Galactic halo

From Wikipedia, the free encyclopedia
https://en.wikipedia.org/wiki/Galactic_halo

A galactic halo is an extended, roughly spherical component of a galaxy which extends beyond the main, visible component. Several distinct components of a galaxy comprise its halo:

• the stellar halo
• the galactic corona (hot gas, i.e. a plasma)
• the dark matter halo

The distinction between the halo and the main body of the galaxy is clearest in spiral galaxies, where the spherical shape of the halo contrasts with the flat disc. In an elliptical galaxy, there is no sharp transition between the other components of the galaxy and the halo.

A halo can be studied by observing its effect on the passage of light from distant bright objects like quasars that are in line of sight beyond the galaxy in question.

Components of the galactic halo

Stellar halo

The stellar halo is a nearly spherical population of field stars and globular clusters. It surrounds most disk galaxies as well as some elliptical galaxies of type cD. A low amount (about one percent) of a galaxy's stellar mass resides in the stellar halo, meaning its luminosity is much lower than other components of the galaxy.

The Milky Way's stellar halo contains globular clusters, RR Lyrae stars with low metal content, and subdwarfs. In our stellar halo, stars tend to be old (most are greater than 12 billion years old) and metal-poor, but there are also halo star clusters with observed metal content similar to disk stars. The halo stars of the Milky Way have an observed radial velocity dispersion of about 200 km/s and a low average velocity of rotation of about 50 km/s. Star formation in the stellar halo of the Milky Way ceased long ago.

Galactic corona

A galactic corona is a distribution of gas extending far away from the center of the galaxy. It can be detected by the distinct emission spectrum it gives off, showing the presence of HI gas (H one, 21 cm microwave line) and other features detectable by X-ray spectroscopy.

Dark matter halo

The dark matter halo is a theorized distribution of dark matter which extends throughout the galaxy extending far beyond its visible components. The mass of the dark matter halo is far greater than the mass of the other components of the galaxy. Its existence is hypothesized in order to account for the gravitational potential that determines the dynamics of bodies within galaxies. The nature of dark matter halos is an important area in current research in cosmology, in particular its relation to galactic formation and evolution.

The Navarro–Frenk–White profile is a widely accepted density profile of the dark matter halo determined through numerical simulations. It represents the mass density of the dark matter halo as a function of ${\displaystyle r}$, the distance from the galactic center:

$${\displaystyle \rho (r)=\frac{{\rho }_{\text{crit}}{\delta }_c}{(r/r_s)(1+r/r_s{)}^2}}$$

where ${\displaystyle r_s}$ is a characteristic radius for the model, ${\displaystyle {\rho }_{\text{crit}}=3H^2/8\pi G}$ is the critical density (with ${\displaystyle H}$ being the Hubble constant), and ${\displaystyle {\delta }_c}$ is a dimensionless constant. The invisible halo component cannot extend with this density profile indefinitely, however; this would lead to a diverging integral when calculating mass. It does, however, provide a finite gravitational potential for all ${\displaystyle r}$. Most measurements that can be made are relatively insensitive to the outer halo's mass distribution. This is a consequence of Newton's laws, which state that if the shape of the halo is spheroidal or elliptical there will be no net gravitational effect from halo mass a distance ${\displaystyle r}$ from the galactic center on an object that is closer to the galactic center than ${\displaystyle r}$. The only dynamical variable related to the extent of the halo that can be constrained is the escape velocity: the fastest-moving stellar objects still gravitationally bound to the Galaxy can give a lower bound on the mass profile of the outer edges of the dark halo.

Formation of galactic halos

The formation of stellar halos occurs naturally in a cold dark matter model of the universe in which the evolution of systems such as halos occurs from the bottom-up, meaning the large scale structure of galaxies is formed starting with small objects. Halos, which are composed of both baryonic and dark matter, form by merging with each other. Evidence suggests that the formation of galactic halos may also be due to the effects of increased gravity and the presence of primordial black holes. The gas from halo mergers goes toward the formation of the central galactic components, while stars and dark matter remain in the galactic halo.

On the other hand, the halo of the Milky Way Galaxy is thought to derive from the Gaia Sausage.

Cosmochemistry

From Wikipedia, the free encyclopedia
https://en.wikipedia.org/wiki/Cosmochemistry

Meteorites are often studied as part of cosmochemistry.

