# Star surface

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As gas balls, stars have no solid surface. Nevertheless - even in the specialist astronomical literature - the term star surface is used very often , and associated terms such as surface temperature, surface gravity and star radius. Since these quantities are fundamental for the physics of the star's interior (see star structure ) as well as the outer layers of the star, the concept of the star surface requires a very careful definition.

Image of the sun from June 7, 1992. In the visible area of ​​light, the sun disk has a well-defined edge.
X-ray image of the sun obtained with the Yohkoh satellite. The sun disk is still clearly recognizable, but surrounded by a diffuse ring, the corona.

In the practice of astronomical observation, instead of the surface, the radius of a star is usually considered, which is, however, equivalent to each other. Accordingly, after the introduction, which presents two typical examples of star surfaces, various definitions and measurement methods for the star radius are discussed.

## Examples of star surfaces

Density and temperature profile of the top layers of the sun

### Surface of the sun

As the only star whose surface can be viewed in detail, the sun is by far the most suitable to introduce the problem. In view of the familiar image of the sharply outlined solar disk, the term surface seems to be unproblematic for our central star, despite the lack of a solid crust.

But if you observe the sun during a total solar eclipse (or with an instrument that largely blocks out the light from the solar disk, a so-called coronograph ), or even in other wavelength ranges, you can see that it by no means "stops" at the edge that is usually so clearly visible. Without the dominant light from its disc, the sun appears to be surrounded by a far-reaching, diffuse ring interspersed with rays. A comparable picture is presented in the X-ray range, and in the radio range the sun also appears much “larger” than in visible light.

This very complex appearance is due to the fact that the outer layers of the sun consist of several plasma layers in which very different physical conditions prevail. The fact that our central star has a sharp edge in the visible can be explained by the fact that the photosphere is opaque to electromagnetic waves, light, and deeper layers and, to a good approximation, has the properties of a black body , while all layers above in the visible area are so transparent , that one thinks that the visible light arises from a thin layer, the thickness of which is very small compared to the radius of the actual solar body. In addition, the visible photosphere is much brighter than the adjacent areas, which are known as the chromosphere and corona . Due to their weakness of light, the latter are only visible in the optical if the much brighter solar disk, which represents the photosphere, is covered by the moon or artificially.

The fact that the corona is not outshone by the actual solar body in the X-ray range is due to its very high temperature (up to 2 million K compared to 4400-6600 K in the photosphere; see also the compilation of the solar state variables on the NASA website given under web links Sun Fact Sheet). The solar radio radiation is subject to strong fluctuations. In addition to the constant “calm” radiation, short-term violent outbreaks of radiation occur again and again in the context of solar flares.

Despite the image of the clear solar disk, the definition of the term surface causes considerable difficulties even for our central star. Since the photosphere provides the lion's share of the total luminosity of the sun compared to the chromosphere and corona , it seems sufficient to equate the surface of a star with its photosphere.

### Surface of Betelgeuse

High-resolution image of Betelgeuse obtained using Lucky Imaging with the VLT. In contrast to the sun, Betelgeuse appears as a diffuse object.

However, there are stars for which equating surface = photosphere leads to difficulties. These include, for example, objects such as red giants , red supergiants , luminous blue variables and Wolf-Rayet stars . Detailed investigations show that in such stars the thickness of the photosphere is no longer small compared to the radius of the actual star body.

A particularly intensively studied star with extended photosphere is Betelgeuse . Despite its distance of 600 light years , its huge size disk can still be resolved as a flat object. This is directly possible with the Hubble space telescope (see e.g. Gilliband and Dupree (1996)). With the help of aperture synthesis (an extension of classic interferometry , in which not just two, but at least three beams are brought to interference) or so-called lucky imaging , Betelgeuse can also be resolved as an extended body with earth-based telescopes such as the Very Large Telescope located in the Atacama Desert (see e.g. Haubois et al. (2009) or Ohnaka et al. (2009)).

Even if the angular diameter of Betelgeuse is only about 5 to 10 times larger than the angular resolution that can be achieved using the above-mentioned techniques , the images obtained with the Hubble space telescope or the VLT make the decisive difference to the sun clear. The Betelgeuse disk has no sharp edge, not even in the wavelength ranges in which the radiation of the photosphere dominates. The diffuse appearance is also a result of the still rather limited resolution compared to the angular diameter. A precise analysis of Betelgeuse's atmosphere confirms, however, that the impression of an expanded envelope compared to the actual star body is true.

