Strömgren sphere
Encyclopedia
In theoretical astrophysics, a Strömgren sphere is the sphere of ionized hydrogen (H II) around a young star of the spectral classes O or B
. Its counterpart in the real world are the H II-regions
, a type of emission nebula
, the most prominent of which is the Rosette Nebula
. It was derived by Bengt Strömgren
in 1937 and later named after him.
s, those of the spectral classes O or B, emit very energetic radiation, especially ultraviolet radiation, which is able to ionize
the neutral hydrogen
(H I) of the surrounding interstellar medium
(ISM), i.e. the hydrogen atom loses its single electron. This state of hydrogen is called H II. After a while free electrons recombine with hydrogen ions. When that happens energy is re-emitted, but not as a single photon
but rather as a series of photons of lesser energy. That way photons lose energy as they travel outward from the star's surface and are no longer energetic enough to contribute to ionization, which explains why the entire ISM has not been ionized in this way. The Strömgren sphere is a theoretical construct which describes these ionized regions.
in 1939, the model examines the effects of the electromagnetic radiation
of a single star (or a tight cluster
of similar stars) of a given surface temperature
and luminosity on the surrounding interstellar medium of a given density. To simplify calculations the latter is taken to be homogeneous and consisting entirely of hydrogen.
The formula derived by Strömgren describes the relationship between the luminosity and temperature of the exciting star on the one hand and the density of the surrounding hydrogen gas
on the other. Using it, the size of the idealized ionized region can be calculated: the Strömgren radius. Strömgren's model also shows that there is a very sharp cut-off of the degree of ionization at the edge of the Strömgren sphere. This is caused by the fact that the transition region between gas that is highly ionized and neutral hydrogen is very narrow compared to the overall size of the Strömgren sphere.
The above-mentioned relationships are as follows:
In Strömgren's model, the sphere named now Strömgren's sphere is made almost exclusively of free protons and electrons. Only close to the surface, a very small amount of atoms appear, whose density increases nearly exponentially to the surface. Outside the sphere, the radiation of atom's frequencies cools the gas strongly, so that it appears a relatively thin region in which the radiation emitted by the star is strongly absorbed by the atoms which loose their energy by radiation in all directions. Thus the Strömgren's system appears as a bright star surrounded by a less emitting , difficult to observe globe.
Strömgren did not know Einstein's theory of optical coherence. The density of excited hydrogen is low, but the paths may be long, so that the hypothesis of a superradiance and other effects observed using lasers must be tested. Supposed superradiant, Strömgren's shell emits space-coherent, time-incoherent beams in the direction for which the path in excited hydrogen is maximal, that is tangentially to the sphere.
More: In Strömgren's explanations, the shell absorbs only the resonant lines of hydrogen, so that the available energy is low. Assuming that the star is a supernova, the radiance of the light it emits corresponds (by Planck's law) to a temperature of several hundreds of kelvins, so that several frequencies may combine to produce the resonance frequencies of hydrogen atom: thus, almost all light emitted by the star is absorbed. Thus, almost all energy radiated by the star amplifies the tangent, superradiant rays.
The Necklace Nebula
is a beautiful Strömgren's sphere. It shows a dotted circle which gives its name. The dot correspond to a competition of the modes emitted by the Strömgren's shell. The star is too weak to be observed.
In supernova remnant 1987A, the Strömgren shell is strangulated into an hourglass whose limbs are three pearls necklaces.
Both Strömgren's original model and the modified one by McCullough do not take into account the effects of dust, clumpiness, multiple stars, detailed radiative transfer, or dynamical effects.
and Chris T. Elvey published their observations of emission nebulae in the constellations Cygnus and Cepheus, most of which are not concentrated toward individual bright stars (in contrast to planetary nebulae). They suggested the UV radiation of the O and B-stars to be the required energy source.
In 1939 Bengt Strömgren took up the problem of the ionization and excitation of the interstellar hydrogen. This is the paper identified with the concept of the Strömgren sphere. It draws, however, on earlier similar efforts of his published in 1937.
