Temperature gradient in stars

Temperature gradient in stars

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It is a well known fact that in stars, there exists a temperature gradiënt. The observational reason is because we perceived spectral lines in the otherwise continuous spectrum of a star. If this wasn't the case, and thus if the temperature were uniform, then the absorption and the emission would cancel each other out and we would observe a net effect that results in a continuous spectrum. Now my question is: what is the physical explanation for this temperature gradient?

You're talking about the temperature gradients in the atmosphere of a star, that's where the emission and absorption lines are originating. This is a much more complicated matter than the star itself.

The temperature of a star must increase from its surface downwards.
This is a simple result of the fact that the stellar structure is given by pressure gradients balancing the force of gravity. We can express this via the law of hydrostatic equilibrium
$$frac{partial P}{partial r} = -g(r) ho $$ which is a pretty good approximation to reality in most cases.

The pressure gradient then translates into a temperature gradient, as the pressure originates in the thermal motion of particles, formalized as the law of ideal gases
$$ P = frac{ ho k_B T}{mu} $$.

So wherever there's non-zero gravity and no other force in a gaseous medium, there will be temperature gradients.

In an atmosphere, things depend on how efficient the gas is at cooling. In very thin media, like in Earths or any planets / stars exosphere cooling is inefficient enough to make temperature gradients vanish.

Force balance only tells you that the pressure must decrease outward, the cause of the decreasing temperature is the nature of heat transport and the requirement that it increase entropy. Hence heat is always transported from higher T to lower T, and a star must transport heat outward because its surface is losing heat to the coldness of space. That requires the temperature must drop as you go out, as long as you are just passing the same heat from layer to layer and the thermodynamics of heat transport rules the situation.

It was pointed out that the low-density atmospheres of stars can have their temperature rise with height, as the Earth's stratosphere also does, yet the net energy transport is still outward. To avoid entropy decrease, this requires that more heat be deposited in the hot layers than is extracted from the cooler ones below, so the heat dumped high up in the atmosphere has to come from somewhere else, other than the cool layers. That's what can't happen deep in the star-- the heat gets no "Free ride" to pass certain layers, it is passed from layer to layer, the same heat. But when there are ways to dump heat into higher layers without extracting it from cooler ones below, then the temperature can rise. (In the Earth's stratosphere, the extra heat comes from sunlight absorbed by ozone, and in the Sun's chromosphere, the extra heat comes from magnetic fields and plasma motions, which pass through the lower layers without being absorbed.)

What is a Temperature Gradient? (with picture)

A temperature gradient is the gradual variance in temperature with distance. The slope of the gradient is consistent within a material. A gradient is established anytime two materials at different temperatures are in physical contact with each other. Units of measure of temperature gradients are degrees per unit distance, such as °F per inch or °C per meter.

Many temperature gradients exist naturally, while others are created. The largest temperature gradient on Earth is the Earth itself. The temperature of the Earth’s core is estimated at about 9,000°F (5,000°C) it is 6,650°F (3,700°C) at the boundary between the core and the mantle, while the crust temperature is about 200°F (93°C). Each layer has a temperature gradient of a different slope, depending on the heat conductivity of the layer.

No temperature gradient exists between the Earth and the sun because there is not an atmosphere between them. Heat capacity is the ability for a material to hold heat. A vacuum has zero heat capacity.

Convection destroys a thermal gradient. In heating a pot of sauce, the liquid closest to the burner becomes the hottest. When stirred, the hot liquid mixes with cooler liquid, the heat becomes evenly distributed, and the temperature gradient is nullified.

If left unstirred, convective heat transfer will cause warm liquid to rise and cold to fall, and some circulation will occur, although it will not be as effective as active stirring. Over time, the conduction forces transferring heat from the bottom will establish equilibrium with the convective forces causing the water to circulate. If the heat source is low, circulation will be slow, a steep temperature gradient may exist, and the sauce may be burned on the bottom. If the heat is high, the sauce will boil, heat transfer by convection will be high, and the temperature gradient will be near zero.

Insulation is used to retard heat transfer by putting material with low heat conductivity next to the heat source. The insulation helps maintain the thermal gradient between the insulated object and ambient conditions. Coffee will stay warmer in a foam cup than in an aluminum cup because the foam conducts heat less readily. Likewise, the coffee drinker may burn a few fingers picking up the aluminum cup because the thermal gradient is near zero and the temperature of the outside of the cup is nearly the same as the inside of the cup.

To be stable, a thermal gradient must have a constant heat source and an available heat sink. Maintaining constant gradients is seldom important, except when conducting chemical reactions. Many industrial processes require careful heat control. The living cell also must maintain careful heat controls for optimum performance. While scientists understand how the human body as a whole maintains a temperature gradient between its core and the outside world, the options available to individual cells are less clear.