## Definitions of the term star radius

The differences between the sun and Betelgeuse can also be illustrated by the following thought experiment. If one could penetrate into the solar atmosphere, as long as one was in the corona or chromosphere, one would still have a largely undisturbed view to the outside, since these layers only absorb (and scatter) a small proportion of the light passing through them. After immersing in the photosphere, however, at a very short distance of only a few hundred kilometers one would get the impression of being surrounded by a glistening fog, since the layers now also located above the hypothetical sun traveler absorb and scatter almost all incident light. If you penetrated the Beteigeuzes photosphere, on the other hand, the view to the outside would only gradually deteriorate; you could cover several tens of millions of kilometers before it becomes completely foggy.

### With the help of optical depth

Absorption spectrum of the sun with the Fraunhofer lines
Spectrum of the red supergiant HD 94599

The thought experiment outlined here allows a first definition of the term star radius. It describes the distance from the center of the star at which the layers above absorb a certain proportion of the light passing through them.

The absorption behavior of star atmospheres is often described in terms of optical depth . If a material layer has the optical depth , it only allows the part e - of the incident light to pass through. ${\ displaystyle \ tau}$${\ displaystyle \ tau}$

Unfortunately, the attempt to arrive at an objective definition of the radius by means of the optical depth leads to considerable difficulties. The first difficulty is determining the value of the optical depth that the layers beyond the star's radius should have. Usually  = 1 is required, which means that the outer layers only allow about 37 percent of the light passing through them. Often, however, the definition  = 2/3 is also used, which corresponds to a passage of 51 percent of the incident energy. In the case of stars with extended atmospheres in particular, such different definitions alone have considerable consequences. Bessell et al. showed as early as 1989 that in so-called Mira stars (variable red giants with very strong periodic fluctuations in brightness) the radii differ by up to 10 percent, depending on whether  = 1 or  = 2/3 is required for the outer layers . ${\ displaystyle \ tau}$${\ displaystyle \ tau}$ ${\ displaystyle \ tau}$${\ displaystyle \ tau}$

The second difficulty arises from the wavelength dependence of absorption. The dark Fraunhofer lines of the solar spectrum clearly show that the photosphere is more or less transparent depending on the wavelength. In the areas of the lines one cannot look as deep into the star as between them. The definition discussed here provides a larger radius if one uses wavelength ranges that are occupied by spectral lines . Again, stars with extended atmospheres are particularly affected. Bessell and his co-workers (1989) found radius differences of up to 100 percent for Mira stars in the visible range, depending on the wavelength in question.

What makes red giants and red supergiants more difficult is that their spectra have practically no line-free areas. Like the example by Malyuto et al. (1997) shows that in the visible the spectra of such stars are dominated by numerous broad absorption bands , between which there is only room for a few narrow, low-absorption areas.

Another disadvantage of the optical depth as a radius criterion is that it cannot be handled. The absorption behavior of a stellar atmosphere cannot be measured directly, because even in the line-free areas there is still continuous absorption due to scattering of free electrons , i.e. an absorption-free level cannot be observed. The absorption can only be inferred indirectly by comparing the spectra predicted by simulated model atmospheres with the observed spectra.

### Based on the enclosed mass

If you want to construct a model of the inner structure of a star (see star structure), you have to determine the distance from the star center at which it should "stop". Using the optical depth here is problematic because, although a good approximation of local thermal equilibrium is achieved inside the star , it is not in the outer layers. Instead, the star radius is often defined as the distance from the center, outside of which there is only a very small fraction of the total star mass, which can only marginally influence the pressure and density of the star layers below.

This definition also has its downsides. As the solar model discussed under star structure shows, the mass fraction “outside” of the actual star body must not be set too large for main sequence stars , because the mass of such stars is very strongly concentrated towards the center. If you ignore 1 percent of the total mass of the sun (i.e. if you include 99 percent of its mass within the reference distance from the center), then, according to Abraham and Iben (1971), you are only 0.82 solar radii away from the center, i.e. still far below the photosphere! Detailed studies by Baschek et al. (1991), who link the star's interior with the photosphere, show an image for red giants that differs significantly from that of the sun. The radius, which surrounds 99 percent of the star's mass, already protrudes into the photosphere. Although red giants have a much denser core than the sun, the mass of the heavily inflated layers is far less concentrated towards the center.

With regard to observability, the radius based on the enclosed mass is to be regarded as at least as critical as that determined by the optical depth. Again, one has to rely on simulations, the predictions of which have to be compared with the observation.

### With the help of temperature

Inconsistent temperature distribution inside the sun. The values ​​differ by a factor of 1000!

The radius definitions discussed so far not only have the disadvantage that they are beyond direct observation. Since a certain optical depth or mass must be required for the matter still present beyond the radius, these definitions are also somewhat arbitrary. The method now discussed is at least free from this deficiency.

