium.”

According to the radiative-convective equilibrium concept, the equation for determining global average surface temperature is ΔQ = ΔF – ΔT/ λ   (1), where ΔQ is the radiative imbalance, ΔF is the radiative forcing, and ΔT is the change in temperature over the same time period. The quantity λ is referred to as the radiative feedback parameter which has been used to relate temperature response to a change in radiative forcing (Gregory et al. 2002, NRC 2005). As such, it has been used as the primary global metric for assessing global warming due to anthropogenic changes in radiative forcing. The quantity ΔT is typically defined as the near-surface global average surface air temperature.

While perhaps conceptually useful, the actual implementation of the equation can be difficult. First, the measurement of ΔT has been shown to have issues with its accurate quantification. In the equation, ΔT is meant to represent both the radiative temperature of the Earth system and the accumulation of heat through the temperature change that would occur as a radiative imbalance occurs. However, changes in temperature at the surface can occur due to a vertical redistribution of heat not necessarily due to an accumulation of heat (McNider et al. 2012), site location issues (Pielke et al. 2007; Fall et al. 2011), as well as due to regional changes in surface temperatures from land-use change, aerosol deposition, and atmospheric aerosols (e.g., Christy et al. 2006, 2009; Strack et al. 2007; Mahmood et al. 2013). Even more importantly, as shown in recent studies (Levitus et al. 2012), a significant fraction of the heat added to the climate system is at depth in the oceans, and thus cannot be sampled completely by ΔT (Spencer and Braswell 2013).

Computing the radiative imbalance ΔQ as a residual from large positive and negative values in the radiative flux budget introduces a large uncertainty. Stephens et al. (2012) reports a value of the global average radiative imbalance (which Stephens et al. calls the “surface imbalance”) as 0.70 Watts per meter squared, but with the uncertainty of 17 W m^-2!

We propose an alternate approach based on the analysis of the accumulation rate of heat in the Earth system in Joules per time. We believe the radiative imbalance can much more accurately be diagnosed by the ocean heat update since the ocean, because of the ocean’s density, area, and depth (i.e., its mass and heat capacity), is by far the dominate reservoir of climate system heat changes ( Pielke, 2003, 2005; Levitus et al. 2012; Trenberth and Fasullo 2013). Thus, the difference in ocean heat content at two different time periods largely accounts for the global average radiative imbalance over that time (within the uncertainty of the ocean heat measurements). Once the annual global annual average radiative imbalance is defined by the ocean accumulation of heat (adjusted for the smaller added heating from our parts of the climate system), we can form an equation that drives this imbalance as

Global annual average radiative imbalance [GAARI] = Global annual average radiative forcing [GAARF] + Global annual average radiative feedbacks [GAARFB] (2), where the units are in Joules per time period (and can be expressed as Watts per area).

Levitus et al. (2012) reported that since 1955, the layer from the surface to 2000 m depth had a warming rate of 0.39 W m^-2 ± 0.031 W m^-2 per unit area of the Earth’s surface which accounts for approximately 90% of the warming of the climate system. Thus, if we add the 10%, the 1955-2010 GAARI= 0.43 W m^-2 ± 0.031 W m^-2.

The radiative forcing can be obtained from the 2013 IPCC SPM WG1 report (unfortunately, they do not give the values for specific time periods but give a difference from 1750 to 1950, 1980 and 2011). Presumably, some of this forcing has been accommodated by warming over the time period, but the IPCC does not address this.

Figure SPM.5 in IPCC (2013) [reproduced below] yields the net radiative forcing = 2.29 (1.13 to 3.33) W m^-2 for the net change in the annual average global radiative forcing from 1750 to 2011.   The report on the change of radiative heating from 1750 to 1950 is 0.57 (0.29 to 0.85) W m^-2. If we assume that all of the radiative forcing up to 1950 has already resulted in feedbacks which remove this net positive forcing, the remaining mean estimate for the current GAARF is 1.72 W m^-2.

For GAARFB, Wielicki et al. (2013; their figure 1; reproduced below) has radiative feedbacks  =  -4.2 W m^-2 K^-1 (from temperature increases) + water vapor feedback (1.9 W m^-2 K^-1) + the albedo feedback (0.30 W m^-2 K^-1) + the cloud feedback (0.79 W m^-2 K^-1)   =  -1.21 W m^-2 K^-1.

It needs to be recognized that deep ocean heating is an unappreciated effective negative temperature feedback, at least in terms of how this heat can significantly influence other parts of the climate system on multi-decadal time scales. Nonetheless, we have retained this heating in our analysis.

Over the time period 1955 to 2010, the global surface temperatures supposedly increased by about 0.6 K (Figure SPM1 from IPCC, 2013 and reproduced below).

Thus, GAARFB = -1.21 W m^-2 K^-1 x 0.6K = -0.73 W m^-2.

Using the IPCC GAARF of 1.72 W m^-2 and the GAARFB of -0.73 W m^-2 in equation (2) yields GAARF + GAARFB = 1.72– 0.73 = 0.99 W m^-2 = GAARI.  This, however, is more than twice as large as the ocean diagnosed GAARI of 0.43 W m^-2 ± 0.031 based on Levitus et al. (2012).

Even the IPCC agrees that the radiative imbalance is relatively smaller than the 0.99 W m^-2 calculated above. They report that the global average radiative imbalance is 0.59 W m^-2 for 1971-2010 while for 1993-2010 it is 0.71 W m^-2. Trenberth and Fasullo (2013) state that the imbalance is 0.5–1W m⁻² over the 2000s.

Rather, than using the IPCC (Wielicki, 2013) GAARFB, we can use equation (2) to solve for the radiative feedbacks with the ocean heat data as a real world constraint, i.e. GAARFB = GAARI – GAARF (3).