In 2000 Peter R. McCullough published a modified model allowing for an evacuated, spherical cavity either centered on the star or with the star displaced with respect to the evacuated cavity. Such cavities might be created by stellar wind
s and supernovae. The resulting images more closely resemble many actual H II-regions than the original model.
so the numerical density
of protons equals the density of electrons (
). Then the Strömgren radius will be the region where the recombination rate equals the ionization rate. We will consider the recombination rate 
of all energy levels, which is
is the recombination rate of the n-th energy level. The reason we have excluded n=1 is that if an electron recombines directly to the ground level the hydrogen atom will release another photon capable of ionizing up from the ground level. This is important, as the electric dipole mechanism always makes the ionization up from the ground level, so we exclude n=1 to add these ionizing field effects. Now, the recombination rate of a particular energy level 
is (with 
):
where
is the recombination coefficient of the nth energy level in a unitary volume at a temperature 
which is the temperature
of the electrons and is usually the same of the sphere. So after doing the sum we arrive to:
Where
is the total recombination rate and has an approximate value of:
.
Using
as the number of nucleons (in this case, protons), we can introduce the degree of ionization 
so 
, and the numerical density of neutral hydrogen is 
. With a cross section
(which has units of area) and the number of ionizing photons per area per second 
the ionization rate 
is:
For simplicity we will consider only the geometric effects on
as we get further from the ionizing source (a source of flux 
), so we have an inverse square law:
We are now in position to calculate the Stromgren Radius
from the balance between the recombination and ionization
and finally, remembering that the region is considered as fully ionized (x=1):
This is the radius of a region ionized by a type O-B star
.
OB star
OB stars are hot, massive stars of spectral types O or B which form in loosely organized groups called OB associations. They are short lived, and thus don't move very far from where they were formed within their life. During their lifetime, they will emit copious amounts of ultraviolet radiation...
. Its counterpart in the real world are the H II-regions
H II region
An H II region is a large, low-density cloud of partially ionized gas in which star formation has recently taken place. The short-lived, blue stars forged in these regions emit copious amounts of ultraviolet light, ionizing the surrounding gas...
, a type of emission nebula
Emission nebula
An emission nebula is a cloud of ionized gas emitting light of various colors. The most common source of ionization is high-energy photons emitted from a nearby hot star...
, the most prominent of which is the Rosette Nebula
Rosette Nebula
The Rosette Nebula is a large, circular H II region located near one end of a giant molecular cloud in the Monoceros region of the Milky Way Galaxy...
. It was derived by Bengt Strömgren
Bengt Strömgren
Bengt Georg Daniel Strömgren was a Danish astronomer and astrophysicist.Bengt Strömgren was born in Gothenburg. His parents were Hedvig Strömgren and Svante Elis Strömgren, who was professor of astronomy at the University of Copenhagen and director of the University Observatory in Copenhagen...
in 1937 and later named after him.
The physics
Very hot starStar
A star is a massive, luminous sphere of plasma held together by gravity. At the end of its lifetime, a star can also contain a proportion of degenerate matter. The nearest star to Earth is the Sun, which is the source of most of the energy on Earth...
s, those of the spectral classes O or B, emit very energetic radiation, especially ultraviolet radiation, which is able to ionize
Ionization
Ionization is the process of converting an atom or molecule into an ion by adding or removing charged particles such as electrons or other ions. This is often confused with dissociation. A substance may dissociate without necessarily producing ions. As an example, the molecules of table sugar...
the neutral hydrogen
Hydrogen
Hydrogen is the chemical element with atomic number 1. It is represented by the symbol H. With an average atomic weight of , hydrogen is the lightest and most abundant chemical element, constituting roughly 75% of the Universe's chemical elemental mass. Stars in the main sequence are mainly...
(H I) of the surrounding interstellar medium
Interstellar medium
In astronomy, the interstellar medium is the matter that exists in the space between the star systems in a galaxy. This matter includes gas in ionic, atomic, and molecular form, dust, and cosmic rays. It fills interstellar space and blends smoothly into the surrounding intergalactic space...
