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Jupiter

 

Jupiter contains most of the mass accumulated by planets in our solar system.

The first figure is a gif that shows 48 low color resolution images of a slide lecture on Jupiter with a frame rate of three seconds per frame.

The entire slide presentation will be incorporated below in the accordion tables along with lecture notes.

The last accordion contains all of the lecture notes as about eight pages of text.

The current 48 chart version of the PowerPoint slide lecture with lecture notes can be downloaded by clicking HERE.

© Bob Field 2016-0225

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Composition, Structure, Formation and Evolution of Jupiter

Is Jupiter like the Sun?

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Approximately 99.99% of the mass of the solar system is concentrated in three giant balls of gas, the Sun, Jupiter, and Saturn. They are mostly hydrogen and helium but contain small amounts of heavier elements, mostly oxygen, carbon, nitrogen, iron, silicon, and magnesium. Everything else in the solar system comprises about 0.01% of its mass. Jupiter and his little brother Saturn are similar, so we will focus on Jupiter.

The composition and structure of Jupiter differ from the Sun and from the other giant planets. After we investigate the differences, we will consider the formation and post-formation evolutionary processes that account for the differences.

The Sun and Jupiter are the two largest objects in our solar system. They have some interesting similarities including composition, being predominately giant self-gravitating balls of hydrogen and helium gas, but they are strikingly different in many ways as we shall see. The composition and structure of Jupiter differ from the Sun and from the other giant planets. After we investigate the differences, we will consider the formation and post-formation evolutionary processes that account for the differences.

The composition of the Sun, planets, and smaller objects in the solar system is related to the composition of the primordial solar nebula from which they formed. The composition of the Sun and the two gas giant planets looks very much like the primordial nebula, but the sequence of physical processes that formed everything else varied the composition of planets. The two ice giant planets formed from dust and ice with very little hydrogen and helium gas and the four tiny inner planets look like rocks covered with thin layers of fluids.

The composition and density of Jupiter happens to be very similar to the Sun, but the enormous difference in their masses produces bodies with vastly different properties. In the absence of thermonuclear fusion, the Sun would have cooled and contracted to form a very dense small dim object.

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Remarkably the Sun would not be dense and hot enough for nucleosynthesis to transform hydrogen into helium were it not for a quantum tunneling process that allows protons to overcome their Coulombic repulsion long enough to fuse. And even with that advantage, fusion would not proceed if the Sun was not so opaque that it traps heat in the dense core for tens of millions of years. The opacity itself is due to the extreme interior temperatures that have fully ionized the hydrogen and helium atoms producing free electrons that are very quick to absorb and scatter photons.

The opacity does not prevent photons in the lower 70% of the Sun’s radius from slowly transporting energy from the core toward the surface because the temperatures and temperature gradients are very large. The photons in the last 30% of the distance to the surface are not hot enough to overcome the opacity, so convective heat flow is triggered to maintain a nearly adiabatic temperature gradient, resulting in a very rapid energy flow for the remaining journey to the surface where the thin relatively transparent photosphere enables radiant energy to flow to space.

As a result of the fusion process, the core has actually grown hotter and denser as the Sun has evolved over billions of years, resulting in an increase in fusion over time despite an overall decrease in available hydrogen fuel. The growing outflow of energy from the core elevates the temperature of the Sun outside of the core causing it to expand and increase the Sun’s surface area and temperature over time. So the Sun gets bigger and brighter over time – and its average density decreases as its core density grows.

Unlike the Sun, Jupiter is cooling and contracting because it does not have a powerful internal energy source like the Sun’s core fusion. The atmosphere of Jupiter controls the rate at which the planet cools. Clouds formed from condensed gases absorb radiant energy from deeper hotter layers of the atmosphere so that only the coolest layers of the atmosphere radiate into space. The surface temperature is maintained by an efficient convective heat transport that is slightly steeper than the adiabatic temperature gradient that would be present if the convective instability were not triggered.

Why is the interior of Jupiter so opaque to radiant energy? Its interior is 1000 times cooler than the Sun’s interior and not sufficient to ionize the gases, but the pressures are so high – more than a million times the Earth’s atmosphere (but far less than the pressures in the Sun) – that the gases are pressure ionized rather than thermally ionized – they are squeezed out of their atomic orbitals and form – not a plasma like the Sun – but a metal with free electrons. Metals are opaque. Metallic hydrogen is opaque like other metals but it is not a very good thermal conductor because the free electrons scatter before they can carry thermal energy very far. So convection is the mode of heat transport throughout Jupiter.

The Sun’s interior is very dense but it is still an ideal gas because the particles are too hot that to interact for very long. This is very convenient for study because the ideal gas law can be applied. The metallic hydrogen and helium which forms most of the mass of Jupiter is not an ideal gas, it is not even a gas, it is such a dense fluid that it is often called a liquid – so the gas giant planets are mostly liquid not gas.

 

 

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Polytrope models of Jupiter's interior

Polytrope models content

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There is a set of four differential equations along with suitable boundary conditions that characterize the state of the interior of stars like the Sun. Several of the equations can describe interiors of planets. Mass M is a function of radius within an object. The mass enclosed within a given radius from the center of a sphere is the integral of the radially varying density function over the volume of the sphere enclosed by the given radius. The boundary condition at the surface is that M equals the entire mass of the sphere, a measurable quantity. The mass conservation equation is simply the differential equation form of the integral.

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A self-gravitating spherical mass is in equilibrium when the hydrostatic support from the pressure gradient at a given radius equals the gravitational force on a volume of mass at that radius. The integral form is from the surface down to the radius (rather than from the center to the radius) to account for all of the force acting at a radius. These two equations have three unknown functions, mass, density, and pressure, so they cannot be solved without adding another equation. For the Sun, the ideal gas can be added, but that introduces a fourth unknown function, which then requires another equation. Jupiter’s interior is not an ideal gas, so the equation of state requires some additional thought.

 

Way back in 1870 (according to Wikipedia), Jonathan Homer Lane “performed the first mathematical analysis of the Sun as a gaseous body. His investigations demonstrated the thermodynamic relations between pressure, temperature, and density of the gas within the Sun, and formed the foundation of what would in the future become the theory of stellar evolution (see Lane-Emden equation).”

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The term polytrope refers to a solution to the Lane-Emden equation in the form P = kp ρ1+1/n, where n is the polytropic index which is typically 3 for the Sun, 1.5 for fully convecting stars, and 1 for fully convecting planets like Jupiter. The polytrope definition provides the third equation in three unknowns which together with some boundary conditions provides a reasonable model of some self-gravitating objects. This animated Wikipedia gif shows the solutions of the Lane-Emden equation for polytropic index from 0.01 to 4.03. The ordinate is density normalized by the central density. For constant mass, the larger the polytropic index the denser the core and the steeper the change in density with radius. The animation pauses for ten seconds at or near polytropic index n = 0, 1 (Jupiter), 1.5 (fully convecting star), 2, 3 (partially convecting Sun), and 4.