Inserting the heat changes in the ocean to diagnose GAARI and the IPCC GAARF in (3) 0.43 W m^-2 ± 0.031 W m^-2 [GAARI] – 1.72 [-1.13 to -3.33] W m^-2 [GAARF], then results in the estimate of GAARFB of – 1.29 W m^-2 with an uncertainty range from the IPCC and Levitus (2012) yielding -1.10 to -3.36 W m^-2.

Thus, even assuming that the fraction of the global average radiative forcing change from 1750 to 1955 has already equilibrated through increasing surface temperatures, the global average radiative imbalance, GAARI, is significantly less than the sum of the global average radiative forcings and feedbacks – GAARF + GAARFB (the use of 1950 and 1955 as a time period should not introduce much added uncertainty).

Also, since there has been little if any temperature increase for a decade or more (nor, apparently little if any recent water vapor increase; Vonder Haar et al. 2012), the disparity between the imbalance and the forcings and feedbacks is even more stark. While including the uncertainty around each of the best estimates of the radiative forcings and feedbacks, and of the radiative imbalance, could still result in a claim that they are not out of agreement, the lack of proper closure of equation (1) in terms of the mean values that are available needs further explanation.

Thus as the next step, the uncertainties in each of the estimates needs to be defined for each of the values in equation (2). The estimates need to be made for the current time (2014). The recognition and explanation for this apparent discrepancy between observed global warming and the radiative forcings and feedbacks needs a higher level of attention than was given in the 2013 IPCC report.

In order to aid in the analyses of equation (2), the combined effects of the radiative forcings and feedbacks over specified time periods (e.g., decades) could be estimated by running the climate models with a set of realizations with and without specific radiative forcings (e.g., CO₂).   One could also do assessments of each vertical profile in a global model at snapshots in time with the added forcings since the last snapshot to estimate the radiative forcing change.

Parabolic reflector

From Wikipedia, the free encyclopedia

Circular paraboloid

A parabolic (or paraboloid or paraboloidal) reflector (or dish or mirror) is a reflective surface used to collect or project energy such as light, sound, or radio waves. Its shape is part of a circular paraboloid, that is, the surface generated by a parabola revolving around its axis. The parabolic reflector transforms an incoming plane wave traveling along the axis into a spherical wave converging toward the focus. Conversely, a spherical wave generated by a point source placed in the focus is reflected into a plane wave propagating as a collimated beam along the axis.

Parabolic reflectors are used to collect energy from a distant source (for example sound waves or incoming star light) and bring it to a common focal point, thus correcting spherical aberration found in simpler spherical reflectors. Since the principles of reflection are reversible, parabolic reflectors can also be used to project energy of a source at its focus outward in a parallel beam,^[1] used in devices such as spotlights and car headlights.

One of the world's largest solar parabolic dishes at the Ben-Gurion National Solar Energy Center in Israel

Theory

Strictly, the three-dimensional shape of the reflector is called a paraboloid. A parabola is the two-dimensional figure. (The distinction is like that between a sphere and a circle.) However, in informal language, the word parabola and its associated adjective parabolic are often used in place of paraboloid and paraboloidal.

If a parabola is positioned in Cartesian coordinates with its vertex at the origin and its axis of symmetry along the y-axis, so the parabola opens upward, its equation is , where is its focal length. (See "Parabola#Equation in Cartesian coordinates".) Correspondingly, the dimensions of a symmetrical paraboloidal dish are related by the equation: where is the focal length, is the depth of the dish (measured along the axis of symmetry from the vertex to the plane of the rim), and is the radius of the rim. All units must be the same. If two of these three quantities are known, this equation can be used to calculate the third.

A more complex calculation is needed to find the diameter of the dish measured along its surface. This is sometimes called the "linear diameter", and equals the diameter of a flat, circular sheet of material, usually metal, which is the right size to be cut and bent to make the dish. Two intermediate results are useful in the calculation: (or the equivalent: and where and are defined as above. The diameter of the dish, measured along the surface, is then given by: where means the natural logarithm of , i.e. its logarithm to base "e".

The volume of the dish, the amount of liquid it could hold if the rim were horizontal and the vertex at the bottom (e.g. the capacity of a paraboloidal wok), is given by where the symbols are defined as above. This can be compared with the formulae for the volumes of a cylinder a hemisphere where and a cone is the aperture area of the dish, the area enclosed by the rim, which is proportional to the amount of sunlight the reflector dish can intercept.

Parallel rays coming in to a parabolic mirror are focused at a point F. The vertex is V, and the axis of symmetry passes through V and F.

The parabolic reflector functions due to the geometric properties of the paraboloidal shape: any incoming ray that is parallel to the axis of the dish will be reflected to a central point, or "focus". (For a geometrical proof, click here.) Because many types of energy can be reflected in this way, parabolic reflectors can be used to collect and concentrate energy entering the reflector at a particular angle. Similarly, energy radiating from the focus to the dish can be transmitted outward in a beam that is parallel to the axis of the dish.

In contrast with spherical reflectors, which suffer from a spherical aberration that becomes stronger as the ratio of the beam diameter to the focal distance becomes larger, parabolic reflectors can be made to accommodate beams of any width. However, if the incoming beam makes a non-zero angle with the axis (or if the emitting point source is not placed in the focus), parabolic reflectors suffer from an aberration called coma. This is primarily of interest in telescopes because most other applications do not require sharp resolution off the axis of the parabola.

The precision to which a parabolic dish must be made in order to focus energy well depends on the wavelength of the energy. If the dish is wrong by a quarter of a wavelength, then the reflected energy will be wrong by a half wavelength, which means that it will interfere destructively with energy that has been reflected properly from another part of the dish. To prevent this, the dish must be made correctly to within about ¹⁄₂₀ of a wavelength. The wavelength range of visible light is between about 400 and 700 nanometres (nm), so in order to focus all visible light well, a reflector must be correct to within about 20 nm. For comparison, the diameter of a human hair is usually about 50,000 nm, so the required accuracy for a reflector to focus visible light is about 2500 times less than the diameter of a hair.