2 Answers 2

For a planet which has a temperature gradient, hot in the center and cooler on the surface, why do we see absorption lines?

The hot center sends photons within the black body spectrum with appropriate energies to excite surface cold atoms ,so the black body curve will have holes, where energy of the photons has been absorbed in exciting surface molecules.

Similarly, why do we see emission lines if the planet is hot on the surface and gets cooler as you move to the center?

The black body spectrum is a continuous spectrum from thermal excitations. There exists though a probability that from the high energy tail of the black body energy spectrum , electrons from atoms on the surface are taken to a higher energy level and then relax back to the ground state emitting the specific line of that atom.

I'm trying to understand why convection is an efficient mode of energy transport in the outer layers of the solar interior.

Could anyone give me a little bit of knowledge?

The details do depend on opacity, but the basic phenomenon can be understood without reference to what is happening with the opacity (as explained, for example, in Kippenhahn and Wiegert, Stellar Structure and Evolution, pg. 75). The main point is, if you consider the interior structure, it supports a radiative diffusion rate that determines a luminosity that the envelope of the star will simply have to handle somehow. A main-sequence star has nothing drastically unusual happening to its envelope, so you cannot really alter the stellar radius much. Given the luminosity and the radius, the surface temperature is more or less handed to the envelope by the Stefan-Boltzmann law, so the envelope just has to deal with that somehow. When the surface temperature handed to the envelope is large, there's no problem, but when it's smaller, there is a problem as we shall see.

Now, if you assume the stellar envelope transports heat predominantly by radiative diffusion, the imposed temperature structure encounters no particular difficulty if the surface temperature is allowed to be rather high (say above about 10,000 K when you put in the details of the opacity). Above that temperature, it turns out that the structure of the envelope is pretty insensitive to what that temperature is, and the envelope happily diffuses out whatever luminosity is required because radiative energy diffuses easily when the temperature is high. (This is a consequence of the fact that the energy density stays high if the temperature stays high, so you don't need much in the way of a diffusion speed to get out the luminosity.) However, if your requirement is that the surface temperature be well below 10,000 K, then you have a serious problem, because at those low temperatures, radiative energy does not diffuse easily-- it requires a high diffusion speed because the energy density, which scales like T^4, is so low. In fact, you need a steeper temperature gradient than is stable to convection in order to get the luminosity out. So the star finds a different mode for transporting the heat, it goes convectively unstable and moves hot parcels of gas upward instead of diffusing radiation. This also reduces the temperature gradient to something that keeps the temperature from going to zero before you get to the surface of the star (a problem that radiative diffusion has when it becomes inefficient at lower T).

Convection vs Radiation at the giant stage of stars

The statement is misleading because the logic is essentially backward-- it seems to claim that red giants are more luminous than dwarfs because they are convective, but in fact they are convective because they are more luminous. So convection does not determine the luminosity of a red giant, but red giants are convective because their luminosity has been determined to be so high that radiative diffusion can't carry it, it must be carried by convection. What's more, convection does not have a particular efficiency at carrying heat, it can carry heat with a wide range of possible efficiencies, whatever it needs to do given the other physics that is actually setting the luminosity. There is a maximum possible efficiency, which is when the gas is convecting at the sound speed, but few stars ever need to be that convective over their whole interior, they just aren't that luminous. Radiation, on the other hand, is limited by the speed that light can diffuse out, which you might think would be very fast given how fast the speed of light is, but the diffusion speed takes the speed of light and divides it by the optical depth, so when the optical depth is huge, it can take a long time to diffuse out.

But convection has a problem-- it only happens when the temperature gradient is steep enough that buoyancy can produce an unstable "turning over" effect, the "rolling boil" effect you see in convecting gas. If the temperature gradient is not steep enough, convection won't happen, and radiation will carry the luminosity of the star. That's where the surface temperature a bit above 2500 K comes in-- when you have that, you get a special kind of opacity where neutral hydrogen picks up a second bound electron, making the "H minus" ion, which is quite good at absorbing light (the extra electron is very weakly bound and easy to knock out of the ion by photon capture). That opacity "bottles up" the radiation, and helps enforce a steep enough temperature gradient where you will get convection. Lower surface temperatures for an ideal gas would be unstable-- rises in T would create more H minus opacity which would absorb more light and raise the temperature. (You can get lower surface temperatures in degenerate material, like brown dwarfs and planets.)