(ISM), i.e. the hydrogen atom loses its single electron. This state of hydrogen is called H II. After a while free electrons recombine with hydrogen ions. When that happens energy is re-emitted, but not as a single photon
Photon
In physics, a photon is an elementary particle, the quantum of the electromagnetic interaction and the basic unit of light and all other forms of electromagnetic radiation. It is also the force carrier for the electromagnetic force...
but rather as a series of photons of lesser energy. That way photons lose energy as they travel outward from the star's surface and are no longer energetic enough to contribute to ionization, which explains why the entire ISM has not been ionized in this way. The Strömgren sphere is a theoretical construct which describes these ionized regions.
The model
In its first and simplest form, derived by the Danish astrophysicist Bengt StrömgrenBengt Strömgren
Bengt Georg Daniel Strömgren was a Danish astronomer and astrophysicist.Bengt Strömgren was born in Gothenburg. His parents were Hedvig Strömgren and Svante Elis Strömgren, who was professor of astronomy at the University of Copenhagen and director of the University Observatory in Copenhagen...
in 1939, the model examines the effects of the electromagnetic radiation
Electromagnetic radiation
Electromagnetic radiation is a form of energy that exhibits wave-like behavior as it travels through space...
of a single star (or a tight cluster
Star cluster
Star clusters or star clouds are groups of stars. Two types of star clusters can be distinguished: globular clusters are tight groups of hundreds of thousands of very old stars which are gravitationally bound, while open clusters, more loosely clustered groups of stars, generally contain less than...
of similar stars) of a given surface temperature
Temperature
Temperature is a physical property of matter that quantitatively expresses the common notions of hot and cold. Objects of low temperature are cold, while various degrees of higher temperatures are referred to as warm or hot...
and luminosity on the surrounding interstellar medium of a given density. To simplify calculations the latter is taken to be homogeneous and consisting entirely of hydrogen.
The formula derived by Strömgren describes the relationship between the luminosity and temperature of the exciting star on the one hand and the density of the surrounding hydrogen gas
Gas
Gas is one of the three classical states of matter . Near absolute zero, a substance exists as a solid. As heat is added to this substance it melts into a liquid at its melting point , boils into a gas at its boiling point, and if heated high enough would enter a plasma state in which the electrons...
on the other. Using it, the size of the idealized ionized region can be calculated: the Strömgren radius. Strömgren's model also shows that there is a very sharp cut-off of the degree of ionization at the edge of the Strömgren sphere. This is caused by the fact that the transition region between gas that is highly ionized and neutral hydrogen is very narrow compared to the overall size of the Strömgren sphere.
The above-mentioned relationships are as follows:
- The hotter and more luminous the exciting star, the larger the Strömgren sphere.
- The denser the surrounding hydrogen gas, the smaller the Strömgren sphere.
In Strömgren's model, the sphere named now Strömgren's sphere is made almost exclusively of free protons and electrons. Only close to the surface, a very small amount of atoms appear, whose density increases nearly exponentially to the surface. Outside the sphere, the radiation of atom's frequencies cools the gas strongly, so that it appears a relatively thin region in which the radiation emitted by the star is strongly absorbed by the atoms which loose their energy by radiation in all directions. Thus the Strömgren's system appears as a bright star surrounded by a less emitting , difficult to observe globe.
Strömgren did not know Einstein's theory of optical coherence. The density of excited hydrogen is low, but the paths may be long, so that the hypothesis of a superradiance and other effects observed using lasers must be tested. Supposed superradiant, Strömgren's shell emits space-coherent, time-incoherent beams in the direction for which the path in excited hydrogen is maximal, that is tangentially to the sphere.
More: In Strömgren's explanations, the shell absorbs only the resonant lines of hydrogen, so that the available energy is low. Assuming that the star is a supernova, the radiance of the light it emits corresponds (by Planck's law) to a temperature of several hundreds of kelvins, so that several frequencies may combine to produce the resonance frequencies of hydrogen atom: thus, almost all light emitted by the star is absorbed. Thus, almost all energy radiated by the star amplifies the tangent, superradiant rays.
The Necklace Nebula
Necklace Nebula
The Necklace Nebula is a planetary nebula located about 15,000 light-years away in the northern constellation Sagitta. It was discovered in 2005 from the Isaac Newton Telescope Photometric H-alpha Survey , a ground-based H-alpha planetary nebula study of the North Galactic Plane...
is a beautiful Strömgren's sphere. It shows a dotted circle which gives its name. The dot correspond to a competition of the modes emitted by the Strömgren's shell. The star is too weak to be observed.