Microwaves, such as are used for satellite-TV signals, have wavelengths of the order of ten millimetres, so dishes to focus these waves can be wrong by half a millimetre or so and still perform well.

Focus-balanced reflector

It is sometimes useful if the centre of mass of a reflector dish coincides with its focus. This allows it to be easily turned so it can be aimed at a moving source of light, such as the Sun in the sky, while its focus, where the target is located, is stationary. The dish is rotated around axes that pass through the focus and around which it is balanced. If the dish is symmetrical and made of uniform material of constant thickness, and if F represents the focal length of the paraboloid, this "focus-balanced" condition occurs if the depth of the dish, measured along the axis of the paraboloid from the vertex to the plane of the rim of the dish, is 1.8478 times F. The radius of the rim is 2.7187 F.^[a] The angular radius of the rim as seen from the focal point is 72.68 degrees.

Scheffler reflector

The focus-balanced configuration (see above) requires the depth of the reflector dish to be greater than its focal length, so the focus is within the dish. This can lead to the focus being difficult to access. An alternative approach is exemplified by the Scheffler Reflector, named after its inventor, Wolfgang Scheffler. This is a paraboloidal mirror which is rotated about axes that pass through its centre of mass, but this does not coincide with the focus, which is outside the dish. If the reflector were a rigid paraboloid, the focus would move as the dish turns. To avoid this, the reflector is flexible, and is bent as it rotates so as to keep the focus stationary. Ideally, the reflector would be exactly paraboloidal at all times. In practice, this cannot be achieved exactly, so the Scheffler reflector is not suitable for purposes that require high accuracy. It is used in applications such as solar cooking, where sunlight has to be focused well enough to strike a cooking pot, but not to an exact point.^[2]

Off-axis reflectors

Off-axis satellite dish. The vertex of the paraboloid is below the bottom edge of the dish. The curvature of the dish is greatest near the vertex. The axis, which is aimed at the satellite, passes through the vertex and the receiver module, which is at the focus.

A circular paraboloid is theoretically unlimited in size. Any practical reflector uses just a segment of it. Often, the segment includes the vertex of the paraboloid, where its curvature is greatest, and where the axis of symmetry intersects the paraboloid. However, if the reflector is used to focus incoming energy onto a receiver, the shadow of the receiver falls onto the vertex of the paraboloid, which is part of the reflector, so part of the reflector is wasted. This can be avoided by making the reflector from a segment of the paraboloid which is offset from the vertex and the axis of symmetry. For example, in the above diagram the reflector could be just the part of the paraboloid between the points P₁ and P₃. The receiver is still placed at the focus of the paraboloid, but it does not cast a shadow onto the reflector. The whole reflector receives energy, which is then focused onto the receiver. This is frequently done, for example, in satellite-TV receiving dishes, and also in some types of astronomical telescope (e.g., the Green Bank Telescope).

Accurate off-axis reflectors, for use in telescopes, can be made quite simply by using a rotating furnace, in which the container of molten glass is offset from the axis of rotation. To make less accurate ones, suitable as satellite dishes, the shape is designed by a computer, then multiple dishes are stamped out of sheet metal.

History

The principle of parabolic reflectors has been known since classical antiquity, when the mathematician Diocles described them in his book On Burning Mirrors and proved that they focus a parallel beam to a point.^[3]
Archimedes in the third century BC studied paraboloids as part of his study of hydrostatic equilibrium,^[4] and it has been claimed that he used reflectors to set the Roman fleet alight during the Siege of Syracuse.^[5] This seems unlikely to be true, however, as the claim does not appear in sources before the 2nd century AD, and Diocles does not mention it in his book.^[6] Parabolic mirrors were also studied by the physicist Ibn Sahl in the 10th century.^[7] James Gregory, in his 1663 book Optica Promota (1663), pointed out that a reflecting telescope with a mirror that was parabolic would correct spherical aberration as well as the chromatic aberration seen in refracting telescopes.
The design he came up with bears his name: the "Gregorian telescope"; but according to his own confession, Gregory had no practical skill and he could find no optician capable of actually constructing one.^[8] Isaac Newton knew about the properties of parabolic mirrors but chose a spherical shape for his Newtonian telescope mirror to simplify construction.^[9] Lighthouses also commonly used parabolic mirrors to collimate a point of light from a lantern into a beam, before being replaced by more efficient Fresnel lenses in the 19th century. In 1888, Heinrich Hertz, a German physicist, constructed the world's first parabolic reflector antenna.^[10]

Applications

Lighting the Olympic Flame

The most common modern applications of the parabolic reflector are in satellite dishes, reflecting telescopes, radio telescopes, parabolic microphones, solar cookers, and many lighting devices such as spotlights, car headlights, PAR lamps and LED housings.^[11]

The Olympic Flame is traditionally lit at Olympia, Greece, using a parabolic reflector concentrating sunlight, and is then transported to the venue of the Games. Parabolic mirrors are one of many shapes for a burning-glass.

Parabolic reflectors are popular for use in creating optical illusions. These consist of two opposing parabolic mirrors, with an opening in the center of the top mirror. When an object is placed on the bottom mirror, the mirrors create a real image, which is a virtually identical copy of the original that appears in the opening. The quality of the image is dependent upon the precision of the optics. Some such illusions are manufactured to tolerances of millionths of an inch.

Antennas of the Atacama Large Millimeter Array on the Chajnantor Plateau.^[12]

A parabolic reflector pointing upward can be formed by rotating a reflective liquid, like mercury, around a vertical axis. This makes the liquid mirror telescope possible. The same technique is used in rotating furnaces to make solid reflectors.

Parabolic reflectors are also a popular alternative for increasing wireless signal strength. Even with simple ones, users have reported 3 dB or more gains.^[13][14]

Parabola

From Wikipedia, the free encyclopedia

A parabola (/pəˈræbələ/; plural parabolas or parabolae, adjective parabolic, from Greek: παραβολή) is a two-dimensional, mirror-symmetrical curve, which is approximately U-shaped when oriented as shown in the diagram below, but which can be in any orientation in its plane. It fits any of several superficially different mathematical descriptions which can all be proved to define curves of exactly the same shape.