Now, although convection has a higher upper limit on how much luminosity it can carry, it never sets the luminosity of the star, it just carries what luminosity some other process in the star tells it to carry. That "other process" can be of two flavors: outside-in, where the surface layers of the star determine the luminosity and the interior simply provides that luminosity (via convection), or inside-out, where some internal engine determines the luminosity and convection carries that out, and the surface just has to deal with whatever it is. The outside-in case is when you have a protostar that is first forming, which has a history that determines its radius. As mentioned, the surface temperature will always be above 2500 K, typically more like about 4000 K actually, so if we take the surface temperature as known, we can determine the luminosity from the radius of the protostar. The radius is set by the history of whatever stage of contraction the star is currently in, so that's "outside-in" luminosity.

But you are asking about red giants, which have their luminosity determined in a completely different way, they are "inside-out." Their luminosity is determined by the fact that they contain at their centers a ball of degenerate gas, which is a lot like a tiny white dwarf living inside a giant ball of gravitationally bound ideal gas. That is a very special structure, and it gives the star essentially three different pieces-- the white dwarf at the center has a mass that is controlled by the history of adding nuclear burned "ash" to the white dwarf, and that rises with time. Its radius is set by degeneracy physics. Then you have a layer sitting on top of that white dwarf which is an ideal gas, but its temperature is set by the gravity of the white dwarf (via something called the virial theorem). That temperature gets very high as the mass of the white dwarf grows, and indeed it is high enough to have fusion. The rate of fusion in that layer is the internal engine that sets the luminosity of the red giant, and it grows with time simply because the white dwarf mass is growing with time, so the temperature is rising-- and fusion likes high temperature.

Then we come at last to the convective envelope, the third piece of the red giant. This is a rather passive player, it just carries the luminosity set by that internal engine without having essentially any effect on that engine. It must puff out a lot to get down to the necessary cool temperatures to have H minus opacity, and that's why the star is a "giant" (and remember that the luminosity must be carried at the surface by the surface temperature to the fourth power times the radius squared, so if the surface temperature needs to be low to get the convective instability, the radius needs to be huge). So we should say that radiation is too slow to carry the huge luminosity generated by the central engine, and convection will appear and can carry almost any luminosity it needs to, but that convection only constrains the surface temperature-- the luminosity is set by the fusion physics (unlike in main-sequence stars, whose luminosity is set by radiative diffusion), but that's another story). Then the luminosity and surface T get together to determine the radius, and it comes out very large-- hence a red giant.

Dissecting Bad Models - Solar Temperature Gradient Paradox by Michael Gmirkin

Over time, theories tend to cease being thought of as theoretical and begin to be thought of as unassailable, unquestionable fact. However, unquestioned fundamental assumptions will one day be science's downfall.

One such theory (that has solidified into rarely questioned "fact") is the thermonuclear model of the sun. In said model, a star is a ball of gas so massive that it crushes itself under its own weight and begins to undergo fusion reactions in its core.

This model traces in large part back to Sir Arthur Eddington, a British astrophysicist prominent in the early 20 th century.

A cagey critic might point out that according to Karl Popper, it only requires one substantiated 'fatal objection' to falsify a model.

Is the observable temperature profile of the sun at odds with the theoretically expected temperature profile? If so, does this constitute a direct contradiction and thus falsification of existing stellar theory?

Put simply, the thermonuclear model of stars put forth by Eddington, et al requires nuclear fusion in the core of the star. That fusion liberates energy, generates extraordinarily high temperatures and thus balances the gases of a star thermally against collapse due to self-gravitation, in theory.

Thus, astrophysicists currently expect of the Sun:

But do simple real-world observations back up the grandiose theories currently in vogue?

They do not.

Leaving aside any specific models of yore, let us objectively observe what we can of the solar atmosphere's temperature profile.

Sunspots are produced when vast magnetic fields push through the surface of the sun, pushing the uppermost layer of the sun's body (the photosphere) away, exposing the cooler (thus darker) solar interior.

("Cooler" is a misnomer, nothing on the sun can be termed "cool." But when compared with the photospheric temperature of approximately 6,000 Kelvin, a sunspot interior can be as low as 3,000 Kelvin.)

Why sunspots should reveal a darker and 'cooler' inside of the sun, when the inside of the sun is supposed to be hotter than the surface (on account of the thermonuclear furnace generating extreme temperatures which should diffuse outward), is a mystery.

If sunspots open a hole to a deeper level of the sun, and deeper levels are supposed to be hotter, should not sunspots be brighter and hotter than the surrounding photosphere?

That sunspots expose a darker and cooler interior appears to belie the thermonuclear sun.