Most polytropes do not have an analytical solution, but fortunately the sinc function provides an exact solution for the n = 1 case. In the 19th century and early 20th century, analytical solutions were particularly valuable because computers were not available to provide numerical solutions. Plus it is nice to visualize a function rather than a string of numbers.

Mathcad can solve systems of differential equations in cases that are not too pathological, but Mathcad is especially convenient to evaluate and graph the n = 1 polytrope solution for Jupiter. Given that density has the form of a sinc function and given the total mass enclosed at the surface radius Mrs and surface radius rs of Jupiter, it is easy to calculate the central density ρ0 of Jupiter. Since Jupiter radiates energy to space from a region with non-zero density, pressure, and temperature, we have chosen a sinc function that goes to zero at r = rz which is slightly greater than rs. In fact, rz can be chosen to match a boundary condition such as the density at rs or the central temperature. Since ρ0 is a constant, it can be factored out of the integral of the mass conservation equation. For polytropic index n = 1, the integral can be done symbolically to get an analytical function for M(r). The graph shows the density and mass functions vs. radius once ρ0 is calculated. The internal structure of Jupiter alters the actual density function from the ideal polytrope.

The pressure function P can be plotted as the square of density or sinc squared given the surface pressure Prs and surface density ρrs. The pressure Pg can be calculated from the integral form of the hydrostatic support equation and the graph shows that P = Pg everywhere or in other words, the polytrope satisfies the hydrostatic support condition as expected from a solution to the Lane-Emden equation. The model shows a central pressure of 40 megabars which is in the ball park of the official value of about 80 megabars.

The Gruneisen parameter characterizes temperature variations along an isentrope (reversible adiabat) and defines the thermal state and the convection instability. The temperature profile of the index n = 1 polytrope varies as the density function raised to the Gruneisen parameter ΓG, which is approximately 2/3 for the interior of Jupiter. Conduction and radiation transfer heat no matter how small the temperature gradient is, but convection only occurs in regions where the actual temperature gradient is steeper than the adiabatic temperature gradient. This case is called convective instability.

The adiabatic temperature gradient also depends on two boundary conditions, surface temperature and surface density. These conditions make it possible to estimate the central temperature. The difference between the adiabat and the actual temperature is generally extremely tiny, so the adiabat is a good estimate of the interior temperature except when calculating the actual flow of energy due to convection.

By normalizing the density, pressure, and temperature to their central values, the shapes of the three thermodynamic functions can be compared on a single graph. Compared to the density sinc function, temperature decreases slowly near the center and pressure decreases rapidly near the center.

The energy density of a small volume of material is the product of the specific heat, the mass density, and the temperature. The total energy E enclosed within an interior radius of a sphere is the integral of the energy density. The specific heat of hydrogen is assumed to be 20,000 joules per kg-K.

Since density and temperature lineshapes do not change over time, the energy density lineshape does not change either. Therefore luminosity L which measures the heat flow from the interior at any given radius is proportional to enclosed thermal energy. Luminosity is graphed by scaling surface luminosity Lrs with the normalized energy E, given the total enclosed thermal energy from the previous graph. Lrs is simply the surface area times the Stefan-Boltzmann flux which varies as the fourth power of surface temperature.

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Jupiter's interior and phase diagrams

 

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Dr. Victor Castillo a computational physicist at LLNL says that all models are wrong, but some are useful. So how useful is the polytrope model for the interior of Jupiter? The answer depends on the detailed structure of the planet, not just its average composition. It is believed that most of the interior of Jupiter is not really a molecular gas but a liquid metallic form of hydrogen and helium because there is a plasma phase transition caused by pressure ionization at pressures above about one megabar. Furthermore, there may be a rock and ice core that formed before the planet was massive enough to capture the vast amount of hydrogen and helium gas that it currently contains.

If the rocky icy core formed and still exists, then the actual density profile has the complicated structure graphed by this Mathcad model rather than the sinc function solution of the simpler polytrope model. Most of the volume is similar to the isotrope but the small volume in the central core is about four times denser.

Saturn has one third as much gas as Jupiter even though the core is about the same size indicating that it was unable to capture as much gas in the limited time that gas was abundant in the early solar system. There are also slightly different distributions of materials in the outer layers. The density of Saturn is about half the density of Jupiter, so its surface radius is only slightly smaller. This is a result of the fact that a smaller mass exerts less pressure and therefore a lower density. The lower pressure also produces a proportionally smaller liquid metallic layer. Apart from that the two gas giant planets are quite similar.

The temperature vs. pressure phase diagram shows the presence of a metallic layer at pressures greater than one megabar for intermediate temperatures typical of large regions within Jupiter and Saturn. When the gas giant planets were much hotter, they probably had less mass in the metallic layer. If they had cooled rapidly, they would have acquired a solid hydrogen volume.

The diagram also shows that the smaller ice planets Uranus and Neptune have little if any liquid metal. Since the interior of the Sun is three orders of magnitude hotter than Jupiter, its interior is off the chart and there is no metallic layer where pressures are high. The cooler parts of the Sun do not have high enough pressure to form a metallic layer. The Sun is mostly thermally ionized plasma except near the surface where it is cool enough for molecular or atomic hydrogen.

The blue shaded rectangle is two orders of magnitude high and six orders of magnitude wide. Aligning it with the curve representing Jupiter shows that the interior of Jupiter has a slope of 2/6 or 1/3. This confirms that pressure P varies as temperature T cubed, consistent with the polytrope index and Gruneisen parameter.

The graph is from a 2005 paper by Guillot and the figure caption provides additional description.

The density versus pressure phase diagram shows relationships for several materials in addition to a model of Jupiter’s interior. Note “PPT?” marks the step change where the plasma phase transition is presumed to occur and the “core?” marks one of the step changes for the icy rocky cores are presumed to occur. Nothing beats a published graph with question marks on it to instill confidence in a model!

This blue shaded rectangle is two orders of magnitude high and four orders of magnitude wide. Aligning it with the curve representing Jupiter shows that the interior of Jupiter has a slope of 2/4 or 1/2. This confirms that pressure P varies as density ρ squared, consistent with the polytrope index n = 1.

There are still mysteries about Jupiter that planetary scientists are investigating. The year 2016 promises to be an especially exciting year as the Juno spacecraft arrives at Jupiter around the 4 of July five years after launch. Some of the things we are discussing are related to this mission.

Juno was launched on August 5, 2011 and will arrive at Jupiter around July 4, 2016. Juno will study the planet's composition, gravity field, magnetic field, and magnetosphere. The Greco-Roman god Jupiter drew a veil of clouds around himself to hide his mischief, but his wife, the goddess Juno, was able to peer through the clouds and see Jupiter's true nature. Juno will search for clues about how Jupiter formed, whether the planet has a rocky core, and how the planet's mass is distributed, according to Wikipedia.