Cosmochemistry (from Ancient Greek κόσμος (kósmos) 'universe', and χημεία (khēmeía) 'chemistry') or chemical cosmology is the study of the chemical composition of matter in the universe and the processes that led to those compositions. This is done primarily through the study of the chemical composition of meteorites and other physical samples. Given that the asteroid parent bodies of meteorites were some of the first solid material to condense from the early solar nebula, cosmochemists are generally, but not exclusively, concerned with the objects contained within the Solar System.

History

In 1938, Swiss mineralogist Victor Goldschmidt and his colleagues compiled a list of what they called "cosmic abundances" based on their analysis of several terrestrial and meteorite samples. Goldschmidt justified the inclusion of meteorite composition data into his table by claiming that terrestrial rocks were subjected to a significant amount of chemical change due to the inherent processes of the Earth and the atmosphere. This meant that studying terrestrial rocks exclusively would not yield an accurate overall picture of the chemical composition of the cosmos. Therefore, Goldschmidt concluded that extraterrestrial material must also be included to produce more accurate and robust data. This research is considered to be the foundation of modern cosmochemistry.

During the 1950s and 1960s, cosmochemistry became more accepted as a science. Harold Urey, widely considered to be one of the fathers of cosmochemistry, engaged in research that eventually led to an understanding of the origin of the elements and the chemical abundance of stars. In 1956, Urey and his colleague, German scientist Hans Suess, published the first table of cosmic abundances to include isotopes based on meteorite analysis.

The continued refinement of analytical instrumentation throughout the 1960s, especially that of mass spectrometry, allowed cosmochemists to perform detailed analyses of the isotopic abundances of elements within meteorites. in 1960, John Reynolds determined, through the analysis of short-lived nuclides within meteorites, that the elements of the Solar System were formed before the Solar System itself which began to establish a timeline of the processes of the early Solar System.

Meteorites

Meteorites are one of the most important tools that cosmochemists have for studying the chemical nature of the Solar System. Many meteorites come from material that is as old as the Solar System itself, and thus provide scientists with a record from the early solar nebula. Carbonaceous chondrites are especially primitive; that is they have retained many of their chemical properties since their formation 4.56 billion years ago, and are therefore a major focus of cosmochemical investigations.

The most primitive meteorites also contain a small amount of material (< 0.1%) which is now recognized to be presolar grains that are older than the Solar System itself, and which are derived directly from the remnants of the individual supernovae that supplied the dust from which the Solar System formed. These grains are recognizable from their exotic chemistry which is alien to the Solar System (such as matrixes of graphite, diamond, or silicon carbide). They also often have isotope ratios which are not those of the rest of the Solar System (in particular, the Sun), and which differ from each other, indicating sources in a number of different explosive supernova events. Meteorites also may contain interstellar dust grains, which have collected from non-gaseous elements in the interstellar medium, as one type of composite cosmic dust ("stardust").

Recent findings by NASA, based on studies of meteorites found on Earth, suggests DNA and RNA components (adenine, guanine and related organic molecules), building blocks for life as we know it, may be formed extraterrestrially in outer space.

Comets

On 30 July 2015, scientists reported that upon the first touchdown of the Philae lander on comet 67/P's surface, measurements by the COSAC and Ptolemy instruments revealed sixteen organic compounds, four of which were seen for the first time on a comet, including acetamide, acetone, methyl isocyanate and propionaldehyde.

Research

See also: List of molecules in interstellar space

In 2004, scientists reported detecting the spectral signatures of anthracene and pyrene in the ultraviolet light emitted by the Red Rectangle nebula (no other such complex molecules had ever been found before in outer space). This discovery was considered a confirmation of a hypothesis that as nebulae of the same type as the Red Rectangle approach the ends of their lives, convection currents cause carbon and hydrogen in the nebulae's core to get caught in stellar winds, and radiate outward. As they cool, the atoms supposedly bond to each other in various ways and eventually form particles of a million or more atoms. The scientists inferred that since they discovered polycyclic aromatic hydrocarbons (PAHs)—which may have been vital in the formation of early life on Earth—in a nebula, by necessity they must originate in nebulae.

In August 2009, NASA scientists identified one of the fundamental chemical building-blocks of life (the amino acid glycine) in a comet for the first time.

In 2010, fullerenes (or "buckyballs") were detected in nebulae. Fullerenes have been implicated in the origin of life; according to astronomer Letizia Stanghellini, "It's possible that buckyballs from outer space provided seeds for life on Earth."

In August 2011, findings by NASA, based on studies of meteorites found on Earth, suggests DNA and RNA components (adenine, guanine and related organic molecules), building blocks for life as we know it, may be formed extraterrestrially in outer space.

