Source: Hydrostatic Equilibrium on Wikipedia A volume of fluid, which can be a gas or a liquid, is said to be in hydrostatic equilibrium when the downward force exerted by gravity is balanced by an upward force exerted by the pressure of the fluid. For example, the Earth’s atmosphere is pulled downwards by gravity, but toward the surface the air is compressed by the weight of all the air above, so the air’s density increases from the top of the atmosphere to the Earth’s surface. This density difference means that air pressure decreases with altitude so that the upward pressure from below is greater than the downward pressure from above and this net upward force balances the downward force of gravity, keeping the atmosphere at a more or less constant height. When a volume of fluid is not in hydrostatic equilibrium it must contract if the gravitational force exceeds the pressure, or expand if the internal pressure is greater. This concept can be expressed as the hydrostatic equilibrium equation. It is usually stated as dp/dz = −gρ and applies to a layer of fluid within a larger volume in hydrostatic equilibrium, where dp is the change in pressure within the layer, dz is the thickness of the layer, g is the acceleration due to gravity and ρ is the density of the fluid. The equation can be used to calculate, for example, the pressure within a planetary atmosphere at a given height above the surface. A volume of gas in space, such as a large cloud of hydrogen, will initially contract due to gravity, with its pressure increasing toward the center. The contraction will continue until there is an outward force equal to the inward gravitational force. This is normally the point when the pressure at the center is so great that the hydrogen nuclei fuse together to produce helium in a process called nuclear fusion that releases huge amounts of energy, giving birth to a star. The resulting heat increases the pressure of the gas, producing an outward force to balance the inward gravitational force, so that the star will be in hydrostatic equilibrium. In the event of gravity increasing, perhaps through more gas falling into the star, the density and temperature of the gas will also increase, providing more outward pressure and maintaining the equilibrium. Stars remain in hydrostatic equilibrium over long periods, typically several billion years, but eventually they will run out of hydrogen and begin fusing progressively heavier elements. These changes temporarily put the star out of equilibrium, causing expansion or contraction until a new equilibrium is established. Iron cannot be fused into heavier elements, as this would require more energy than the process would produce, so when all the star’s nuclear fuel has eventually transformed into iron, no further fusion can take place and the star collapses. This might leave a solid iron core, a neutron star or a black hole, depending on the mass of the star. In the case of a black hole, no known physical process can generate sufficient internal pressure to halt gravitational collapse, so hydrostatic equilibrium cannot be achieved and it is thought that the star contracts to a point of infinite density known as a singularity. Solar system objects in hydrostatic equilibrium This entry was posted in Uncategorized. Bookmark the permalink.
By: Mike Luciuk, Jeremy Carlo INTRODUCTION During formation of the solar system 4.6 billion years ago, a cold molecular gas and dust cloud began to contract around its denser portions. As the cloud contracted, its rotation rate increased and it evolved to a disk‐like configuration. Over time, the proto‐Sun and proto‐ planets formed. Figure 1 illustrates planet compositions and temperatures versus proto‐Sun distance. As illustrated, four terrestrial planets and many smaller rocky bodies were formed closest to the proto‐Sun, then as temperatures fell with distance, the four gas giants. Finally, beyond Neptune, many small bodies formed composed of a substantial percentage of various ices. The temperature decrease with proto‐Sun distance resulted in significant composition variations of these three classes of solar system bodies. The resulting variations in mass, composition, and density produced major differences in states of hydrostatic equilibrium and differentiation. Microsoft Word – tut38 Hydrostatic Equilibrium And Planetary Differentiation.doc HYDROSTATIC EQUILIBRIUM An astronomical body is in a state of hydrostatic equilibrium (HE) when its self gravitational force is balanced by its internal pressure; the body is neither expanding nor contracting. From a technical perspective, a body in HE will assume a spherical shape to minimize gravitational potential since any deviations from sphericity increases gravitational energy. The size of deviations is affected by the body’s mass; the larger the mass, the smaller the deviations must be, all else being equal. However, its actual form depends on its rotation rate, mass, material strength, geologic activity and its history. Figure 2 illustrates the balance between the two forces with an equation quantifying their magnitudes.  