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Gas Giants vs. Ice Giants?

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The two gas giant planets are far more massive than the two ice giant planets. The giant planets dwarf the four inner rocky planets.

For a constant density self-gravitating sphere, radius should increase as the cube root of the mass. Hydrostatic pressure compresses the interior of a sphere, increasing its density so that radius increases slower than the cube root. In fact the radius of gas giant exoplanets much larger than Jupiter may be the same or even slightly smaller than Jupiter’s. The Sun and other stars are exceptions because of outflows of energy from thermonuclear fusion. The graph shows the four giant planets and three curves for massive spheres composed of the solar mixture of gases, water, or rock.

Neptune and Uranus are regarded as ice giants rather than gas giants because they only captured modest amounts of hydrogen and helium. They both have rocky cores and layers that consist of H₂O⁺ + NH₄⁺ + OH⁻ ions and molecular H₂ + He + CH₄.

Lodders and Fegley’s book, “The Planetary Scientist’s Companion” shows radial slices of the gas giants complete with composition, structure, temperature, and pressure. “The Planetary Scientist’s Companion” also shows radial slices of the ice giants complete with composition, structure, temperature, and pressure (not the same scale as previous illustration).

Lodders and Fegley’s book, “The Planetary Scientist’s Companion” also shows radial slices of the ice giants complete with composition, structure, temperature, and pressure (not the same scale as previous illustration).

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How could Jupiter form?

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The differences in the mass, composition and structure of planets is related to the environment in which their formation occurred. The first case to consider is Jupiter. The big mystery is how such a massive gas giant could form at all. Jupiter had less than ten million years to capture all of its gas because the Sun expelled most of the gas in the solar system shortly after thermonuclear fusion was initiated in its core. If we could not see Jupiter with our own eyes, we might not believe that such a planet could exist anywhere.

This animation provides an oversimplified explanation of the core accretion gas capture process which is believed to have formed Jupiter in an incredibly brief time. The core accretion gas capture theory assumes that one embryo was much more massive than other nearby planetesimals. Gas capture does not begin until the core accretion exceeds about 10 Earth masses. The growing rocky icy core raises the escape velocity above the thermal velocity of the nebular gas.

As the embryonic planet captures cold gases, they get hot and lose energy. The growing gas envelope compresses the gas to form a layer of “liquid” metallic atomic hydrogen and helium. Gas capture lasted 20,000 years and Jupiter was a million times brighter than it is today due to vast amounts of energy radiated during formation as per the virial theorem. If an object does not radiate energy into space during formation, then the hot gases will not be gravitationally bound and will simply escape.

The graph shows the Hubickyj model predictions of the growth of Jupiter’s mass during its 2.5 million year formation period based on the core-accretion gas-capture theory. The green curve shows the early accretion of rock and ice with no ability to capture and retain gas until the core reached about ten Earth masses which appears to occur after 500 thousand years. At that point, the core continues to grow much more slowly because it has depleted the local supply of rock and ice.

Remarkably the model shows that the gas envelope begins to form leading to a growing total mass over the next two million years. After a total of 2.5 million years, the total mass is so great that an incredibly rapid acceleration in gas capture begins as shown by the nearly vertical line. In order to show the early core accretion process clearly, the graph only displays the curve up to 40 Earth masses.

The graph shows the Hubickyj model predictions of the growth of Jupiter’s mass during its 2.5 million year formation period based on the core-accretion gas-capture theory on a scale that clearly shows the 318 Earth masses of the fully formed Jupiter. This graph shows more clearly than the previous exactly how steep the gas capture curve is.

The graph shows the Hubickyj model predictions of the growth of Jupiter’s mass during the critical 100 thousand year surrounding the period of rapid gas capture. This shows that nearly 300 Earth masses of gas are captured in about 20,000 years at a rate that is nearly linear with time.

The graph shows the Hubickyj model predictions of the luminosity of Jupiter’s surface due to energy loss during its 2.5 million year formation period based on the core-accretion gas-capture theory. The vertical scale is logarithmic and is normalized to the luminosity of the Sun which would be zero on this scale. In other words, the peak gas capture period is characterized by a glow that is only a few hundred times less intense than the Sun. The numbers in the box are some of my calculations to quantify the energy processes in joules.

The graph shows the Hubickyj model predictions of Jupiter’s surface temperature due to energy loss during its 2.5 million year formation period based on the core-accretion gas-capture theory. The vertical scale is once again linear and in Kelvin degrees. The peak gas capture period is characterized by a surface temperature that is nearly 60% of the Sun’s surface temperature. The surface luminosity depends on surface area and the fourth power of the surface temperature.

The graph shows the Hubickyj model predictions of Jupiter’s surface temperature during the critical 100 thousand year surrounding the period of rapid gas capture. The jagged steps in the graph presumably represent the finite resolution of the model simulation.

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Is Jupiter still evolving?

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Is Jupiter still evolving? The formation of Jupiter by core accretion and gas capture ended within ten million years of the formation of the Sun, perhaps as quickly as the 2.5 million years illustrated in the Hubickyj simulation. At this point, the contracting gases had radiated a great deal of energy into space, but the surface was cooling rapidly and the volume was shrinking rapidly as the cooling gases contracted. This process of cooling and contracting released additional energy into space, a process that has continued at an ever slower pace for the entire 4.5 billion year evolutionary history of Jupiter. So yes, Jupiter is still evolving.

The radius of the new born planet was 40% larger than today because the gas envelope was so hot. The internal pressure was lower so the liquid metallic region was somewhat smaller.

As Jupiter cooled, the radius of the planet contracted to 70% of its initial value, the surface area decreased to half its initial value, the volume decreased to one third of its initial value, and the average density increased by a factor of three.

Hubickyj is a co-author of a 2007 paper entitled “On the Luminosity of Young Jupiters “, which modeled the post-formation evolution of the surface radius, surface temperature, and surface luminosity of gas giant planets with masses ranging from one to ten times the value of Jupiter for two different formation scenarios. The three graphs are from this paper. This model has potential value in helping the search for additional exoplanets such as those found by co-author Jonathan Fortney.

This graph shows a model of luminosity vs. time on a log-log plot starting at 0.1 million years and ending at four billion years. As previously shown, the luminosity during peak gas capture is only a few orders of magnitude less than the Sun.

After formation, the luminosity is about one millionth of the Sun and it has since decreased by more than three more orders of magnitude as the planet has cooled and contracted. On the log-log scale, the post-formation luminosity appears to decrease nearly linearly, indicating a sharp decrease in the first 100 million years and tapering off to a very gradual decline over the last four billion years.