Models of the star's interior not only indicate which mass is enclosed with increasing distance from the center, but also which luminosity . In the case of non-pulsating stars, the enclosed luminosity is constant outside the core, the energy flow density decreases purely geometrically with the radius. Using the Stefan-Boltzmann law , the energy flux density can be converted into the so-called effective temperature by assuming unhindered radiation of the associated black body radiation . In the interior of the star, the actual temperature (of the radiation as well as the particles) is much higher, since the radiation field there is almost isotropic, contrary to what is assumed. On the surface, however, the gas temperature drops steeply and finally intersects the effective temperature (see diagram). This intersection defines the star radius and the surface temperature. The gas temperature is well defined there, as the particles (electrons, ions, atoms, molecules) are locally in equilibrium with each other far beyond the surface due to collisions. ${\ displaystyle r}$${\ displaystyle L (r)}$ ${\ displaystyle T _ {\ mathrm {eff}}}$

The question arises again to what extent the radius determined in this way is consistent with that from the optical depth. For main sequence stars this is actually true. In the case of stars with extended atmospheres, however, z. Sometimes there are considerable deviations which, according to Bessell and co-workers (1989), account for up to 40 percent - the temperature in red giants systematically provides larger radii.

### Based on the edge darkening

The above image of the solar disk shows that it is much brighter in the middle than at the edge. This so-called edge darkening is due to the fact that in the center of the solar disk one goes into deeper, i.e. H. hotter and thus, according to the Stefan-Boltzmann law, more luminous areas of the photosphere can look into than at its edge.

The edge darkening can also be used to define the star radius. The radius is the distance from the center of the star at which the intensity of the star disk has dropped to a certain fraction compared to the center (in practice it is usually set to 1 percent). As with the optical depth and the enclosed mass, a subjective threshold value must be defined. Of all radius criteria, however, the edge darkening is the clearest and also plays by far the most important role for practical radius measurements. The difficulties that arise are discussed along with the measurements in the next section.

## Surface temperature

The surface temperature of stars is the measurable temperature of the star, but says little about its internal temperature. It is higher there than that on the surface because large amounts of heat are released inside through nuclear fusion . This heat is then transported to the outside along the temperature gradient and is radiated there, whereby the relationship between the wavelength of the emitted light and the surface temperature of the star is given by Wien's law of displacement .

Since one is not able to measure the temperature of the star directly, but the wavelength of the radiation released by it, one can determine the surface temperature of the star indirectly in this way. For example, solar radiation has a spectrum with a maximum at 500  nm wavelength (blue-green light), from which a temperature of its photosphere of around 5,800 Kelvin can be deduced.

## Measurements of the star radius

### Interferometric method

Despite their great distances, stars are not point sources. Interferometric techniques make it possible to resolve a star as a flat object, i.e. H. in particular to determine its angular diameter. If the distance is also known, the physical radius can be derived directly from the angular diameter. Several interferometric methods have been developed over the past century, all of which are explained in the relevant articles, so only a brief summary is given here.

#### Procedure

Michelson star interferometer
Two bundles of rays are generated by two primary telescopes at a distance , which are then superimposed so that an interference pattern is created. To measure the angular diameter of a star, start with a small one and pull the primary telescopes apart until the interference pattern disappears. The angular diameter is then given directly by , where is the wavelength used. The interference pattern is significantly disturbed by the air turbulence, so that adaptive optics must be used to achieve sufficient measurement accuracy .${\ displaystyle D}$${\ displaystyle D}$${\ displaystyle \ lambda / D}$${\ displaystyle \ lambda}$
Intensity interferometer
In this case, the two primary beams are converted into photocurrents that are correlated with one another. If the value is small , the currents are strongly correlated with one another. With increasing , the correlation gets smaller and smaller and finally sinks to zero. The larger the angular diameter of a star, the smaller the distance that is required to achieve a correlation drop.${\ displaystyle D}$${\ displaystyle D}$${\ displaystyle D}$
The correlation between the photocurrents is due to the Bose-Einstein statistics on correlated arrival times of the photons . In practice, it is superimposed by much stronger, uncorrelated fluctuations in the currents that are caused by the air turbulence. In order to average them away, very long exposure times (often several days!) Are required, so that nowadays the Michelson star interferometer equipped with adaptive optics is preferred due to its far better sensitivity.
Speckle interferometry
As a result of the unrest in the air, the light from a star reaches the telescope in slightly different ways. These paths are superimposed on one another, so that a complex interference pattern consisting of individual spots (speckles) is created, which is also constantly changing rapidly. With very short exposure times (often only 1/100 of a second!) Interference patterns are obtained, which can be regarded as approximately stationary. If these are combined with one another through a special process, a reconstructed image of the star is created, which is almost entirely subject to diffraction , but is largely free of the unrest in the air. The sensitivity is now limited by short exposure times, so that again modern Michelson star interferometers perform better.