Solving the equation in Figure 2 for the pressure at the center of a body in HE, Pc = 3GM 2 / 8πR4 (1) Where Pc is the pressure at the center in pascals, newtons/m2. (1 atmosphere = 101,325 pascals) G is the gravitational constant, 6.673 x 10‐11 m3/kg s2 M is the body’s mass in kg. R is the body’s radius in m. Microsoft Word – tut38 Hydrostatic Equilibrium And Planetary Differentiation.doc Equation 1 only gives approximate central pressure because it assumes a constant density throughout the body. It can offer rough results for terrestrial bodies, but badly underestimates pressures in gaseous planets or the Sun. The equation forecasts Earth’s central pressure to be about 1.7 million atmospheres, which is about half the value estimated by more sophisticated methods. The equation predicts Pluto’s central pressure to be about 6,440 atmospheres, about 1/250th forecasted for Earth. Wikipedia has an interesting article listing all known solar system bodies in HE (http://en.wikipedia.org/wiki/List_of_Solar_System_objects_in_hydrostatic_equilibrium). They include the Sun, eight major planets, five dwarf planets, and nineteen planetary satellites. Also listed are ten plutoids, possibly large enough to be in HE, but not yet recognized by the IAU as dwarf planets. The largest solar system body listed is the Sun, of course. The smallest is Mimas, a satellite of Saturn whose spheroidal form has been shaped by the planet’s tidal forces. Mimas is an icy body with a diameter of about 400 km and density slightly greater than water. It can be difficult to determine the shape of distant KBOs by optical imaging. In 2008, the IAU announced that a TNO with an absolute magnitude of +1 or brighter has sufficient size to be in HE and be classified as a plutoid. Recall that a body’s absolute magnitude H, is the visual magnitude an observer would record if the asteroid was located 1 astronomical unit (AU) from Earth and 1 AU from the Sun at a zero phase angle. Astronomers can estimate TNO sizes based on H magnitudes and assumptions of albedo. Table 1 shows diameter (km) by absolute magnitude and differing albedo assumptions.
TABLE 1. Asteroid Sizes (km) Let’s examine some factors that determine whether an astronomical body can achieve a spherical shape, the hallmark of HE. One key factor is the bulk modulus, the strength of materials that make up an astronomical body. Bulk modulus measures the ability of a substance to resist compressive forces. The bulk modulus of rocky material is about an order of magnitude larger than ice, which means it’s easier for an icy body to achieve HE than a rocky body. Mimas is in HE with a 400 km diameter, while Ceres, the smallest dwarf planet body of mostly rocky composition in HE has a mean diameter of about 940 km. Pallas (~610 km) and Vesta (~540 km) are moderately elongated. The most important HE requirement is mass. Comets’ low mass (~1015 kg) gravitational forces hold constituent ice and rock material together like a rubble pile. Mimas’ mass of 3.7×1019 kg mass coupled with ice’s lower bulk modulus permits its HE form. Ceres’ 9.43×1020 kg mass is sufficient to overcome its higher rocky bulk modulus and achieve HE. Vesta and Pallas have masses 3.5x – 4.5x respectively smaller than Ceres, resulting in definite ellipsoid shapes. In a recent arxiv paper Lineweaver and Norman), the authors determined that icy bodies can achieve HE with about 400 km diameters, and rocky asteroids at about 600 km based in first principles. To summarize, a non‐rotating astronomical body with sufficient mass so its self‐gravity can overcome its materials’ strength will acquire a spheroidal shape and achieve hydrostatic equilibrium. This tends to occur as bodies made of ice approach sizes of about 400 km and those of about 600 km made of rock. Vesta and Pallas were in transition toward HE during their formation, but fell somewhat short of full sphericity. DIFFERENTIATION A good description of planetary differentiation similar to what has occurred on Earth (Figure 3) follows (http://wapi.isu.edu/Geo_Pgt/Mod03_PlanetaryEvo/mod3_pt1.htm): Planetary differentiation … refers to the processes that cause an essentially homogeneous accreted body that is made up of primordial solar material to become separated into layers having different chemical and/or physical properties. If a planetary body is large enough it will develop a core, mantle and crust each of which may be further subdivided. Differentiation operates as materials of varying density are separated by a body’s self gravity, with those