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Current surface luminosity is estimated in watts based on the Stefan-Boltzmann equation and current surface radius and temperature. Hubickyj provided the data for my graphs of the formation of Jupiter as well as my graphs of the post-formation evolution in an unpublished private communication similar to work previously published. Four critical events in Jupiter’s history are labeled and their values are estimated in this exercise for students.

This graph shows the model’s plot of temperature vs. time on the same log log plot starting at 0.1 million years and ending at four billion years. The surface temperature during peak gas capture is deliberately very far off scale in order to highlight post-formation temperatures. The surface temperature appears to cool from 550 K to 100 K.

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How does Jupiter's interior lose heat?

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Jupiter receives about 27 times less incident sunlight than Earth. Some of the light is reflected and 8.1 w/m is absorbed and then re-radiated. Unlike Earth, Jupiter radiates much more infrared energy than it absorbs from the Sun – about 5.4 w/m more. The extra energy from the internal radial heat flow as Jupiter cools. In the absence of absorbed sunlight, the surface temperature would be about 100 K and would radiate about 3.5x1017 watts.

Jupiter radiates energy to space that has flowed to the surface from its interior as it cools and contracts. Unlike Earth, most of the infrared energy comes from cooling rather than radioactive decay. How does the surface lose heat to space? Blackbody radiation! In order to maintain the surface temperature that is necessary to radiate energy to space, energy must flow from the interior – does that involve thermal conduction, radiative transport, or thermal convection?

We can rule out thermal conduction and radiative transport as primary contributors to the radial flow of energy in the interior of Jupiter due to low thermal conductivity and high opacity respectively.

Luminosity or energy flow in the interior of a sphere by thermal conduction is proportional to the temperature gradient dT/dr, the thermal conductivity, and the cross sectional area. The thermal conductivity of Jupiter is too low everywhere to transport much energy for any plausible temperature gradient.

Radiative transport is proportional to the gradient of the Stefan-Boltzmann blackbody radiation factor which varies as the fourth power of temperature. The luminosity is also proportional to the area and inversely proportional to opacity as well as to the density of mass which absorbs photons. Unlike the Sun, Jupiter’s interior opacity is too high everywhere to transport much energy for any plausible temperature gradient.

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Even the free electrons in the metallic layer cannot conduct heat efficiently because they scatter before they can travel very far.

Opacity in general is a function of many factors including composition, density, and temperature.

Even if these heat transfer processes were significant today, they could not have been significant billions of years ago when the outflow from the interior was three orders of magnitude greater. For thermal conduction, a temperature gradient three orders of magnitude greater would correspond to a central temperature one thousand times larger and that much more heat to dissipate in order to cool down, not to mention no way to produce such a heat from gravitational gas capture. For radiative transport, the problem is the same although the temperature gradient and central temperature would “only” need to be roughly ten times greater to produce a three order of magnitude greater outflow. All of these arguments are somewhat oversimplified as other properties also vary with temperature, but you get the picture.

 

Fortunately thermal convection provided a powerful mechanism for the radial flow of energy in the interior of Jupiter during its formation and early thermal evolution. Thermal convection continues to provide the energy to the surface that maintains the current surface temperature which radiates significantly more infrared radiation to space than would be expected from the re-radiation of absorbed sunlight alone.

Unlike thermal conduction and radiative transport, luminosity in the interior of a sphere by thermal convection is NOT proportional to the temperature gradient dT/dr. Unlike other forms of energy flow, convective heat flow can be zero even in the presence of a temperature gradient.

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As long as the temperature gradient is less than the adiabatic temperature gradient, the interior fluid is stable against convective flow. However a very slight gradient in excess of the adiabat is enough to trigger a convection instability that transports significant amounts of thermal energy. As the excess gradient grows, the flow increases rapidly even though the absolute temperature gradient and temperature are barely different than the adiabat. Because the convective heat flow is proportional to the difference in two gradients to the 3/2 power, the flow can vary by several orders of magnitude with very little change in temperature. So a hot newly formed planet can supply enormous amounts of interior heat to maintain a very high surface temperature and luminosity, but as it cools, its outflows decrease dramatically, altering the surface temperature and therefore the optical properties of the atmosphere (as clouds condense) leading to a dramatic decrease in luminosity. The luminosity equation in this chart is based on an equation on page 44 in the book “Jupiter” edited by Bagenal et al 2006. The del operator uses the Arabic nabla symbol ∇ which means arrow (or some say harp). The difference in gradients is actually the difference in del operators which is related to the difference in gradients by the equation that involves mixing length theory. The del function itself is actually the derivative on the log of temperature with respect to the log of pressure. The form of the equation also depends on the area, Gruneisen parameter, specific heat, polytrope coefficient, density, and absolute temperature. The temperature gradient differs from the del operator by a factor of temperature divided by mixing length which is roughly comparable to the radius of the planet.

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Jupiter's atmosphere controls global cooling

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The semi-log graph shows that the vertical structure of the atmosphere of Jupiter exhibits a temperature gradient with increasing altitude in the convecting troposphere from about 430K to about 130K above the cloud tops as pressure decreases from about 20 bars at the bottom of the troposphere to about 0.1 bar at the 50 km altitude tropopause. Convection keeps temperatures from deviating much from the adiabatic gradient. Clouds typically scatter infrared radiation keeping deeper warmer atmospheric layers from radiating their energy to space and trapping heat to reduce planetary cooling rates. Large molecules like ammonia, ammonium hydrosulfide, and water condense to form clouds at three distinct temperatures and altitudes. At extremely high altitudes (hundreds of kilometers), the temperature of the very thin (microbars) thermosphere is elevated by absorption of incident ultraviolet sunlight.

The atmosphere influences the surface temperature and radiative cooling rate by trapping infrared heat that cannot escape from lower altitudes due to clouds and other atmospheric constituents. The first graph shows a more detailed plot of the temperatures, pressures, and altitudes of four cloud layers in the troposphere of Jupiter.

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summary

At this point you should be able to answer the key questions addressed in this slide lecture:

1 - Why is a gas giant planet mostly liquid metallic hydrogen and helium?

2 - Why does Jupiter radiate more heat than it absorbs from the Sun?

3 - How did Jupiter form before the solar system lost most of its gases?

4 - How did Jupiter lose heat quickly in its youth and slowly as it evolved?

The short answers are:

1 - pressure ionization freeing electrons in a moderately hot volume

2 - internal heat flows to the surface of the still cooling planet

3 - rapid gas capture once Jupiter reached a critical size of about twenty Earth masses following rapid growth of a ten Earth mass core accreted from nearby rock and ices.

4 - convective heat flow rates scale as the three halves power of the difference between the actual temperature gradient and the adiabatic temperature gradient.