Put frankly, the observable temperatures of the solar atmosphere invert the theoretically expected temperature profile of the sun. Whereas the thermonuclear model of the sun expects fusion in the (extremely hot) core and a steeply declining temperature gradient extending outward, observations show precisely the opposite! The outermost layer of the sun (directly observable) is the hottest, while the innermost layer of the sun (again, directly observable) is the coolest.

One wonders whether Sir Arthur Eddington would have speculated upon the thermonuclear model of stars had modern observations of temperatures in the solar atmosphere been available during his day and age.

The observable temperature profile, juxtaposed with the theoretical wranglings of the thermonuclear model of the Sun and stars leads to a paradox of thermodynamic proportions!

That is to say, real-world observations show the solar atmosphere to be hottest on the outside and coolest on the inside. The thermonuclear sun is expected to be hottest on the inside and coolest on the outside, in theory. If the two models are superimposed, you arrive at a contradictory state with a hot core and a hot corona and a temperature minimum at the photosphere (or possibly just below it). That appears to be precisely the predicament in which astrophysicists find themselves mired.

How can such a temperature minimum be maintained between two extraordinarily hot regions? Should not heat diffuse from both adjacent hot regions into the cold region, thus warming it until it disappears completely?

Precisely this question was raised by the electrical engineer Ralph Juergens in 1972:

To date, there does not appear to be any good answer as to why such a contradictory temperature minimum should exist, let alone persist, under a thermonuclear model of the sun. Its very existence would seem to be a piece of falsifying data for the thermonuclear model that spawned it.

Returning to Sir Arthur Eddington, he once framed the debate about the constitution of stars thus:

Sir Arthur Eddington chose the former path, believing that stars harbor internally the vast majority of the energy that they expend over their lifetimes. He dismissed, seemingly without prejudice or much further thought, the alternative.

As if in answer to Eddington, Ralph Juergens' quote (above) continues as follows:

Perhaps Eddington took the garden path when conjecturing the internal thermonuclear furnace as that which makes the stars shine. Is it possible that &ldquosome subtle radiation traversing space, which the stars pick up&rdquo is responsible for keeping them alight instead?

Rethinking the energy source for the sun and stars may end up having far-reaching implications for astronomy and cosmology. Perhaps it's time to rethink some foundational assumptions underlying modern astrophysical theories and see what shakes loose in the light of current observations.

Temperature gradient in stars - Astronomy

  1. radiation---photons (energy packets) leak outward by scattering off gas particles. Nature prefers this way.
  2. conduction---fast-moving atoms collide with other atoms imparting some of their motion to them. This is used by metals like copper or aluminum to transfer heat (e.g., from your stove element to the food), but it is not used by a gas since the gas molecules are so far apart from one another. The process of conduction is too inefficient in a gas to worry about. (This is why you can stick your hand into your oven while something is baking and not immediately burn your hand if it does not touch anything, especially the metal sides and rack.)
  3. convection---big pieces of the atmosphere cycle between cold regions and warm regions. Hot air below expands and its density decreases so it rises. Cooler, denser air falls and displaces the hot air. As a hot bubble rises, it cools by giving up its heat energy to the cool surroundings. The gas will then fall and heat up when it comes into contact with the warm surface or interior.

In addition to transporting energy outward to space, convection also distributes the heat across the planet, from the warm daylit equatorial regions to the cooler latitudes closer to the poles and to the night side of the planet. The warm air at the equatorial regions rises and the cooler air from other parts of the planet flows across the surface toward the equator to replace the rising air. All of the winds in a planet's atmosphere are due to convective processes. If the planet is rotating quickly enough, the motion of the air can be deflected sideways by the Coriolis effect (see also the Galileo section in the history chapter).

If a pocket of air from the pole moves toward the equator without changing direction, the Earth will rotate beneath it. The packet of air has a sideways motion equal to the rotation speed at the pole, but the parts of the Earth's surface closer to the equator have a greater rotational speed because they are farther from the rotation axis. To an observer on the ground, the path appears deflected to the west. The Coriolis effect on a spherical body is actually a bit more complicated than just the east or west deflection described above but a more complete treatment of the Coriolis effect requires higher level physics beyond the scope of this textbook. For our purposes, it is sufficient to say that objects will be deflected to the right in the northern hemisphere and to the left in the southern hemisphere, even for objects traveling due east or due west. The Coriolis deflections produce the spiral patterns of cyclonic storms (winds spiraling inward counter-clockwise in the northern hemisphere and clockwise in the southern hemisphere) and air flow away from high-pressure regions (winds rotate clockwise in the northern hemisphere and counter-clockwise in the southern hemisphere).

Some nice animations of air circulation around low and high pressure regions are available from NASA's Visible Earth site: low-pressure circulation animation -- high-pressure circulation animation.