of the highest density moving to its center. Melting or partial melting of material is required for the process to occur, which takes place over long time scales. The rate of differentiation depends on buoyancy forces and heat generated in the materials which may be solid, but exhibit fluid properties over geologic time. This can be quantified by the Rayleigh number, a complex fluid mechanics relationship, the value of which determines the onset of convection, the prime mover of differentiation. http://www.didcotgirls.oxon.sch.uk/depts/geog/bcs_geography/images/earth%20structure3.gif Microsoft Word – tut38 Hydrostatic Equilibrium And Planetary Differentiation.doc There are several sources of heat relating to differentiation, but planetary scientists think that most important are those from radioactive elements. Relatively short half life (0.7– 1.5 million years) isotopes like 26Al and 60Fe were responsible for initial internal heating creating differentiation early in planetary evolution, and especially important for smaller bodies like asteroids. This means that dust in the solar system molecular cloud contained material from nearby red giant, nova or supernova events occurring at planetoid formation time. Long lived isotopes like 238U and 40K with half lives in the billions of years were important differentiation sources for larger bodies like Earth, and continue to heat interiors to the present day. During differentiation, as denser material descended to the center, heat was generated by the exchange of gravitational potential energy to heat. An important differentiation heat source comes from tidal heating of some planetary satellites. The volcanism exhibited by Jupiter’s Io comes from tidal effects on the satellite. Of course, heat also occurs from solar radiation and impacts from space detritus, but these sources tend to mainly affect surface temperatures, and are mostly unrelated to aiding differentiation. There is ample meteorite evidence that asteroids have undergone differentiation. Spectral characteristics of differentiated meteorites such as achondrites have been matched to Vesta, which confirms its differentiated history as well as of other asteroids. Figure 4 shows an achondrite meteorite found in Antarctica. The heat sources were short half life isotopes based on analysis of the meteorites, which reveal the presence of the stable daughter products of 26Al and 60Fe isotopes. It’s thought that during formation, many asteroids reached internal temperatures (~1,500 K) capable of melting minerals resulting in a differentiated structure. The Dawn Mission reaching Vesta in 2011, and Ceres in 2015 will reveal surface detail of these asteroids and evidence of past geologic activity. http://www.solarviews.com/cap/meteor/achndrit.htm Microsoft Word – tut38 Hydrostatic Equilibrium And Planetary Differentiation.doc Less well known is the capability of KBOs to achieve that state given their different composition and great solar distances. KBOs are thought to be made up of about 50% of various ices and are located in ~50 K temperatures. There is no reason to believe that KBOs in HE couldn’t become differentiated, since in their formation it’s likely they were also heated by short and long half life isotopes. The prominence of crystalline ice on some KBOs, which forms at 110 K versus amorphous ice, is a possible indication of geologic activity associated with differentiation. We’ll have to wait for images from New Horizons in 2015 for surface confirmation of KBO differentiation. CONCLUDING REMARKS Are hydrostatic equilibrium and differentiation related? Can we assume that a body in HE has achieved differentiation? When a body made of ice (Mimas) achieves HE, there is nothing to differentiate since it’s made up of a single compound, but may have different phases internally. Mark Sykes, the well known planetary scientist claims in support for Pluto’s planetary status that Objects become “round” when they are so massive that their gravity crushes them into a shape that is in hydrostatic equilibrium. Heat from formation and the decay of radionuclides increases interior temperatures to the point where differentiation, mantle convection, and other processes occur. See: http://www.sciencemag.org/cgi/content/full/319/5871/1765?ijkey=exNqd2F83NhIM&k eytype=ref&siteid=sci Extreme pressures at the center of a body in HE may result in sufficient heat generation to create melting and adding to heat from radioactive isotopes promoting differentiation. On the other hand, we cannot assume that all differentiated bodies achieve HE. Even if small asteroids are differentiated, it’s unlikely they have sufficient mass to reach spherical form. So differentiation does not necessarily imply a body has achieved HE. REFERENCES
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