Notes for 48 Jupiter lecture slides

The composition, structure, formation, and evolution of the Sun and Jupiter

© Bob Field 2016-0225

Approximately 99.99% of the mass of the solar system is concentrated in three giant balls of gas, the Sun, Jupiter, and Saturn. They are mostly hydrogen and helium but contain small amounts of heavier elements, mostly oxygen, carbon, nitrogen, iron, silicon, and magnesium. Everything else in the solar system comprises about 0.01% of its mass. Jupiter and his little brother Saturn are similar, so we will focus on Jupiter.

The composition and structure of Jupiter differ from the Sun and from the other giant planets. After we investigate the differences, we will consider the formation and post-formation evolutionary processes that account for the differences.

The Sun and Jupiter are the two largest objects in our solar system. They have some interesting similarities including composition, being predominately giant self-gravitating balls of hydrogen and helium gas, but they are strikingly different in many ways as we shall see. The composition and structure of Jupiter differ from the Sun and from the other giant planets. After we investigate the differences, we will consider the formation and post-formation evolutionary processes that account for the differences.

The composition of the Sun, planets, and smaller objects in the solar system is related to the composition of the primordial solar nebula from which they formed. The composition of the Sun and the two gas giant planets looks very much like the primordial nebula, but the sequence of physical processes that formed everything else varied the composition of planets. The two ice giant planets formed from dust and ice with very little hydrogen and helium gas and the four tiny inner planets look like rocks covered with thin layers of fluids.

The composition and density of Jupiter happens to be very similar to the Sun, but the enormous difference in their masses produces bodies with vastly different properties. In the absence of thermonuclear fusion, the Sun would have cooled and contracted to form a very dense small dim object.Remarkably the Sun would not be dense and hot enough for nucleosynthesis to transform hydrogen into helium were it not for a quantum tunneling process that allows protons to overcome their Coulombic repulsion long enough to fuse. And even with that advantage, fusion would not proceed if the Sun was not so opaque that it traps heat in the dense core for tens of millions of years. The opacity itself is due to the extreme interior temperatures that have fully ionized the hydrogen and helium atoms producing free electrons that are very quick to absorb and scatter photons.

The opacity does not prevent photons in the lower 70% of the Sun’s radius from slowly transporting energy from the core toward the surface because the temperatures and temperature gradients are very large. The photons in the last 30% of the distance to the surface are not hot enough to overcome the opacity, so convective heat flow is triggered to maintain a nearly adiabatic temperature gradient, resulting in a very rapid energy flow for the remaining journey to the surface where the thin relatively transparent photosphere enables radiant energy to flow to space.

As a result of the fusion process, the core has actually grown hotter and denser as the Sun has evolved over billions of years, resulting in an increase in fusion over time despite an overall decrease in available hydrogen fuel. The growing outflow of energy from the core elevates the temperature of the Sun outside of the core causing it to expand and increase the Sun’s surface area and temperature over time. So the Sun gets bigger and brighter over time – and its average density decreases as its core density grows.

Unlike the Sun, Jupiter is cooling and contracting because it does not have a powerful internal energy source like the Sun’s core fusion. The atmosphere of Jupiter controls the rate at which the planet cools. Clouds formed from condensed gases absorb radiant energy from deeper hotter layers of the atmosphere so that only the coolest layers of the atmosphere radiate into space. The surface temperature is maintained by an efficient convective heat transport that is slightly steeper than the adiabatic temperature gradient that would be present if the convective instability were not triggered.

Why is the interior of Jupiter so opaque to radiant energy? Its interior is 1000 times cooler than the Sun’s interior and not sufficient to ionize the gases, but the pressures are so high – more than a million times the Earth’s atmosphere (but far less than the pressures in the Sun) – that the gases are pressure ionized rather than thermally ionized – they are squeezed out of their atomic orbitals and form – not a plasma like the Sun – but a metal with free electrons. Metals are opaque. Metallic hydrogen is opaque like other metals but it is not a very good thermal conductor because the free electrons scatter before they can carry thermal energy very far. So convection is the mode of heat transport throughout Jupiter.

The Sun’s interior is very dense but it is still an ideal gas because the particles are too hot that to interact for very long. This is very convenient for study because the ideal gas law can be applied. The metallic hydrogen and helium which forms most of the mass of Jupiter is not an ideal gas, it is not even a gas, it is such a dense fluid that it is often called a liquid – so the gas giant planets are mostly liquid not gas.

There is a set of four differential equations along with suitable boundary conditions that characterize the state of the interior of stars like the Sun. Several of the equations can describe interiors of planets. Mass M is a function of radius within an object. The mass enclosed within a given radius from the center of a sphere is the integral of the radially varying density function over the volume of the sphere enclosed by the given radius. The boundary condition at the surface is that M equals the entire mass of the sphere, a measurable quantity. The mass conservation equation is simply the differential equation form of the integral.

A self-gravitating spherical mass is in equilibrium when the hydrostatic support from the pressure gradient at a given radius equals the gravitational force on a volume of mass at that radius. The integral form is from the surface down to the radius (rather than from the center to the radius) to account for all of the force acting at a radius.

These two equations have three unknown functions, mass, density, and pressure, so they cannot be solved without adding another equation. For the Sun, the ideal gas can be added, but that introduces a fourth unknown function, which then requires another equation. Jupiter’s interior is not an ideal gas, so the equation of state requires some additional thought.

Way back in 1870 (according to Wikipedia), Jonathan Homer Lane “performed the first mathematical analysis of the Sun as a gaseous body. His investigations demonstrated the thermodynamic relations between pressure, temperature, and density of the gas within the Sun, and formed the foundation of what would in the future become the theory of stellar evolution (see Lane-Emden equation).”

The term polytrope refers to a solution to the Lane-Emden equation in the form P = k ρ1+1/n, where n is the polytropic index which is typically 3 for the Sun, 1.5 for fully convecting stars, and 1 for fully convecting planets like Jupiter. The polytrope definition provides the third equation in three unknowns which together with some boundary conditions provides a reasonable model of some self-gravitating objects.

This animated Wikipedia gif shows the solutions of the Lane-Emden equation for polytropic index from 0.01 to 4.03. The ordinate is density normalized by the central density. For constant mass, the larger the polytropic index the denser the core and the steeper the change in density with radius. The animation pauses for ten seconds at or near polytropic index n = 0, 1 (Jupiter), 1.5 (fully convecting star), 2, 3 (partially convecting Sun), and 4.

Most polytropes do not have an analytical solution, but fortunately the sinc function provides an exact solution for the n = 1 case. In the 19th century and early 20th century, analytical solutions were particularly valuable because computers were not available to provide numerical solutions. Plus it is nice to visualize a function rather than a string of numbers.