The rapid rotation of a planet will also complicate the convective flow of energy from the warm equator to the cool poles. On a planet with little or no rotation (Venus, for example), the air circulation is very simple: warm air rises along the equator, flows at high altitudes toward the poles, and near the surface returns to the equator. On a planet with rapid rotation (Earth or the jovian planets, for example), the surface winds from the poles are deflected into large-scale eddies with belts of wind and calm. At high altitudes narrow bands of high-speed winds called jet streams are formed and they play an important role on the surface weather. Land masses sticking up into the air flow disrupt the spiraling circulation and provide a place for storms to expend their energy.

The rapidly rotating jovian planets have much greater Coriolis effects. The powerful, narrow jet streams deflect the clouds into belts moving parallel to the planet equators. The winds in a belt move in the opposite direction of the belt next to it. Large vortices can form from the interplay of the belts. A large vortex can last for decades, even centuries or longer because the jovian planets do not have a solid surface for storms to expend their energy. Jupiter's Great Red Spot is an example of a large vortex. Twice the size of the Earth, it is at least 400 years old.

Image from the Voyager spacecraft courtesy of NASA

For most of the planets, the Sun provides the energy to maintain the temperature (and surface temperature for the terrestrial planets) and to drive the convective motions of the atmosphere. But Jupiter, Saturn, and Neptune generate over twice as much heat than they receive from the Sun. Most of this energy is leftover heat from when the planets formed 4.6 billion years ago. As material collected onto the forming planets, it heated up when energy was released by the material falling in the planet's gravity field. All of the planets were hot enough to be liquid. The heavier, denser materials (like iron and nickel) separated from the lighter materials (like silicon, hydrogen, and helium) and fell toward the planet cores. The process called differentiation released more gravitational energy and heated up the planets further. Due to their large size, the jovian planets still retain a lot of their initial formation heat and that energy is responsible for the spectacular clouds patterns. In the case of Saturn, the differentiation process may still be going on as the helium in the interior separates from the hydrogen and sinks toward the core, a "helium rain". The helium rain is probably why there is a smaller percentage of helium in Saturn's atmosphere than in Jupiter's atmosphere.

The much blander atmosphere of Uranus is a result of its lower heat emission. Most of the heat inside the much smaller Earth and Venus is produced from radioactivity in the rocky material (in fact, the higher radioactive heating long ago may have been necessary for the terrestial planets to undergo differentiation). However, the heat of Venus' and Earth's interior has little to zero effect on their atmospheres because the crust is such a poor conductor of heat (though convection in their interiors is responsible for the geologic processes seen on their surfaces). Sunlight energy is what determines their surface temperatures and drives their weather.

Atmospheres moderate the heat lost to space at night and shield the planet surface from energetic radiation like solar ultraviolet and X-rays and the high-speed charged particles in solar wind and most cosmic rays (extremely high-energy particles from space, mostly protons). The planet Mercury has almost no atmosphere and so there is a difference of several hundred degrees between places in the shade and sunlit areas! The planet Mars has a very thin atmosphere, so it experiences a temperature drop of over a 100 degrees when night comes. Humans landing on the martian surface will need to contend with the extreme cold of the night and will need to protect themselves from the harmful solar radiation during the day. The Earth's atmosphere is thick enough that the temperature difference between night and day is at most a few tens of degrees. Our atmosphere also blocks high-energy light like UV and X-rays and solar wind particles. Some cosmic ray particles have high enough energy to penetrate the atmosphere and even several meters of rock! If a cosmic ray strikes the DNA in the cells, the DNA structure can be altered. Cosmic rays are responsible for some of the genetic mutations in life.

Acoustics, Dr. William Robertson

Overview: Computer simulations and experiments in acoustic band gap and acoustic metamaterials.

Activities: Experimental acoustic measurements using the impulse response technique. The experiments generally explore arrayed systems of resonators that are designed to manipulate the properties of sound wave propagation including extraordinary acoustic transmission, acoustic lensing, and realization of fast and slow acoustic group velocities. The experiments are designed and interpreted using computer simulations in MATLAB and COMSOL.

Minimum student background: Enrolled in Modern Physics. Programming experience a plus.

Model II

In this model there is the creation of heat within the star matter. There is a balance between forces due to gravitation and the pressure gradient and the ideal gas law is assumed to apply as in Model I but the temperature is not uniform. Steady state conditions require that the heat created within the star be transferred to the surface and this requires a radial temperature gradient.