In supernova remnant 1987A, the Strömgren shell is strangulated into an hourglass whose limbs are three pearls necklaces.
Both Strömgren's original model and the modified one by McCullough do not take into account the effects of dust, clumpiness, multiple stars, detailed radiative transfer, or dynamical effects.
The history
In 1938 the American astronomers Otto StruveOtto Struve
Otto Struve was a Russian astronomer. In Russian, his name is sometimes given as Otto Lyudvigovich Struve ; however, he spent most of his life and his entire scientific career in the United States...
and Chris T. Elvey published their observations of emission nebulae in the constellations Cygnus and Cepheus, most of which are not concentrated toward individual bright stars (in contrast to planetary nebulae). They suggested the UV radiation of the O and B-stars to be the required energy source.
In 1939 Bengt Strömgren took up the problem of the ionization and excitation of the interstellar hydrogen. This is the paper identified with the concept of the Strömgren sphere. It draws, however, on earlier similar efforts of his published in 1937.
In 2000 Peter R. McCullough published a modified model allowing for an evacuated, spherical cavity either centered on the star or with the star displaced with respect to the evacuated cavity. Such cavities might be created by stellar wind
Stellar wind
A stellar wind is a flow of neutral or charged gas ejected from the upper atmosphere of a star. It is distinguished from the bipolar outflows characteristic of young stars by being less collimated, although stellar winds are not generally spherically symmetric.Different types of stars have...
s and supernovae. The resulting images more closely resemble many actual H II-regions than the original model.
The mathematics
Let's suppose the region is exactly spherical, fully ionized (x=1), and composed only of hydrogenHydrogen
Hydrogen is the chemical element with atomic number 1. It is represented by the symbol H. With an average atomic weight of , hydrogen is the lightest and most abundant chemical element, constituting roughly 75% of the Universe's chemical elemental mass. Stars in the main sequence are mainly...
so the numerical density
Density
The mass density or density of a material is defined as its mass per unit volume. The symbol most often used for density is ρ . In some cases , density is also defined as its weight per unit volume; although, this quantity is more properly called specific weight...
of protons equals the density of electrons (
). Then the Strömgren radius will be the region where the recombination rate equals the ionization rate. We will consider the recombination rate of all energy levels, which is is the recombination rate of the n-th energy level. The reason we have excluded n=1 is that if an electron recombines directly to the ground level the hydrogen atom will release another photon capable of ionizing up from the ground level. This is important, as the electric dipole mechanism always makes the ionization up from the ground level, so we exclude n=1 to add these ionizing field effects. Now, the recombination rate of a particular energy level is (with ):where
is the recombination coefficient of the nth energy level in a unitary volume at a temperature which is the temperatureTemperature
Temperature is a physical property of matter that quantitatively expresses the common notions of hot and cold. Objects of low temperature are cold, while various degrees of higher temperatures are referred to as warm or hot...
of the electrons and is usually the same of the sphere. So after doing the sum we arrive to:
Where
is the total recombination rate and has an approximate value of:.Using
as the number of nucleons (in this case, protons), we can introduce the degree of ionization so , and the numerical density of neutral hydrogen is . With a cross sectionCross section (physics)
A cross section is the effective area which governs the probability of some scattering or absorption event. Together with particle density and path length, it can be used to predict the total scattering probability via the Beer-Lambert law....
(which has units of area) and the number of ionizing photons per area per second the ionization rate is:For simplicity we will consider only the geometric effects on
as we get further from the ionizing source (a source of flux ), so we have an inverse square law:We are now in position to calculate the Stromgren Radius
from the balance between the recombination and ionizationand finally, remembering that the region is considered as fully ionized (x=1):
This is the radius of a region ionized by a type O-B star
OB star
OB stars are hot, massive stars of spectral types O or B which form in loosely organized groups called OB associations. They are short lived, and thus don't move very far from where they were formed within their life. During their lifetime, they will emit copious amounts of ultraviolet radiation...
.