Mathcad can solve systems of differential equations in cases that are not too pathological, but Mathcad is especially convenient to evaluate and graph the n = 1 polytrope solution for Jupiter. Given that density has the form of a sinc function and given the total mass enclosed at the surface radius Mrs and surface radius rs of Jupiter, it is easy to calculate the central density ρ0 of Jupiter. Since Jupiter radiates energy to space from a region with non-zero density, pressure, and temperature, we have chosen a sinc function that goes to zero at r = rz which is slightly greater than rs. In fact, rz can be chosen to match a boundary condition such as the density at rs or the central temperature. Since ρ0 is a constant, it can be factored out of the integral of the mass conservation equation. For polytropic index n = 1, the integral can be done symbolically to get an analytical function for M(r). The graph shows the density and mass functions vs. radius once ρ0 is calculated. The internal structure of Jupiter alters the actual density function from the ideal polytrope.

The pressure function P can be plotted as the square of density or sinc squared given the surface pressure Prs and surface density ρrs. The pressure Pg can be calculated from the integral form of the hydrostatic support equation and the graph shows that P = Pg everywhere or in other words, the polytrope satisfies the hydrostatic support condition as expected from a solution to the Lane-Emden equation. The model shows a central pressure of 40 megabars which is in the ball park of the official value of about 80 megabars.

The Gruneisen parameter characterizes temperature variations along an isentrope (reversible adiabat) and defines the thermal state and the convection instability. The temperature profile of the index n = 1 polytrope varies as the density function raised to the Gruneisen parameter ΓG, which is approximately 2/3 for the interior of Jupiter. Conduction and radiation transfer heat no matter how small the temperature gradient is, but convection only occurs in regions where the actual temperature gradient is steeper than the adiabatic temperature gradient. This case is called convective instability.

The adiabatic temperature gradient also depends on two boundary conditions, surface temperature and surface density. These conditions make it possible to estimate the central temperature. The difference between the adiabat and the actual temperature is generally extremely tiny, so the adiabat is a good estimate of the interior temperature except when calculating the actual flow of energy due to convection.

By normalizing the density, pressure, and temperature to their central values, the shapes of the three thermodynamic functions can be compared on a single graph. Compared to the density sinc function, temperature decreases slowly near the center and pressure decreases rapidly near the center.

The energy density of a small volume of material is the product of the specific heat, the mass density, and the temperature. The total energy E enclosed within an interior radius of a sphere is the integral of the energy density. The specific heat of hydrogen is assumed to be 20,000 joules per kg-K.

Since density and temperature lineshapes do not change over time, the energy density lineshape does not change either. Therefore luminosity L which measures the heat flow from the interior at any given radius is proportional to enclosed thermal energy. Luminosity is graphed by scaling surface luminosity Lrs with the normalized energy E, given the total enclosed thermal energy from the previous graph. Lrs is simply the surface area times the Stefan-Boltzmann flux which varies as the fourth power of surface temperature.

Dr. Victor Castillo a computational physicist at LLNL says that all models are wrong, but some are useful. So how useful is the polytrope model for the interior of Jupiter? The answer depends on the detailed structure of the planet, not just its average composition. It is believed that most of the interior of Jupiter is not really a molecular gas but a liquid metallic form of hydrogen and helium because there is a plasma phase transition caused by pressure ionization at pressures above about one megabar. Furthermore, there may be a rock and ice core that formed before the planet was massive enough to capture the vast amount of hydrogen and helium gas that it currently contains.

If the rocky icy core formed and still exists, then the actual density profile has the complicated structure graphed by this Mathcad model rather than the sinc function solution of the simpler polytrope model. Most of the volume is similar to the isotrope but the small volume in the central core is about four times denser.

Saturn has one third as much gas as Jupiter even though the core is about the same size indicating that it was unable to capture as much gas in the limited time that gas was abundant in the early solar system. There are also slightly different distributions of materials in the outer layers. The density of Saturn is about half the density of Jupiter, so its surface radius is only slightly smaller. This is a result of the fact that a smaller mass exerts less pressure and therefore a lower density. The lower pressure also produces a proportionally smaller liquid metallic layer. Apart from that the two gas giant planets are quite similar.

The temperature vs. pressure phase diagram shows the presence of a metallic layer at pressures greater than one megabar for intermediate temperatures typical of large regions within Jupiter and Saturn. When the gas giant planets were much hotter, they probably had less mass in the metallic layer. If they had cooled rapidly, they would have acquired a solid hydrogen volume.

The diagram also shows that the smaller ice planets Uranus and Neptune have little if any liquid metal. Since the interior of the Sun is three orders of magnitude hotter than Jupiter, its interior is off the chart and there is no metallic layer where pressures are high. The cooler parts of the Sun do not have high enough pressure to form a metallic layer. The Sun is mostly thermally ionized plasma except near the surface where it is cool enough for molecular or atomic hydrogen.

The blue shaded rectangle is two orders of magnitude high and six orders of magnitude wide. Aligning it with the curve representing Jupiter shows that the interior of Jupiter has a slope of 2/6 or 1/3. This confirms that pressure P varies as temperature T cubed, consistent with the polytrope index and Gruneisen parameter.

The graph is from a 2005 paper by Guillot and the figure caption provides additional description.

The density versus pressure phase diagram shows relationships for several materials in addition to a model of Jupiter’s interior. Note “PPT?” marks the step change where the plasma phase transition is presumed to occur and the “core?” marks one of the step changes for the icy rocky cores are presumed to occur. Nothing beats a published graph with question marks on it to instill confidence in a model!

This blue shaded rectangle is two orders of magnitude high and four orders of magnitude wide. Aligning it with the curve representing Jupiter shows that the interior of Jupiter has a slope of 2/4 or 1/2. This confirms that pressure P varies as density ρ squared, consistent with the polytrope index n = 1.

There are still mysteries about Jupiter that planetary scientists are investigating. The year 2016 promises to be an especially exciting year as the Juno spacecraft arrives at Jupiter around the 4 of July five years after launch. Some of the things we are discussing are related to this mission.

Juno was launched on August 5, 2011 and will arrive at Jupiter around July 4, 2016. Juno will study the planet's composition, gravity field, magnetic field, and magnetosphere. The Greco-Roman god Jupiter drew a veil of clouds around himself to hide his mischief, but his wife, the goddess Juno, was able to peer through the clouds and see Jupiter's true nature. Juno will search for clues about how Jupiter formed, whether the planet has a rocky core, and how the planet's mass is distributed, according to Wikipedia.

The two gas giant planets are far more massive than the two ice giant planets. The giant planets dwarf the four inner rocky planets.

For a constant density self-gravitating sphere, radius should increase as the cube root of the mass. Hydrostatic pressure compresses the interior of a sphere, increasing its density so that radius increases slower than the cube root. In fact the radius of gas giant exoplanets much larger than Jupiter may be the same or even slightly smaller than Jupiter’s. The Sun and other stars are exceptions because of outflows of energy from thermonuclear fusion. The graph shows the four giant planets and three curves for massive spheres composed of the solar mixture of gases, water, or rock.