In October 2011, scientists reported that cosmic dust contains complex organic matter ("amorphous organic solids with a mixed aromatic-aliphatic structure") that could be created naturally, and rapidly, by stars.

On August 29, 2012, astronomers at Copenhagen University reported the detection of a specific sugar molecule, glycolaldehyde, in a distant star system. The molecule was found around the protostellar binary IRAS 16293-2422, which is located 400 light years from Earth. Glycolaldehyde is needed to form ribonucleic acid, or RNA, which is similar in function to DNA. This finding suggests that complex organic molecules may form in stellar systems prior to the formation of planets, eventually arriving on young planets early in their formation.

In September 2012, NASA scientists reported that polycyclic aromatic hydrocarbons (PAHs), subjected to interstellar medium (ISM) conditions, are transformed, through hydrogenation, oxygenation and hydroxylation, to more complex organics—"a step along the path toward amino acids and nucleotides, the raw materials of proteins and DNA, respectively". Further, as a result of these transformations, the PAHs lose their spectroscopic signature which could be one of the reasons "for the lack of PAH detection in interstellar ice grains, particularly the outer regions of cold, dense clouds or the upper molecular layers of protoplanetary disks."

In 2013, the Atacama Large Millimeter Array (ALMA Project) confirmed that researchers have discovered an important pair of prebiotic molecules in the icy particles in interstellar space (ISM). The chemicals, found in a giant cloud of gas about 25,000 light-years from Earth in ISM, may be a precursor to a key component of DNA and the other may have a role in the formation of an important amino acid. Researchers found a molecule called cyanomethanimine, which produces adenine, one of the four nucleobases that form the "rungs" in the ladder-like structure of DNA. The other molecule, called ethanamine, is thought to play a role in forming alanine, one of the twenty amino acids in the genetic code. Previously, scientists thought such processes took place in the very tenuous gas between the stars. The new discoveries, however, suggest that the chemical formation sequences for these molecules occurred not in gas, but on the surfaces of ice grains in interstellar space. NASA ALMA scientist Anthony Remijan stated that finding these molecules in an interstellar gas cloud means that important building blocks for DNA and amino acids can 'seed' newly formed planets with the chemical precursors for life.

In January 2014, NASA reported that current studies on the planet Mars by the Curiosity and Opportunity rovers will now be searching for evidence of ancient life, including a biosphere based on autotrophic, chemotrophic and/or chemolithoautotrophic microorganisms, as well as ancient water, including fluvio-lacustrine environments (plains related to ancient rivers or lakes) that may have been habitable. The search for evidence of habitability, taphonomy (related to fossils), and organic carbon on the planet Mars is now a primary NASA objective.

In February 2014, NASA announced a greatly upgraded database for tracking polycyclic aromatic hydrocarbons (PAHs) in the universe. According to scientists, more than 20% of the carbon in the universe may be associated with PAHs, possible starting materials for the formation of life. PAHs seem to have been formed shortly after the Big Bang, are widespread throughout the universe, and are associated with new stars and exoplanets.

Relativistic beaming

From Wikipedia, the free encyclopedia

https://en.wikipedia.org/wiki/Relativistic_beaming

Only a single jet is visible in M87.

Two jets are visible in 3C 31.

In physics, relativistic beaming (also known as Doppler beaming, Doppler boosting, or the headlight effect) is the process by which relativistic effects modify the apparent luminosity of emitting matter that is moving at speeds close to the speed of light. In an astronomical context, relativistic beaming commonly occurs in two oppositely-directed relativistic jets of plasma that originate from a central compact object that is accreting matter. Accreting compact objects and relativistic jets are invoked to explain x-ray binaries, gamma-ray bursts, and, on a much larger scale, active galactic nuclei (quasars are also associated with an accreting compact object, but are thought to be merely a particular variety of active galactic nuclei, or AGNs).

Beaming affects the apparent brightness of a moving object. Consider a cloud of gas moving relative to the observer and emitting electromagnetic radiation. If the gas is moving towards the observer, it will be brighter than if it were at rest, but if the gas is moving away, it will appear fainter. The magnitude of the effect is illustrated by the AGN jets of the galaxies M87 and 3C 31 (see images at right). M87 has twin jets aimed almost directly towards and away from Earth; the jet moving towards Earth is clearly visible (the long, thin blueish feature in the top image), while the other jet is so much fainter it is not visible. In 3C 31, both jets (labeled in the lower figure) are at roughly right angles to our line of sight, and thus, both are visible. The upper jet actually points slightly more in Earth's direction and is therefore brighter.