One description of a parabola involves a point (the focus) and a line (the directrix). The focus does not lie on the directrix. The parabola is the locus of points in that plane that are equidistant from both the directrix and the focus. Another description of a parabola is as a conic section, created from the intersection of a right circular conical surface and a plane which is parallel to another plane which is tangential to the conical surface.^[a] A third description is algebraic. A parabola is a graph of a quadratic function, such as

The line perpendicular to the directrix and passing through the focus (that is, the line that splits the parabola through the middle) is called the "axis of symmetry". The point on the axis of symmetry that intersects the parabola is called the "vertex", and it is the point where the curvature is greatest. The distance between the vertex and the focus, measured along the axis of symmetry, is the "focal length". The "latus rectum" is the chord of the parabola which is parallel to the directrix and passes through the focus. Parabolas can open up, down, left, right, or in some other arbitrary direction. Any parabola can be repositioned and rescaled to fit exactly on any other parabola — that is, all parabolas are geometrically similar.

Parabolas have the property that, if they are made of material that reflects light, then light which travels parallel to the axis of symmetry of a parabola and strikes its concave side is reflected to its focus, regardless of where on the parabola the reflection occurs. Conversely, light that originates from a point source at the focus is reflected into a parallel ("collimated") beam, leaving the parabola parallel to the axis of symmetry. The same effects occur with sound and other forms of energy. This reflective property is the basis of many practical uses of parabolas.

The parabola has many important applications, from a parabolic antenna or parabolic microphone to automobile headlight reflectors to the design of ballistic missiles. They are frequently used in physics, engineering, and many other areas.

Strictly, the adjective parabolic should be applied only to things that are shaped as a parabola, which is a two-dimensional shape. However, as shown in the last paragraph, the same adjective is commonly used for three-dimensional objects, such as parabolic reflectors, which are really paraboloids. Sometimes, the noun parabola is also used to refer to these objects. Though not perfectly correct, this usage is generally understood.

Part of a parabola (blue), with various features (other colours). The complete parabola has no endpoints. In this orientation, it extends infinitely to the left, right, and upward.

Introductory images

• Parabolic Cartesian graph of the function y=6x²+4x-8
• 

  A parabola obtained as the intersection of a cone with a (red) plane parallel to a (checkered) plane which is tangential to the cone's surface.
• 

  Parabolic curve showing directrix (L) and focus (F). The distance from any point on the parabola to the focus (P_nF) equals the perpendicular distance from the same point on the parabola to the directrix (P_nQ_n).
• 

  For description, see text below.

Description of final image

Parabolic curve showing chord (L), focus (F), and vertex (V). L is an arbitrary chord of the parabola perpendicular to its axis of symmetry, which passes through V and F. (The ends of the chord are not shown here.) The lengths of all paths Q_n - P_n - F are the same, equalling the distance between the chord L and the directrix. (See previous image.) This is similar to saying that a parabola is an ellipse, but with one focal point at infinity. It also directly implies, by the wave nature of light, that parallel light arriving along the lines Q_n - P_n will be reflected to converge at F. A linear wavefront along L is concentrated, after reflection, to the one point where all parts of it have travelled equal distances and are in phase, namely F. No consideration of angles is required

History

Parabolic compass designed by Leonardo da Vinci

The earliest known work on conic sections was by Menaechmus in the fourth century BC. He discovered a way to solve the problem of doubling the cube using parabolas. (The solution, however, does not meet the requirements imposed by compass and straightedge construction). The area enclosed by a parabola and a line segment, the so-called "parabola segment", was computed by Archimedes via the method of exhaustion in the third century BC, in his The Quadrature of the Parabola. The name "parabola" is due to Apollonius who discovered many properties of conic sections. It means "application", referring to "application of areas" concept, that has a connection with this curve, as Apollonius had proved.^[1] The focus–directrix property of the parabola and other conics is due to Pappus.
Galileo showed that the path of a projectile follows a parabola, a consequence of uniform acceleration due to gravity.

The idea that a parabolic reflector could produce an image was already well known before the invention of the reflecting telescope.^[2] Designs were proposed in the early to mid seventeenth century by many mathematicians including René Descartes, Marin Mersenne,^[3] and James Gregory.^[4] When Isaac Newton built the first reflecting telescope in 1668, he skipped using a parabolic mirror because of the difficulty of fabrication, opting for a spherical mirror. Parabolic mirrors are used in most modern reflecting telescopes and in satellite dishes and radar receivers.^[5]

Equation in Cartesian coordinates

Let the directrix be the line x = −p and let the focus be the point (p, 0). If (x, y) is a point on the parabola then, by definition of a parabola, it is the same distance from the directrix as the focus; in other words:

Squaring both sides and simplifying produces

as the equation of the parabola. By interchanging the roles of x and y one obtains the corresponding equation of a parabola with a vertical axis as

The equation can be generalized to allow the vertex to be at a point other than the origin by defining the vertex as the point (h, k). The equation of a parabola with a vertical axis then becomes

The last equation can be rewritten

so the graph of any function which is a polynomial of degree 2 in x is a parabola with a vertical axis.

More generally, a parabola is a curve in the Cartesian plane defined by an irreducible equation — one that does not factor as a product of two not necessarily distinct linear equations — of the general conic form

with the parabola restriction that

where all of the coefficients are real and where A and C are not both zero. The equation is irreducible if and only if the determinant of the 3×3 matrix

is non-zero: that is, if (AC − B²/4)F + BED/4 − CD²/4 − AE²/4 ≠ 0. The reducible case, also called the degenerate case, gives a pair of parallel lines, possibly real, possibly imaginary, and possibly coinciding with each other.^[6]

Conic section and quadratic form

Cone with cross-sections (To enlarge, click on diagram. To shrink, go to previous page.)