Neptune and Uranus are regarded as ice giants rather than gas giants because they only captured modest amounts of hydrogen and helium. They both have rocky cores and layers that consist of H₂O⁺ + NH₄⁺ + OH⁻ ions and molecular H₂ + He + CH₄.

Lodders and Fegley’s book, “The Planetary Scientist’s Companion” shows radial slices of the gas giants complete with composition, structure, temperature, and pressure. “The Planetary Scientist’s Companion” also shows radial slices of the ice giants complete with composition, structure, temperature, and pressure (not the same scale as previous illustration).

Lodders and Fegley’s book, “The Planetary Scientist’s Companion” also shows radial slices of the ice giants complete with composition, structure, temperature, and pressure (not the same scale as previous illustration).

The differences in the mass, composition and structure of planets is related to the environment in which their formation occurred. The first case to consider is Jupiter. The big mystery is how such a massive gas giant could form at all. Jupiter had less than ten million years to capture all of its gas because the Sun expelled most of the gas in the solar system shortly after thermonuclear fusion was initiated in its core. If we could not see Jupiter with our own eyes, we might not believe that such a planet could exist anywhere.

This animation provides an oversimplified explanation of the core accretion gas capture process which is believed to have formed Jupiter in an incredibly brief time. The core accretion gas capture theory assumes that one embryo was much more massive than other nearby planetesimals. Gas capture does not begin until the core accretion exceeds about 10 Earth masses. The growing rocky icy core raises the escape velocity above the thermal velocity of the nebular gas.

As the embryonic planet captures cold gases, they get hot and lose energy. The growing gas envelope compresses the gas to form a layer of “liquid” metallic atomic hydrogen and helium. Gas capture lasted 20,000 years and Jupiter was a million times brighter than it is today due to vast amounts of energy radiated during formation as per the virial theorem. If an object does not radiate energy into space during formation, then the hot gases will not be gravitationally bound and will simply escape.

The graph shows the Hubickyj model predictions of the growth of Jupiter’s mass during its 2.5 million year formation period based on the core-accretion gas-capture theory. The green curve shows the early accretion of rock and ice with no ability to capture and retain gas until the core reached about ten Earth masses which appears to occur after 500 thousand years. At that point, the core continues to grow much more slowly because it has depleted the local supply of rock and ice.

Remarkably the model shows that the gas envelope begins to form leading to a growing total mass over the next two million years. After a total of 2.5 million years, the total mass is so great that an incredibly rapid acceleration in gas capture begins as shown by the nearly vertical line. In order to show the early core accretion process clearly, the graph only displays the curve up to 40 Earth masses.

The graph shows the Hubickyj model predictions of the growth of Jupiter’s mass during its 2.5 million year formation period based on the core-accretion gas-capture theory on a scale that clearly shows the 318 Earth masses of the fully formed Jupiter. This graph shows more clearly than the previous exactly how steep the gas capture curve is.

The graph shows the Hubickyj model predictions of the growth of Jupiter’s mass during the critical 100 thousand year surrounding the period of rapid gas capture. This shows that nearly 300 Earth masses of gas are captured in about 20,000 years at a rate that is nearly linear with time.

The graph shows the Hubickyj model predictions of the luminosity of Jupiter’s surface due to energy loss during its 2.5 million year formation period based on the core-accretion gas-capture theory. The vertical scale is logarithmic and is normalized to the luminosity of the Sun which would be zero on this scale. In other words, the peak gas capture period is characterized by a glow that is only a few hundred times less intense than the Sun. The numbers in the box are some of my calculations to quantify the energy processes in joules.

The graph shows the Hubickyj model predictions of Jupiter’s surface temperature due to energy loss during its 2.5 million year formation period based on the core-accretion gas-capture theory. The vertical scale is once again linear and in Kelvin degrees. The peak gas capture period is characterized by a surface temperature that is nearly 60% of the Sun’s surface temperature. The surface luminosity depends on surface area and the fourth power of the surface temperature.

The graph shows the Hubickyj model predictions of Jupiter’s surface temperature during the critical 100 thousand year surrounding the period of rapid gas capture. The jagged steps in the graph presumably represent the finite resolution of the model simulation.

Is Jupiter still evolving? The formation of Jupiter by core accretion and gas capture ended within ten million years of the formation of the Sun, perhaps as quickly as the 2.5 million years illustrated in the Hubickyj simulation. At this point, the contracting gases had radiated a great deal of energy into space, but the surface was cooling rapidly and the volume was shrinking rapidly as the cooling gases contracted. This process of cooling and contracting released additional energy into space, a process that has continued at an ever slower pace for the entire 4.5 billion year evolutionary history of Jupiter. So yes, Jupiter is still evolving.

The radius of the new born planet was 40% larger than today because the gas envelope was so hot. The internal pressure was lower so the liquid metallic region was somewhat smaller.

As Jupiter cooled, the radius of the planet contracted to 70% of its initial value, the surface area decreased to half its initial value, the volume decreased to one third of its initial value, and the average density increased by a factor of three.

Hubickyj is a co-author of a 2007 paper entitled “On the Luminosity of Young Jupiters “, which modeled the post-formation evolution of the surface radius, surface temperature, and surface luminosity of gas giant planets with masses ranging from one to ten times the value of Jupiter for two different formation scenarios. The three graphs are from this paper. This model has potential value in helping the search for additional exoplanets such as those found by co-author Jonathan Fortney.

This graph shows a model of luminosity vs. time on a log-log plot starting at 0.1 million years and ending at four billion years. As previously shown, the luminosity during peak gas capture is only a few orders of magnitude less than the Sun.

After formation, the luminosity is about one millionth of the Sun and it has since decreased by more than three more orders of magnitude as the planet has cooled and contracted. On the log-log scale, the post-formation luminosity appears to decrease nearly linearly, indicating a sharp decrease in the first 100 million years and tapering off to a very gradual decline over the last four billion years. Current surface luminosity is estimated in watts based on the Stefan-Boltzmann equation and current surface radius and temperature.

Hubickyj provided the data for my graphs of the formation of Jupiter as well as my graphs of the post-formation evolution in an unpublished private communication similar to work previously published.

Four critical events in Jupiter’s history are labeled and their values are estimated in this exercise for students.

This graph shows the model’s plot of temperature vs. time on the same log log plot starting at 0.1 million years and ending at four billion years. The surface temperature during peak gas capture is deliberately very far off scale in order to highlight post-formation temperatures. The surface temperature appears to cool from 550 K to 100 K.