Relativistically moving objects are beamed due to a variety of physical effects. Light aberration causes most of the photons to be emitted along the object's direction of motion. The Doppler effect changes the energy of the photons by red- or blue-shifting them. Finally, time intervals as measured by clocks moving alongside the emitting object are different from those measured by an observer on Earth due to time dilation and photon arrival time effects. How all of these effects modify the brightness, or apparent luminosity, of a moving object is determined by the equation describing the relativistic Doppler effect (which is why relativistic beaming is also known as Doppler beaming).

A simple jet model

The simplest model for a jet is one where a single, homogeneous sphere is travelling towards the Earth at nearly the speed of light. This simple model is also an unrealistic one, but it does illustrate the physical process of beaming quite well.

Synchrotron spectrum and the spectral index

Relativistic jets emit most of their energy via synchrotron emission. In our simple model the sphere contains highly relativistic electrons and a steady magnetic field. Electrons inside the blob travel at speeds just a tiny fraction below the speed of light and are whipped around by the magnetic field. Each change in direction by an electron is accompanied by the release of energy in the form of a photon. With enough electrons and a powerful enough magnetic field the relativistic sphere can emit a huge number of photons, ranging from those at relatively weak radio frequencies to powerful X-ray photons.

The figure of the sample spectrum shows basic features of a simple synchrotron spectrum. At low frequencies the jet sphere is opaque and its luminosity increases with frequency until it peaks and begins to decline. In the sample image this peak frequency occurs at ${\displaystyle \log \nu =3}$. At frequencies higher than this the jet sphere is transparent. The luminosity decreases with frequency until a break frequency is reached, after which it declines more rapidly. In the same image the break frequency occurs when ${\displaystyle \log \nu =7}$. The sharp break frequency occurs because at very high frequencies the electrons which emit the photons lose most of their energy very rapidly. A sharp decrease in the number of high energy electrons means a sharp decrease in the spectrum.

The changes in slope in the synchrotron spectrum are parameterized with a spectral index. The spectral index, α, over a given frequency range is simply the slope on a diagram of ${\displaystyle \log S}$ vs. ${\displaystyle \log \nu }$. (Of course for α to have real meaning the spectrum must be very nearly a straight line across the range in question.)

Beaming equation

In the simple jet model of a single homogeneous sphere the observed luminosity is related to the intrinsic luminosity as

${\displaystyle S_o=S_e{D}^p,}$

where

${\displaystyle p=3-\alpha .}$

The observed luminosity therefore depends on the speed of the jet and the angle to the line of sight through the Doppler factor, ${\displaystyle D}$, and also on the properties inside the jet, as shown by the exponent with the spectral index.

The beaming equation can be broken down into a series of three effects:

• Relativistic aberration
• Time dilation
• Blue- or redshifting

Aberration

Aberration is the change in an object's apparent direction caused by the relative transverse motion of the observer. In inertial systems it is equal and opposite to the light time correction.

In everyday life aberration is a well-known phenomenon. Consider a person standing in the rain on a day when there is no wind. If the person is standing still, then the rain drops will follow a path that is straight down to the ground. However, if the person is moving, for example in a car, the rain will appear to be approaching at an angle. This apparent change in the direction of the incoming raindrops is aberration.

The amount of aberration depends on the speed of the emitted object or wave relative to the observer. In the example above this would be the speed of a car compared to the speed of the falling rain. This does not change when the object is moving at a speed close to ${\displaystyle c}$. Like the classic and relativistic effects, aberration depends on: 1) the speed of the emitter at the time of emission, and 2) the speed of the observer at the time of absorption.

In the case of a relativistic jet, beaming (emission aberration) will make it appear as if more energy is sent forward, along the direction the jet is traveling. In the simple jet model a homogeneous sphere will emit energy equally in all directions in the rest frame of the sphere. In the rest frame of Earth the moving sphere will be observed to be emitting most of its energy along its direction of motion. The energy, therefore, is ‘beamed’ along that direction.

Quantitatively, aberration accounts for a change in luminosity of

${\displaystyle D^2.}$

Time dilation

Time dilation is a well-known consequence of special relativity and accounts for a change in observed luminosity of

${\displaystyle D^1.}$

Blue- or redshifting

Blue- or redshifting can change the observed luminosity at a particular frequency, but this is not a beaming effect.