The diagram represents a cone with its axis vertical.^[b] The point A is its apex. A horizontal cross-section of the cone passes through the points B, E, C, and D. This cross-section is circular, but appears elliptical when viewed obliquely, as is shown in the diagram. An inclined cross-section of the cone, shown in pink, is inclined from the vertical by the same angle, θ, as the side of the cone. According to the definition of a parabola as a conic section, the boundary of this pink cross-section, EPD, is a parabola. The cone also has another horizontal cross-section, which passes through the vertex, P, of the parabola, and is also circular, with a radius which we will call r. Its centre is V, and PK is a diameter. The chord BC is a diameter of the lower circle, and passes through the point M, which is the midpoint of the chord ED. Let us call the lengths of the line segments EM and DM x, and the length of PM y.

Thus:

  (The triangle BPM is isosceles.)

  (PMCK is a parallelogram.)

Using the intersecting chords theorem on the chords BC and DE, we get:

Substituting:

Rearranging:

For any given cone and parabola, r and θ are constants, but x and y are variables which depend on the arbitrary height at which the horizontal cross-section BECD is made. This last equation is a simple quadratic one which describes how x and y are related to each other, and therefore defines the shape of the parabolic curve. This shows that the definition of a parabola as a conic section implies its definition as the graph of a quadratic function. Both definitions produce curves of exactly the same shape.

Focal length

It is proved below that if a parabola has an equation of the form where is a positive constant, then where is its focal length. Comparing this with the last equation above shows that the focal length of the above parabola is .

Position of the focus

In the diagram, the point F is the foot of the perpendicular from the point V to the plane of the parabola.^[c] By symmetry, F is on the axis of symmetry of the parabola. Angle VPF is complementary to θ, and angle PVF is complementary to angle VPF, therefore angle PVF is θ. Since the length of PV is r, the distance of F from the vertex of the parabola is r sin θ. It is shown above that this distance equals the focal length of the parabola, which is the distance from the vertex to the focus. The focus and the point F are therefore equally distant from the vertex, along the same line, which implies that they are the same point. Therefore the position of the focus is at F.

Other geometric definitions

A parabola may also be characterized as a conic section with an eccentricity of 1. As a consequence of this, all parabolas are similar, meaning that while they can be different sizes, they are all the same shape. Another consequence is that the universal parabolic constant is the same for all parabolas.^[7] A parabola can also be obtained as the limit of a sequence of ellipses where one focus is kept fixed as the other is allowed to move arbitrarily far away in one direction. In this sense, a parabola may be considered an ellipse that has one focus at infinity. The parabola is an inverse transform of a cardioid.

A parabola has a single axis of reflective symmetry, which passes through its focus and is perpendicular to its directrix. The point of intersection of this axis and the parabola is called the vertex. A parabola spun about this axis in three dimensions traces out a shape known as a paraboloid of revolution.

The parabola is found in numerous situations in the physical world (see below).

Equations

Cartesian

In the following equations and are the coordinates of the vertex of the parabola and is the distance from the vertex to the focus and the vertex to the directrix.

Vertical axis of symmetry

where

.

Parametric form:

Horizontal axis of symmetry

where

.

Parametric form:

General parabola

The general form for a parabola is

This result is derived from the general conic equation given below:

and the fact that, for a parabola,

.

The equation for a general parabola with a focus point F(u, v), and a directrix in the form

is

Latus rectum, semilatus rectum, and polar coordinates

In Polar coordinate system, a parabola with the focus at the origin and the directrix parallel to the y-axis, is given by the equation

where l is the semilatus rectum: the distance from the focus to the parabola itself, measured along a line perpendicular to the axis of symmetry. Note that this equals the perpendicular distance from the focus to the directrix, and is twice the focal length, which is the distance from the focus to the vertex of the parabola.
The latus rectum is the chord that passes through the focus and is perpendicular to the axis of symmetry. It has a length of 2l.

Dimensions of parabolas with axes of symmetry parallel to the y-axis

These parabolas have equations of the form . By interchanging and the parabolas' axes of symmetry become parallel to the x-axis.

Some features of a parabola

Coordinates of the vertex

The x-coordinate at the vertex can be found by completing the square to put the equation in vertex form, or by differentiating the original equation, setting the resulting equal to zero (a critical point), and solving for . Both methods yield: .
Substituting this into the original equation yields:

These terms can be combined over a common denominator:

, where is the discriminant.

Thus, the vertex is at the point .

Coordinates of the focus

Since the axis of symmetry of this parabola is parallel with the y-axis, the x-coordinates of the focus and the vertex are equal. The coordinates of the vertex are calculated in the preceding section. The x-coordinate of the focus is therefore also

To find the y-coordinate of the focus, consider the point, P, located on the parabola where the slope is 1, so the tangent to the parabola at P is inclined at 45 degrees to the axis of symmetry. Using the reflective property of a parabola, we know that light which is initially travelling parallel to the axis of symmetry is reflected at P toward the focus. The 45-degree inclination causes the light to be turned 90 degrees by the reflection, so it travels from P to the focus along a line that is perpendicular to the axis of symmetry and to the y-axis. This means that the y-coordinate of P must equal that of the focus.

By differentiating the equation of the parabola and setting the slope to 1, we find the x-coordinate of P:

Substituting this value of in the equation of the parabola, we find the y-coordinate of P, and also of the focus:

where is the discriminant, as used in the "Coordinates of the vertex" section.

The focus is therefore the point:

Axis of symmetry, focal length, latus rectum, and directrix

The above coordinates of the focus of a parabola of the form:

can be compared with the coordinates of its vertex, which are derived in the section "Coordinates of the vertex", above, and are:

where

The axis of symmetry is the line which passes through both the focus and the vertex. In this case, it is vertical, with equation:

.

The focal length of the parabola is the difference between the y-coordinates of the focus and the vertex:

It is sometimes useful to invert this equation and use it in the form: See the section "Conic section and quadratic form", above.