Jupiter receives about 27 times less incident sunlight than Earth. Some of the light is reflected and 8.1 w/m is absorbed and then re-radiated. Unlike Earth, Jupiter radiates much more infrared energy than it absorbs from the Sun – about 5.4 w/m more. The extra energy from the internal radial heat flow as Jupiter cools. In the absence of absorbed sunlight, the surface temperature would be about 100 K and would radiate about 3.5x1017 watts.

Jupiter radiates energy to space that has flowed to the surface from its interior as it cools and contracts. Unlike Earth, most of the infrared energy comes from cooling rather than radioactive decay. How does the surface lose heat to space? Blackbody radiation! In order to maintain the surface temperature that is necessary to radiate energy to space, energy must flow from the interior – does that involve thermal conduction, radiative transport, or thermal convection?

We can rule out thermal conduction and radiative transport as primary contributors to the radial flow of energy in the interior of Jupiter due to low thermal conductivity and high opacity respectively.

Luminosity or energy flow in the interior of a sphere by thermal conduction is proportional to the temperature gradient dT/dr and inversely proportional to the resistance to conduction. The inverse of the resistance to conduction is proportional to the thermal conductivity and to the cross sectional area. The thermal conductivity of Jupiter is too low everywhere to transport much energy for any plausible temperature gradient. Even the free electrons in the metallic layer cannot conduct heat efficiently because they scatter before they can travel very far.

Radiative transport is proportional to the gradient of the Stefan-Boltzmann blackbody radiation factor which varies as the fourth power of temperature. The luminosity is also proportional to the area and inversely proportional to opacity as well as to the density of mass which absorbs photons. Unlike the Sun, Jupiter’s interior opacity is too high everywhere to transport much energy for any plausible temperature gradient. Opacity in general is a function of many factors including composition, density, and temperature.

Even if these heat transfer processes were significant today, they could not have been significant billions of years ago when the outflow from the interior was three orders of magnitude greater. For thermal conduction, a temperature gradient three orders of magnitude greater would correspond to a central temperature one thousand times larger and that much more heat to dissipate in order to cool down, not to mention no way to produce such a heat from gravitational gas capture. For radiative transport, the problem is the same although the temperature gradient and central temperature would “only” need to be roughly ten times greater to produce a three order of magnitude greater outflow. All of these arguments are somewhat oversimplified as other properties also vary with temperature, but you get the picture.

Fortunately thermal convection provided a powerful mechanism for the radial flow of energy in the interior of Jupiter during its formation and early thermal evolution. Thermal convection continues to provide the energy to the surface that maintains the current surface temperature which radiates significantly more infrared radiation to space than would be expected from the re-radiation of absorbed sunlight alone.

Unlike thermal conduction and radiative transport, luminosity in the interior of a sphere by thermal convection is NOT proportional to the temperature gradient dT/dr. Unlike other forms of energy flow, convective heat flow can be zero even in the presence of a temperature gradient. As long as the temperature gradient is less than the adiabatic temperature gradient, the interior fluid is stable against convective flow. However a very slight gradient in excess of the adiabat is enough to trigger a convection instability that transports significant amounts of thermal energy.

As the excess gradient grows, the flow increases rapidly even though the absolute temperature gradient and temperature are barely different than the adiabat. Because the convective heat flow is proportional to the difference in two gradients to the 3/2 power, the flow can vary by several orders of magnitude with very little change in temperature. So a hot newly formed planet can supply enormous amounts of interior heat to maintain a very high surface temperature and luminosity, but as it cools, its outflows decrease dramatically, altering the surface temperature and therefore the optical properties of the atmosphere (as clouds condense) leading to a dramatic decrease in luminosity.

The luminosity equation in this chart is based on an equation on page 44 in the book “Jupiter” edited by Bagenal et al 2006. The del operator uses the Arabic nabla symbol ∇ which means arrow (or some say harp). The difference in gradients is actually the difference in del operators which is related to the difference in gradients by the equation that involves mixing length theory. The del function itself is actually the derivative on the log of temperature with respect to the log of pressure. The form of the equation also depends on the area, Gruneisen parameter, specific heat, polytrope coefficient, density, and absolute temperature. The temperature gradient differs from the del operator by a factor of temperature divided by mixing length which is roughly comparable to the radius of the planet.

The semi-log graph shows that the vertical structure of the atmosphere of Jupiter exhibits a temperature gradient with increasing altitude in the convecting troposphere from about 430K to about 130K above the cloud tops as pressure decreases from about 20 bars at the bottom of the troposphere to about 0.1 bar at the 50 km altitude tropopause. Convection keeps temperatures from deviating much from the adiabatic gradient. Clouds typically scatter infrared radiation keeping deeper warmer atmospheric layers from radiating their energy to space and trapping heat to reduce planetary cooling rates. Large molecules like ammonia, ammonium hydrosulfide, and water condense to form clouds at three distinct temperatures and altitudes. At extremely high altitudes (hundreds of kilometers), the temperature of the very thin (microbars) thermosphere is elevated by absorption of incident ultraviolet sunlight.

The atmosphere influences the surface temperature and radiative cooling rate by trapping infrared heat that cannot escape from lower altitudes due to clouds and other atmospheric constituents. The first graph shows a more detailed plot of the temperatures, pressures, and altitudes of four cloud layers in the troposphere of Jupiter.

The second graph shows the flux vs. altitude relative to the tropopause. The atmosphere above the tropopause is relatively transparent to most wavelengths in both the solar spectrum and the infrared blackbody spectrum. Orange arrows mark paths of incident and reflected sunlight in the solar spectrum. The blue arrow shows the upwelling energy due to thermal convection from the cooling interior which goes to zero at the troposphere where infrared energy can escape to space. The red arrow shows the net outgoing infrared flux which rises sharply at the tropopause. The incident and reflected sunlight also decrease sharply below the tropopause. The curves represent global averages; the solar spectrum in particular varies greatly between night and day and from Equator to poles.

At this point you should be able to answer the key questions addressed in this slide lecture:

1 - Why is a gas giant planet mostly liquid metallic hydrogen and helium?

2 - Why does Jupiter radiate more heat than it absorbs from the Sun?

3 - How did Jupiter form before the solar system lost most of its gases?

4 - How did Jupiter lose heat quickly in its youth and slowly as it evolved?

The short answers are:

1 - pressure ionization freeing electrons in a moderately hot volume

2 - internal heat flows to the surface of the still cooling planet

3 - rapid gas capture once Jupiter reached a critical size of about twenty Earth masses following rapid growth of a ten Earth mass core accreted from nearby rock and ices.

4 - convective heat flow rates scale as the three halves power of the difference between the actual temperature gradient and the adiabatic temperature gradient.

 

My final remark:

 

To put the whole thing in perspective, the density of Jupiter is similar to the Sun, but its mass and volume are 1000 times smaller. Saturn is three times smaller than Jupiter and the other giant planets are far smaller even though they dwarf the four rocky inner planets of Mercury, Venus, Earth, and Mars.

 

 

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