Blueshifting accounts for a change in observed luminosity of

${\displaystyle \frac{1}{D^{\alpha }}.}$

Lorentz invariants

A more-sophisticated method of deriving the beaming equations starts with the quantity ${\displaystyle \frac{S}{{\nu }^3}}$. This quantity is a Lorentz invariant, so the value is the same in different reference frames.

Terminology

beamed, beaming

shorter terms for ‘relativistic beaming’

beta

the ratio of the jet speed to the speed of light, sometimes called ‘relativistic beta’

core

region of a galaxy around the central black hole

counter-jet

the jet on the far side of a source oriented close to the line of sight, can be very faint and difficult to observe

Doppler factor

a mathematical expression which measures the strength (or weakness) of relativistic effects in AGN, including beaming, based on the jet speed and its angle to the line of sight with Earth

flat spectrum

a term for a non-thermal spectrum that emits a great deal of energy at the higher frequencies when compared to the lower frequencies

intrinsic luminosity

the luminosity from the jet in the rest frame of the jet

jet (often termed 'relativistic jet')

a high velocity (close to c) stream of plasma emanating from the polar direction of an AGN

observed luminosity

the luminosity from the jet in the rest frame of Earth

spectral index

a measure of how a non-thermal spectrum changes with frequency. The smaller α is, the more significant the energy at higher frequencies is. Typically α is in the range of 0 to 2.

steep spectrum

a term for a non-thermal spectrum that emits little energy at the higher frequencies when compared to the lower frequencies

Physical quantities

angle to the line-of-sight with Earth

${\displaystyle \theta }$

jet speed

${\displaystyle v_j}$

intrinsic luminosity

${\displaystyle S_e}$ (sometimes called emitted luminosity)

observed Luminosity

${\displaystyle S_o}$

spectral index

${\displaystyle \alpha }$ where ${\displaystyle S\propto {\nu }^{\alpha }}$

Speed of light

${\displaystyle c=2.9979\times {10}^8}$ m/s

Mathematical expressions

relativistic beta

${\displaystyle \beta =\frac{v_j}{c}}$

Lorentz factor

${\displaystyle \gamma =\frac{1}{\sqrt{1-{\beta }^2}}}$

Doppler factor

${\displaystyle D=\frac{1}{\gamma (1-\beta \cos \theta )}}$

Hayashi limit

From Wikipedia, the free encyclopedia
https://en.wikipedia.org/wiki/Hayashi_limit

The Hayashi limit is a theoretical constraint upon the maximum radius of a star for a given mass. When a star is fully within hydrostatic equilibrium—a condition where the inward force of gravity is matched by the outward pressure of the gas—the star can not exceed the radius defined by the Hayashi limit. This has important implications for the evolution of a star, both during the formulative contraction period and later when the star has consumed most of its hydrogen supply through nuclear fusion.

A Hertzsprung-Russell diagram displays a plot of a star's surface temperature against the luminosity. On this diagram, the Hayashi limit forms a nearly vertical line at about 3,500 K. The outer layers of low temperature stars are always convective, and models of stellar structure for fully convective stars do not provide a solution to the right of this line. Thus in theory, stars are constrained to remain to the left of this limit during all periods when they are in hydrostatic equilibrium, and the region to the right of the line forms a type of "forbidden zone". Note, however, that there are exceptions to the Hayashi limit. These include collapsing protostars, as well as stars with magnetic fields that interfere with the internal transport of energy through convection.

Red giants are stars that have expanded their outer envelope in order to support the nuclear fusion of helium. This moves them up and to the right on the H-R diagram. However, they are constrained by the Hayashi limit not to expand beyond a certain radius. Stars that find themselves across the Hayashi limit have large convection currents in their interior driven by massive temperature gradients. Additionally, those stars states are unstable so the stars rapidly adjust their states, moving in the Hertzprung-Russel diagram until they reach the Hayashi limit.

When lower mass stars in the main sequence start expanding and becoming a red giant the stars revisit the Hayashi track. The Hayashi limit constrains the asymptotic giant branch evolution of stars which is important in the late evolution of stars and can be observed, for example, in the ascending branches of the Hertzsprung–Russell diagrams of globular clusters, which have stars of approximately the same age and composition.

The Hayashi limit is named after Chūshirō Hayashi, a Japanese astrophysicist.

Despite its importance to protostars and late stage main sequence stars, the Hayashi limit was only recognized in Hayashi’s paper in 1961. This late recognition may be because the properties of the Hayashi track required numerical calculations that were not fully developed before. 