The point where the slope of the parabola is 1 lies at one end of the latus rectum. The length of the semilatus rectum (half of the latus rectum) is the difference between the x-coordinates of this point, which is considered as P in the above derivation of the coordinates of the focus, and of the focus itself. Thus, the length of the semilatus rectum is:

, where is the focal length.

The total length of the latus rectum is therefore four times the focal length.

Measured along the axis of symmetry, the vertex is the midpoint between the focus and the directrix. Therefore, the equation of the directrix is:

Proof of the reflective property

Reflective property of a parabola

The reflective property states that, if a parabola can reflect light, then light which enters it travelling parallel to the axis of symmetry is reflected to the focus. This is derived from the wave nature of light in the caption to a diagram near the top of this article. This derivation is valid, but may not be satisfying to readers who would prefer a mathematical approach. In the following proof, the fact that every point on the parabola is equidistant from the focus and from the directrix is taken as axiomatic.

Consider the parabola Since all parabolas are similar, this simple case represents all others. The right-hand side of the diagram shows part of this parabola.

Construction and definitions
The point E is an arbitrary point on the parabola, with coordinates The focus is F, the vertex is A (the origin), and the line FA (the y-axis) is the axis of symmetry. The line EC is parallel to the axis of symmetry, and intersects the x-axis at D. The point C is located on the directrix (which is not shown, to minimize clutter). The point B is the midpoint of the line segment FC.

Deductions

Measured along the axis of symmetry, the vertex, A, is equidistant from the focus, F, and from the directrix. Correspondingly, since C is on the directrix, the y-coordinates of F and C are equal in absolute value and opposite in sign. B is the midpoint of FC, so its y-coordinate is zero, so it lies on the x-axis. Its x-coordinate is half that of E, D, and C, i.e. The slope of the line BE is the quotient of the lengths of ED and BD, which is which comes to

But is also the slope (first derivative) of the parabola at E. Therefore the line BE is the tangent to the parabola at E.

The distances EF and EC are equal because E is on the parabola, F is the focus and C is on the directrix. Therefore, since B is the midpoint of FC, triangles FEB and CEB are congruent (three sides), which implies that the angles marked are congruent. (The angle above E is vertically opposite angle BEC.) This means that a ray of light which enters the parabola and arrives at E travelling parallel to the axis of symmetry will be reflected by the line BE so it travels along the line EF, as shown in red in the diagram (assuming that the lines can somehow reflect light). Since BE is the tangent to the parabola at E, the same reflection will be done by an infinitesimal arc of the parabola at E.
Therefore, light that enters the parabola and arrives at E travelling parallel to the axis of symmetry of the parabola is reflected by the parabola toward its focus.

The point E has no special characteristics. This conclusion about reflected light applies to all points on the parabola, as is shown on the left side of the diagram. This is the reflective property.

Other consequences

There are other theorems that can be deduced simply from the above argument.

Tangent bisection property

The above proof, and the accompanying diagram, show that the tangent BE bisects the angle FEC. In other words, the tangent to the parabola at any point bisects the angle between the lines joining the point to the focus, and perpendicularly to the directrix.

Intersection of a tangent and perpendicular from focus

Perpendicular from focus to tangent

Since triangles FBE and CBE are congruent, FB is perpendicular to the tangent BE. Since B is on the x-axis, which is the tangent to the parabola at its vertex, it follows that the point of intersection between any tangent to a parabola and the perpendicular from the focus to that tangent lies on the line that is tangential to the parabola at its vertex.
See animated diagram.^[8]

Reflection of light striking the convex side

If light travels along the line CE, it moves parallel to the axis of symmetry and strikes the convex side of the parabola at E. It is clear from the above diagram that this light will be reflected directly away from the focus, along an extension of the segment FE.

Alternative proofs

Parabola and tangent

The above proofs of the reflective and tangent bisection properties use a line of calculus. For readers who are not comfortable with calculus, the following alternative is presented.

In this diagram, F is the focus of the parabola, and T and U lie on its directrix. P is an arbitrary point on the parabola. PT is perpendicular to the directrix, and the line MP bisects angle FPT. Q is another point on the parabola, with QU perpendicular to the directrix. We know that FP=PT and FQ=QU. Clearly, QT>QU, so QT>FQ. All points on the bisector MP are equidistant from F and T, but Q is closer to F than to T. This means that Q is to the "left" of MP, i.e. on the same side of it as the focus. The same would be true if Q were located anywhere else on the parabola (except at the point P), so the entire parabola, except the point P, is on the focus side of MP. Therefore MP is the tangent to the parabola at P. Since it bisects the angle FPT, this proves the tangent bisection property.

The logic of the last paragraph can be applied to modify the above proof of the reflective property. It effectively proves the line BE to be the tangent to the parabola at E if the angles are equal. The reflective property follows as shown previously.

Tangent properties

Two tangent properties related to the latus rectum

Let the line of symmetry intersect the parabola at point Q, and denote the focus as point F and its distance from point Q as f. Let the perpendicular to the line of symmetry, through the focus, intersect the parabola at a point T.
Then (1) the distance from F to T is 2f, and (2) a tangent to the parabola at point T intersects the line of symmetry at a 45° angle.^[9]:p.26

Orthoptic property

Perpendicular tangents intersect on the directrix

If two tangents to a parabola are perpendicular to each other, then they intersect on the directrix. Conversely, two tangents which intersect on the directrix are perpendicular.

Proof

Without loss of generality, consider the parabola Suppose that two tangents contact this parabola at the points and Their slopes are and respectively. Thus the equation of the first tangent is of the form where is a constant. In order to make the line pass through the value of must be so the equation of this tangent is Likewise, the equation of the other tangent is At the intersection point of the two tangents, Thus Factoring the difference of squares, cancelling, and dividing by 2 gives Substituting this into one of the equations of the tangents gives an expression for the y-coordinate of the intersection point: Simplifying this gives

We now use the fact that these tangents are perpendicular. The product of the slopes of perpendicular lines is −1, assuming that both of the slopes are finite. The slopes of our tangents are and , so so Thus the y-coordinate of the intersection point of the tangents is given by This is also the equation of the directrix of this parabola, so the two perpendicular tangents intersect on the directrix.