The heat energy passing through a surface is proportion to the area times the temperature gradient. The proportionality factor is negative because heat is transferred in the direction that temperature is declining. The net flow outflow from an infinitesimal volume is therefore proportional to the divergence of the temperature gradient. But for steady state conditions this net outflow must be equal to the heat generated within the infinitesimal volume. Thus

C&rho - D&nabla 2 T = 0

where c is the rate of production of heat per unit mass and D is the coefficient of heat conduction. As in Model I, &rho and T represent the mass density and temperature of the star material. This equation is of the form of a Poisson equation. The Laplacian of T, &nabla 2 T, for spherical coordinates when there is spherical symmetry is:

&nabla 2 T = (1/r 2 )&part(r 2 &partT/&partr)/&partr).

The full version of Model II is:

GM(r)/r 2 = (1/&rho)&partp/&partr M(r) = &int0 r 4&pi&rho(s)s 2 ds p = &rhoRT (1/r 2 )&part(r 2 &partT/&partr)/&partr) = (c/D)&rho

This last equation can be put into the form

&part(r 2 &partT/&partr)/&partr) = (c/4&piD)&rho4&pir 2

which upon integration with respect to the radius variable gives

R 2 (&partT/&partr) = (c/4&piD)&int0 r &rho4&pis 2 ds = (c/4&piD)M(r)

M(r)/r 2 = (c/4&piD)(&partT/&partr) AND M(r)/r 2 = (1/G)(1/&rho)(&partp/&partr) therefore (Gc/(4&piD))&partT/&partr = (1/&rho)(&partp/&partr)

Since from the ideal gas equation

&partp/&partr = RT&part&rho/&partr + R&rho&partT/&partr and hence (1/&rho)(&partp/&partr) = RT(1/&rho)(&part&rho/&partr) + R&partT/&partr it follows that &gamma&partT/&partr = RT(1/&rho)(&part&rho/&partr) + R(&partT/&partr) and thus (&gamma-1)(1/T)&partT/&partr = (1/&rho)&part&rho/&partr

This last equation above implies that

&rho/&rho0 = (T/T0) &gamma-1

where &rho0 and T0 represents a standardized density and temperature.

Because from the ideal gas equation

(p/p0) = (&rho/&rho0)(T/T0) it follows that (p/p0) = (T/T0) &gamma

The temperature profile is determined thus from the Poisson equation

&nabla 2 T = CT &gamma

Temperature gradient in stars - Astronomy

Mirages: Can Mirages Explain UFO Reports?

From ( on April 29, 2002.

What are mirages and how do they appear?

A mirage is usually defined as a phenomenon where light is reflected from a shallow layer of very hot air in contact with the ground, the appearance being that of pools of water in which inverted images of more distant objects are seen. This is the inferior mirage, which occurs where a very hot plane surface, such as a desert or a roadway, heats a layer of air very close to it. The temperature gradient in the thermocline (the region of rapidly changing temperature) between this hot layer and cooler air above it is so steep as to constitute a discontinuity. This discontinuity acts as a mirror (or caustic) for light striking it above a critical (large) angle to the normal. In this way one can see distant objects such as the sky or vehicles reflected in the surface.

How can this explain UFO reports? It is not well known that these discontinuities can form in the upper air as the result of a temperature inversion - that is where a layer of warm air lies over cold air. Temperature inversions form almost every clear night when the ground cools by radiation more rapidly than the air above. Strong inversions are more likely to form a discontinuity and lead to mirages. These are called superior mirages, that is a mirage seen above the source or object being reflected (see Figure 1). In this way an inverted image of some bright but distant source may be seen in the sky. The definition of a superior mirage needs to be extended to cover one or more displaced images of a very distant but bright light source, usually distorted and brightened. Naturally this must be considered a major alternative to the ETH and a strong contender for explaining UFO reports. Figure 1: How the rays from a source (S) are reflected by the caustic in the thermocline of a temperature inversion if they strike it at or above the critical angle (c). Ray 4 enters at below the critical angle and so penetrates the caustic and undergoes normal gradual refraction.

Where the source is already in the sky, for example, an astronomical object, the image may be elevated, considerably so where the thermocline is curved. Non?horizontal thermoclines may displace the image laterally, and moving thermoclines may produce a moving image. Because an inversion forms in a fluid (air), the image can take various shapes and alter its shape with time. Consequently superior mirages can be unusual and protean.

Not all mirages are reflections some are caused by abnormal refraction. If a temperature inversion forms over a very wide area, say over a cold ocean or ice field, and the temperature gradient is strong enough, light can be ducted around the curvature of the Earth, so allowing one to see an image of an astronomical object that is actually below the horizon. This is the 'Novaya Zemlya' mirage. The light in such a mirage can be ducted for hundreds of kilometres and the image may be distorted. It may also change shape and/or colour and be very bright. Light striking the discontinuity below a critical (large) angle to the normal, will not be reflected, but will pass through it and be refracted (Figure 1). An observer above the thermocline may then see a bright source elevated above its normal position.

