Strömgren sphere

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 word are the H II-regions, a type of an emission nebula, the most prominent of which is the Rosette Nebula. It was derived by Bengt Strömgren in 1937 and later named after him.

The physics

Very hot stars, 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 less 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. This explains why not the entire ISM has been ionized. 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 Strömgren in 1939, the model examines the effects of the electromagnetic radiation of a single star (or a cluster of close but 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.cite journal
author=Strömgren, Bengt.
title=The Physical State of Interstellar Hydrogen
journal=The Astrophysical Journal
year=1939
volume=89
issue=
pages=526–547
url=http://adsabs.harvard.edu/cgi-bin/nph-bib_query?bibcode=1939ApJ....89..526S | doi=10.1086/144074]

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.

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.cite journal
author=McCullough Peter R.
title=Modified Strömgren Sphere
journal=Publications of the Astronomical Society of the Pacific
year=2000
volume=112
issue=
pages=1542-1548
url=http://adsabs.harvard.edu/abs/2000PASP..112.1542M]

The history

In 1938 the American astronomers Otto Struve 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.cite journal
author=Struve Otto, Elvey Chris T.
title=Emission Nebulosities in Cygnus and Cepheus
journal=The Astrophysical Journal
year=1938
volume=88
issue=
pages=364-368
url=http://cdsads.u-strasbg.fr/abs/1938ApJ....88..364S ]

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.cite journal
author=Kuiper Gerard P., Struve Otto, Strömgren Bengt
title=The Interpretation of ε Aurigae
journal=The Astrophysical Journal
year=1937
volume=86
issue=
pages=570-612
url=http://adsabs.harvard.edu/cgi-bin/nph-bib_query?bibcode=1937ApJ....86..570 ]

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 winds and supernovae. The resulting images more closely resemble many actual H II-regions than the original model.

The maths

An idealized calculation is simple, let's suppose the region is exactly spherical and fully ionized (x=1) and composed only of hydrogen so the numerical density of protons equals the density of electrons ($n\_e\; =\; n\_p$), then the Strömgren radius will be the region where the recombination rate equals the ionization rate. We will consider the recombination rate $N\_R$ of all energy levels which is

$N\_R\; =\; sum\_\{n=2\}^\{infty\}N\_n$

$N\_n$ is the recombination rate of the n-th energy level. The reason we have excluded n=1 is that if a photon with enough energy recombines in to the ground level the hydrogen atom will release another photon capable of ionizing up to the ground level. This is important as electric dipole mechanism always make the ionization up to the ground level so we exclude n=1 to add these ionizing field effect. Now, the recombination rate of a particular energy level $N\_n$ is (with $n\_e=n\_p$):

where is the recombination coefficient of the nth energy level in a unitary volume at a temperature $T\_e$ 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 $n$ as the number of nucleons (in this case, protons), we can introduce the degree of ionization $0leq\; x\; leq1$ so $n\_e=xn$, and the numerical density of neutral hydrogen is $n\_h=(1-x)n$. With a cross section $alpha\_0$ (which has units of area) and the number of ionizing photons per area per second $J$ the ionization rate $N\_I$ is:

$N\_I=alpha\_0\; n\_h\; J$

For simplicity we will consider only the geometric effects on $J$ as we get further from the ionizing source flux $S\_*$, so we have an inverse square law:

$alpha\_0\; n\_h\; J(r)=frac\{3\; S\_*\}\{4\; pi\; r^3\}$

We are now in position of calculating the Stromgren Radius $R\_S$, 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.

References