Lambert's theorem

Let three tangents to a parabola form a triangle. Then Lambert's theorem states that the focus of the parabola lies on the circumcircle of the triangle.^{[8]:Corollary 20 [10]}

Tsukerman's converse to Lambert's theorem states that, given three lines that bound a triangle, if two of the lines are tangent to a parabola whose focus lies on the circumcircle of the triangle, then the third line is also tangent to the parabola.^[11]

Facts related to chords

Focal length calculated from parameters of a chord

Suppose a chord crosses a parabola perpendicular to its axis of symmetry. Let the length of the chord between the points where it intersects the parabola be and the distance from the vertex of the parabola to the chord, measured along the axis of symmetry, be The focal length, of the parabola is given by:

Proof

Suppose a system of Cartesian coordinates is used such that the vertex of the parabola is at the origin, and the axis of symmetry is the y-axis. The parabola opens upward. It is shown elsewhere in this article that the equation of the parabola is 4fy=x², where f is the focal length. At the positive-x end of the chord, x=c/2 and y=d. Since this point is on the parabola, these coordinates must satisfy the equation above. Therefore, by substitution, 4fd=(c/2)². From this, f=c²/(16d).

Area enclosed between a parabola and a chord

Parabola (magenta) and line (lower light blue) including a chord (blue). The area enclosed between them is in pink. The chord itself ends at the points where the line intersects the parabola.

The area enclosed between a parabola and a chord (see diagram) is two-thirds of the area of a parallelogram which surrounds it. One side of the parallelogram is the chord, and the opposite side is a tangent to the parabola.^[12][13] The slope of the other parallel sides is irrelevant to the area. Often, as here, they are drawn parallel with the parabola's axis of symmetry, but this is arbitrary.

A theorem equivalent to this one, but different in details, was derived by Archimedes in the 3rd Century BCE. He used the areas of triangles, rather than that of the parallelogram.^[d] See the article "The Quadrature of the Parabola".

If the chord has length b, and is perpendicular to the parabola's axis of symmetry, and if the perpendicular distance from the parabola's vertex to the chord is h, the parallelogram is a rectangle, with sides of b and h. The area, A, of the parabolic segment enclosed by the parabola and the chord is therefore:

This formula can be compared with the area of a triangle: .

In general, the enclosed area can be calculated as follows. First, locate the point on the parabola where its slope equals that of the chord. This can be done with calculus, or by using a line that is parallel with the axis of symmetry of the parabola and passes through the midpoint of the chord. The required point is where this line intersects the parabola.^[e] Then, using the formula given in the article "Distance from a point to a line", calculate the perpendicular distance from this point to the chord. Multiply this by the length of the chord to get the area of the parallelogram, then by to get the required enclosed area.

Corollary concerning midpoints and endpoints of chords

A corollary of the above discussion is that if a parabola has several parallel chords, their midpoints all lie on a line which is parallel to the axis of symmetry. If tangents to the parabola are drawn through the endpoints of any of these chords, the two tangents intersect on this same line parallel to the axis of symmetry.^[f]

Length of an arc of a parabola

If a point X is located on a parabola which has focal length and if is the perpendicular distance from X to the axis of symmetry of the parabola, then the lengths of arcs of the parabola which terminate at X can be calculated from and as follows, assuming they are all expressed in the same units.^[g]

This quantity, , is the length of the arc between X and the vertex of the parabola.

The length of the arc between X and the symmetrically opposite point on the other side of the parabola is

The perpendicular distance, , can be given a positive or negative sign to indicate on which side of the axis of symmetry X is situated. Reversing the sign of reverses the signs of and without changing their absolute values. If these quantities are signed, the length of the arc between any two points on the parabola is always shown by the difference between their values of The calculation can be simplified by using the properties of logarithms:

This can be useful, for example, in calculating the size of the material needed to make a parabolic reflector or parabolic trough.

This calculation can be used for a parabola in any orientation. It is not restricted to the situation where the axis of symmetry is parallel to the y-axis.

Focal length and radius of curvature at the vertex

The focal length of a parabola is half of its radius of curvature at its vertex.

Proof

• 

  Image is inverted. A"B" is x-axis. C is origin. O is centre. A is (x,y). OA=OC=R. PA=x. CP=y. OP=(R-y). Other points and lines are irrelevant for this purpose.
• 

  The radius of curvature at the vertex is twice the focal length. The measurements shown on the above diagram are in units of the latus rectum, which is four times the focal length.
• 

Consider a point on a circle of radius and with centre at the point The circle passes through the origin. If the point is near the origin, the Pythagorean Theorem shows that:

But, if is extremely close to the origin, since the x-axis is a tangent to the circle, is very small compared with so is negligible compared with the other terms. Therefore, extremely close to the origin:

.....(Equation 1)

Compare this with the parabola:

......(Equation 2)

which has its vertex at the origin, opens upward, and has focal length . (See preceding sections of this article.)

Equations 1 and 2 are equivalent if Therefore this is the condition for the circle and parabola to coincide at and extremely close to the origin. The radius of curvature at the origin, which is the vertex of the parabola, is twice the focal length.

Corollary

A concave mirror which is a small segment of a sphere behaves approximately like a parabolic mirror, focusing parallel light to a point which is midway between the centre and the surface of the sphere.

Mathematical generalizations

In algebraic geometry, the parabola is generalized by the rational normal curves, which have coordinates the standard parabola is the case and the case is known as the twisted cubic. A further generalization is given by the Veronese variety, when there is more than one input variable.