Mirage images can consist of double images, with an upright image above the inverted one. This may be due to light penetrating the thermocline and being bent back down towards the observer (as shown in Figure 1). Where the thermocline is low over the source, the separation of the two images will be large. However, as the height of the thermocline increases, the two images can merge, making it difficult to recognize the image (see Figure 2). There is some reason to believe that each mirage image can split in the plane of the inversion, creating two separate images if this occurs when there are already two images, the result will be four images of the same object!

Figure 2: One means by which the twin images of a mirage can be formed. Image Y1 is formed by reflection from the discontinuity in the thermocline (T) of the inversion. At P reflection ceases because the critical angle is not exceeded and the observer sees a refracted (upright) image (Y2). It can be seen that, as the height between the object (X) and the inversion increases, the two images will merge, eventually disappearing. Conversely, as the height decreases, Y1 and Y2 separate. If T is very shallow, Y2 will not appear. Drawn with exaggerated vertical scale for clarity.

Mirage images can be greatly enlarged and/or distorted by atmospheric lens effects: the more distant the object, the greater the magnification (because of the greater size of the atmospheric lens). Sources outside the atmosphere may be subject to the greatest magnification among these, the commonest are astronomical sources. It may be expected therefore that the largest and most common mirages will be those of astronomical objects at low altitude. Magnification also increases as the source aligns with the thermocline. This means that, as the disc of an astronomical object approaches the thermocline, the two images enlarge and merge until they form a classic 'flying saucer' shape (see Figure 3). The two images may not always be the same size. Figure 3: A diagram showing how the two images of an astronomical body in a mirage can appear with different separation. As the images merge and enlarge, they form a classic 'flying saucer'.

Some mirage images of astronomical objects may display clusters of lights, perhaps multiple images of the object, and it is common for mirage images to shimmer. The enlargement of an astronomical object in a mirage will make its intrinsic colour more apparent, although differential refraction may produce several different colours at once, spatially separated. In a statement submitted to a symposium on UFOs organized by a committee of the US House of Representatives in 1968, astronomer Donald Menzel explained how strange an astronomical mirage could appear: "Sometimes a layer of warm air, sandwiched between two layers of cold air, can act as a lens, projecting a pulsating, spinning, vividly colored, saucer?like image of a planet. Pilots, thinking they were dealing with a nearby flying object, have often tried to intercept the image, which evades all attempts to cut it off. The distances may seem to change rapidly, as the star fades or increases in brightness. Actual 'dog fights' have been recorded between confused military pilots and a planet. I myself have observed this phenomenon of star mirage. It is both realistic and frightening." This is a reference to Menzel's own observation of a 'flying saucer' when he was flying over Alaska on a military mission in 1955. The object, which appeared to be flying alongside his aircraft, was complete with flashing red and green lights, a 'lighted propeller' on top and with a silvery metallic sheen. Later he identified it as a mirage of the bright star Sirius although it appears that it was actually a mirage of the planet Saturn.

UFO reports explained by mirages Surprisingly, and significantly, the very first 'flying saucer' report, that by Kenneth Arnold in 1947, can be explained in this way. He reported seeing a chain of nine peculiar 'aircraft' flying near Mount Ranier in Washington state (USA). They all moved together and occasionally flashed very brightly. However analysis shows that the apparent movement was entirely due to his own, just as a low moon will appear to follow you across a stationary landscape. All very distant objects at low altitude will appear to move because their direction does not change as that of a nearer object would. In this case, the source was nine snow-capped peaks in the Cascade Range over 100 kilometres away. In the bright sunlight, mirages of them were formed by temperature inversions over two deep river valleys between Arnold and the mountains. Where the inversions were strong, the mirages of the peaks flashed brightly. It appears that Arnold was not familiar with mirages, but this is true of almost all pilots.

In the right circumstances, any bright surface object can produce a mirage. On 17 November 1986, a Japanese freighter aircraft had crossed the North Pole and was heading SW toward its next stop, Anchorage in Alaska. Suddenly the crew were confronted by clusters of lights just ahead of them. They assumed that the lights were the exhausts of some unidentified aircraft and tried in vain to evade them. Gradually the mysterious lights shifted to port and the captain was sure he could make out the shape of a huge UFO alongside them. The incident was reported to the (US) Federal Aviation Administration (FAA), who issued a report on the incident, but without any explanation.