Derivation of the limit

We can derive the relation between the luminosity, temperature and pressure for a simple model for a fully convective star and from the form of this relation we can infer the Hayashi limit. This is an extremely crude model of what occurs in convective stars, but it has good qualitative agreement with the full model with less complications. We follow the derivation in Kippenhahn, Weigert, and Weiss in Stellar Structure and Evolution.

Nearly all of the interior part of convective stars has an adiabatic stratification (corrections to this are small for fully convective regions), such that

${\displaystyle \frac{\delta lnT}{\delta lnP}={\mathrm{\nabla }}_{adiabatic}=0.4}$, which holds for an adiabatic expansion of an ideal gas.

We assume that this relation holds from the interior to the surface of the star—the surface is called photosphere. We assume \grad_{adiabatic} to be constant throughout the interior of the star with value 0.4. However, we obtain the correct distinctive behavior.

For the interior we consider a simple polytropic relation between P and T:

${\displaystyle P=C{T}^{(1+n)}}$

With the index ${\displaystyle n=3/2}$.

We assume the relation above to hold until the photosphere where we assume to have a simple absorption law

${\displaystyle \kappa ={\kappa }_0{P}^a{T}^b}$

Then, we use the hydrostatic equilibrium equation and integrate it with respect to the radius to give us

${\displaystyle P_0=Constant\ast {\left(\frac{M}{R^2}{T}_{eff}^{-b}\right)}^{\frac{1}{1+a}}}$

For the solution in the interior we set ${\displaystyle P=P_0}$ ; ${\displaystyle T=T_{eff}}$ in the P-T relation and then eliminate pressure of this equation. Luminosity is given by the Stephan-Boltzmann law applied to a perfect black body:

${\displaystyle L=4\pi R^2\sigma T_{eff}^4}$.

Thus, any value of R corresponds to a certain point in the Hertzsprung–Russell diagram.

Finally, after some algebra this is the equation for the Hayashi limit in the Hertzsprung–Russell diagram:

${\displaystyle \log (T_{e}ff)=A\log (L)+B\log (M)+constant}$ 

With coefficients

${\displaystyle A=\frac{0.75a-0.25}{b-5.5a+1.5}}$, ${\displaystyle B=\frac{0.5a-1.5}{b-5.5a+1.5}}$

Takeaways from plugin in ${\displaystyle a\approx 1}$ and ${\displaystyle b\approx 3}$ for a cool hydrogen ion dominated atmosphere opacity model (${\displaystyle T<5000K}$):

• The Hayashi limit must be far to the right in the Hertzsprung–Russell diagram which means temperatures have to be low.
• The Hayashi limit must be very steep. The gradient of Luminosity with respect to temperature has to be large.
• The Hayashi limit shifts slightly to the left in the Hertzsprung–Russell diagram for increasing M.

These predictions are supported by numerical simulations of stars. 

What happens when stars cross the limit

Until now we have made no claims on the stability of locale to the left, right or at the Hayashi limit in the Hertzsprung–Russell diagram. To the left of the Hayashi limit, we have ${\displaystyle \mathrm{\nabla }<\mathrm{\nabla }adiabatic}$ and some part of the model is radiative. The model is fully convective at the Hayashi limit with ${\displaystyle \mathrm{\nabla }=\mathrm{\nabla }adiabatic}$. Models to the right of the Hayashi limit should have ${\displaystyle \mathrm{\nabla }>{\mathrm{\nabla }}_{adiabatic}}$.

If a star is formed such that some region in its deep interior has large ${\displaystyle \mathrm{\nabla }-{\mathrm{\nabla }}_{adiabatic}>0}$ large convective fluxes with velocities ${\displaystyle v_{convective}\approx (\mathrm{\nabla }-{\mathrm{\nabla }}_{adiabatic})/2}$. The convective fluxes of energy cooldown the interior rapidly until ${\displaystyle \mathrm{\nabla }={\mathrm{\nabla }}_{adiabatic}}$ and the star has moved to the Hayashi limit. In fact, it can be shown from the mixing length model that even a small excess can transport energy from the deep interior to the surface by convective fluxes. This will happen within the short timescale for the adjustment of convection which is still larger than timescales for non-equilibrium processes in the star such as hydrodynamic adjustment associated with the thermal time scale. Hence, the limit between an “allowed” stable region (left) and a “forbidden” unstable region (right) for stars of given M and composition that are in hydrostatic equilibrium and have a fully adjusted convection is the Hayashi limit.