In the theory of quadratic forms, the parabola is the graph of the quadratic form (or other scalings), while the elliptic paraboloid is the graph of the positive-definite quadratic form (or scalings) and the hyperbolic paraboloid is the graph of the indefinite quadratic form Generalizations to more variables yield further such objects.

The curves for other values of p are traditionally referred to as the higher parabolas, and were originally treated implicitly, in the form for p and q both positive integers, in which form they are seen to be algebraic curves. These correspond to the explicit formula for a positive fractional power of x. Negative fractional powers correspond to the implicit equation and are traditionally referred to as higher hyperbolas. Analytically, x can also be raised to an irrational power (for positive values of x); the analytic properties are analogous to when x is raised to rational powers, but the resulting curve is no longer algebraic, and cannot be analyzed via algebraic geometry.

Parabolas in the physical world

In nature, approximations of parabolas and paraboloids are found in many diverse situations. The best-known instance of the parabola in the history of physics is the trajectory of a particle or body in motion under the influence of a uniform gravitational field without air resistance (for instance, a baseball flying through the air, neglecting air friction).

The parabolic trajectory of projectiles was discovered experimentally by Galileo in the early 17th century, who performed experiments with balls rolling on inclined planes. He also later proved this mathematically in his book Dialogue Concerning Two New Sciences.^[14][h] For objects extended in space, such as a diver jumping from a diving board, the object itself follows a complex motion as it rotates, but the center of mass of the object nevertheless forms a parabola. As in all cases in the physical world, the trajectory is always an approximation of a parabola. The presence of air resistance, for example, always distorts the shape, although at low speeds, the shape is a good approximation of a parabola. At higher speeds, such as in ballistics, the shape is highly distorted and does not resemble a parabola.

Another hypothetical situation in which parabolas might arise, according to the theories of physics described in the 17th and 18th Centuries by Sir Isaac Newton, is in two-body orbits; for example the path of a small planetoid or other object under the influence of the gravitation of the Sun. Parabolic orbits do not occur in nature; simple orbits most commonly resemble hyperbolas or ellipses. The parabolic orbit is the degenerate intermediate case between those two types of ideal orbit. An object following a parabolic orbit would travel at the exact escape velocity of the object it orbits; objects in elliptical or hyperbolic orbits travel at less or greater than escape velocity, respectively. Long-period comets travel close to the Sun's escape velocity while they are moving through the inner solar system, so their paths are close to being parabolic.

Approximations of parabolas are also found in the shape of the main cables on a simple suspension bridge. The curve of the chains of a suspension bridge is always an intermediate curve between a parabola and a catenary, but in practice the curve is generally nearer to a parabola, and in calculations the second degree parabola is used.^[15][16] Under the influence of a uniform load (such as a horizontal suspended deck), the otherwise catenary-shaped cable is deformed toward a parabola. Unlike an inelastic chain, a freely hanging spring of zero unstressed length takes the shape of a parabola. Suspension-bridge cables are, ideally, purely in tension, without having to carry other, e.g. bending, forces. Similarly, the structures of parabolic arches are purely in compression.

Paraboloids arise in several physical situations as well. The best-known instance is the parabolic reflector, which is a mirror or similar reflective device that concentrates light or other forms of electromagnetic radiation to a common focal point, or conversely, collimates light from a point source at the focus into a parallel beam. The principle of the parabolic reflector may have been discovered in the 3rd century BC by the geometer Archimedes, who, according to a legend of debatable veracity,^[17] constructed parabolic mirrors to defend Syracuse against the Roman fleet, by concentrating the sun's rays to set fire to the decks of the Roman ships. The principle was applied to telescopes in the 17th century. Today, paraboloid reflectors can be commonly observed throughout much of the world in microwave and satellite-dish receiving and transmitting antennas.

In parabolic microphones, a parabolic reflector that reflects sound, but not necessarily electromagnetic radiation, is used to focus sound onto a microphone, giving it highly directional performance.

Paraboloids are also observed in the surface of a liquid confined to a container and rotated around the central axis. In this case, the centrifugal force causes the liquid to climb the walls of the container, forming a parabolic surface.
This is the principle behind the liquid mirror telescope.

Aircraft used to create a weightless state for purposes of experimentation, such as NASA's "Vomit Comet," follow a vertically parabolic trajectory for brief periods in order to trace the course of an object in free fall, which produces the same effect as zero gravity for most purposes.

In the United States, vertical curves in roads are usually parabolic by design.

Gallery

• 

  A bouncing ball captured with a stroboscopic flash at 25 images per second. Note that the ball becomes significantly non-spherical after each bounce, especially after the first. That, along with spin and air resistance, causes the curve swept out to deviate slightly from the expected perfect parabola.
• 

  Parabolic trajectories of water in a fountain.
• 

  The path (in red) of Comet Kohoutek as it passed through the inner solar system, showing its nearly parabolic shape. The blue orbit is the Earth's
• 

  Hercilio Luz Bridge, Florianópolis, Brazil. The supporting cables of suspension bridges follow a curve which is intermediate between a parabola and a catenary.^[15]
• 

  The Rainbow Bridge across the Niagara River, connecting Canada (left) to the United States (right). The parabolic arch is in compression, and carries the weight of the road.
• 

  Parabolic arches used in architecture
• 

  Parabolic shape formed by a liquid surface under rotation. Two liquids of different densities completely fill a narrow space between two sheets of transparent plastic. The gap between the sheets is closed at the bottom, sides and top. The whole assembly is rotating around a vertical axis passing through the centre. (See Rotating furnace)
• 

  Solar cooker with parabolic reflector
• 

  Parabolic antenna
• 

  Parabolic microphone with optically transparent plastic reflector, used to overhear referee conversations at an American college football game.
• 

  Array of parabolic troughs to collect solar energy
• 

  Edison's searchlight, mounted on a cart. The light had a parabolic reflector.
• 

  Physicist Stephen Hawking in an aircraft flying a parabolic trajectory to produce zero-gravity