Because the object's direction appeared to move aft with time, it was obvious that the source lay on the ground only a few hundred kilometres away, and because the crew gave good descriptions and bearings to the lights at various times on their route, it was possible to locate its source. This turned out to be the US Army airfield at Delta Junction. The crew's description of the lights exactly matched that of typical runway lights and the FAA reported that a temperature inversion had existed over the area at the time. The 'UFO' was a mirage of the runway lights.

Aircraft headlights are a typical source of mirages. In May 1996, BBC Scotland showed me a video of mysterious lights seen over Inverness a few months earlier. It turned out that they were multiple mirages of the lights of a Nimrod aircraft which regularly trains from RAF Kinloss on the Moray Firth. This phenomenon explains the lights filmed in 1950 over Great Falls (Montana) two jet aircraft were flying about the area at the time but no one seems to have asked if they had their lights on. It also explains the many lights filmed over Tremonton (Utah) in 1952. In that case, there is evidence of several inversions, one on top of the other. A mirage of aircraft lights also explains a report investigated by physicist Bruce Maccabee in 1975: two bright objects 'like bright stars' were seen to the NE of Cheverly (Maryland), just east of Washington DC. They were seen in the general direction of Baltimore?Washington Airport about 34 kilometres away where a Boeing 707 was due to take off about the time of the sighting. Maccabee never considered mirages as an explanation and so failed to explain the report. Given that distant bright objects are often the source of mirages, astronomical objects at low altitude must be strong candidates.

Although the moon has sometimes been responsible, Venus, the brightest planet is the commonest source of such mirages. Indeed it was the object filmed as a UFO by a film crew in an aircraft off New Zealand in December 1978. In the new year, the film was shown on TV all over the world. Although Venus itself was below the horizon, its mirage image was visible via a Novaya Zemlya effect in which the light was ducted several hundred kilometres around the earth due to a temperature inversion over the cold Southern Ocean. It was also the object seen in daylight by forester Robert Taylor at Livingston (Scotland) in November 1979, a case I investigated on the ground. Mirages of Venus explain very many strange UFO reports, including the 1952 Nash/Fortenberry report (USA), the egg-shaped object seen over Anglesey (Wales) in September 1978 and the object seen and report in Todmorden (England) by policeman Alan Godfrey.

Other bright planets at low altitude have also been the source of UFO reports. The most sensational was the mirage of Jupiter reported and photographed by Almiro Barauna from a Brazilian research ship at Ilha da Trindade in the south Atlantic Ocean in January 1958. These are unique photographs, clearly showing the double image which results from the merging of two mirage images (see photo). A mirage of Jupiter was also the object which Capt. Thomas Mantell followed to his death over Kentucky (USA) in January 1948 and which Lt George Gorman tried to catch over Fargo (N. Dakota) in October the same year.

Two enlargements of the mirage of Jupiter photographed by Almiro Barauna at Ilha da Trindade (APRO). A mirage of Saturn was the object which scared young Ronald Johnson at his parents' farm near Delphos (Kansas) in November 1971. Mirages of Mars and Mercury have also produced strange UFO reports. Sometimes several planets together have been involved, as in the 1959 Gill case from Papua-New Guinea. Bright stars at low altitude can also stimulate mirages, but not necessarily only at night.

Sirius, the brightest star, is often responsible, as at Kirtland AFB in New Mexico in November 1957, when it was thought to be an object trying to land at the base. But it is the second-brightest star, Canopus, which has caused more reports. A mirage of Canopus was the object reported by police patrolman Lonnie Zamora over Socorro (New Mexico) in April 1964. This appears to have been caused by an inversion over the Rio Grande valley, south of the town. Astronomer Allen Hynek frequently challenged sceptics to explain this report, which he regarded as the epitome of the UFO phenomenon, apparently unaware that it had an astronomical explanation. A mirage of Canopus was also responsible for the sensational Cash/Landrum report from Huffman (Texas) in December 1980. The witnesses were convinced that a UFO had landed on the road ahead of them. A mirage of Canopus appears to have been the object which led to the death of pilot Frederick Valentich over the Bass Strait in October 1978. Disorientated by the mirage and convinced that it was on top of him, he seems to have crashed into the sea. There are 20 first magnitude stars, almost all of which at various times and in various places either directly or via mirage have been responsible for UFO reports.

In Conclusion Not only are mirages an 'alternative to the ETH', they explain reports which are otherwise inexplicable, especially the core reports which remain when all other reports have found an explanation. The result is that no UFO report remains unexplained and there is no mysterious phenomenon behind the reports. Furthermore UFO reports have nothing to do with extraterrestrial intelligence.

Steuart Campbell, 2000 References The UFO Mystery Solved Campbell, S. Explicit Books, 1994

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