#### Problems

Edge darkening
The result of the darkening of the edge is that interferometric measurements generally deliver an angle diameter that is too small, since the areas at the center of the star disk contribute more to the interference behavior than those at the edge. In comparison to a uniformly shining disk with the Michelson star interferometer, the two primary telescopes have to be moved a little further apart in order to make the interference pattern disappear. The same applies to the intensity interferometer, where the correlation of the two photocurrents decreases somewhat more slowly with increasing . In the case of speckle interferometry, there is a risk that the edge of the star disk will fall below the sensitivity limit of the reconstructed image.${\ displaystyle D}$
In practice, the measured angular diameter is corrected by multiplying it by a factor that must be derived from models of stellar atmospheres. Fortunately, even for a star like Betelgeuse, the effect is small despite the expansive atmosphere. So Ohnaka et al. (2009) for Betelgeuse a nominal angular diameter of 43.2 milli-arcseconds in the near infrared and taking into account the edge darkening of 43.6 milli-arcseconds. The difference of 0.4 milli-arcseconds therefore corresponds to only 1 percent of the absolute angular diameter. The measurement error for both values ​​was less than 0.1 milli-arcseconds.
Extensive photosphere
An extended photosphere makes a star appear larger than the actual star body below it actually is. However, it is not possible to separate the two components using an interferometric measurement. One is dependent on models of both the star's atmosphere and the star's interior, taking into account all of the uncertainties discussed in the second section of the article.

#### accuracy

The angular diameter of a star - including the correction by the darkening of the edges - can be measured with modern interferometers with an extremely high accuracy of less than 0.1 milli-arcseconds. The main source of error in determining the physical radius is therefore usually the distance from the star. For objects that are accessible to an interferometric measurement, the distance is usually well known, since trigonometric parallaxes measured by the Hipparcos satellite are available. The radii are thus typically affected by uncertainties of only a few percent of the absolute value.

### Radiation-energetic method

#### Procedure

If one knows the luminosity and the effective temperature of a star, one can directly determine its radius according to the Stefan-Boltzmann law : ${\ displaystyle L}$${\ displaystyle T _ {\ mathrm {eff}}}$${\ displaystyle R}$

${\ displaystyle R = {\ sqrt {\ frac {L} {4 \ pi \ sigma T _ {\ mathrm {eff}} ^ {4}}}}}$

Compared to interferometry, this method has the advantage that it can also be used on distant stars whose angular diameter is too small for a direct measurement, e.g. B. on objects in the galactic halo or even nearby neighboring galaxies such as the Magellanic Clouds . Stars are thus accessible, which differ considerably in terms of their chemical composition from the stars in the sun's environment, which usually belong to the galactic disk . Objects that are much weaker in light such as red dwarfs or even brown dwarfs can also be examined.

Since no attempt is made to represent the star as a flat object, the air turbulence is also far less disruptive than with interferometry, so that no adaptive optics are required. No knowledge of the edge darkening is required either.

#### Problems

However, the advantages outlined above are offset by considerable disadvantages. To determine the effective temperature of a star, its spectrum must be recorded and compared with the spectra of simulated stellar atmospheres. Determining the radius with the help of Stefan-Boltzmann's law is therefore much more dependent on theoretical models than the interferometric method, where such models are only included in the already small correction of the edge darkening.

The second serious handicap is that the luminosity of the star must be known (which is why we speak of the radiation-energetic method). In order to determine it, the intensity of the stellar radiation must be measured over the largest possible wavelength range, whereby not only the distance to the star, but also the attenuation of the light by the interstellar dust , the so-called interstellar extinction , must be known. Stars where the interferometric method fails could usually not be measured by Hipparcos either, so that no precise distance information is available.

In the case of earth-based observations, it is not even possible to obtain a spectrum that extends over all relevant wavelength ranges, because both the ultraviolet and the infrared portion of the starlight is largely filtered out by the atmosphere. Very hot and cool stars (the former mainly radiate in the ultraviolet, the latter mainly in the infrared) are particularly affected. Again, one has to rely on simulated stellar atmospheres in order to be able to draw conclusions from the observed spectrum about the hidden parts.

#### accuracy

While the effective temperature can usually be determined quite precisely (the uncertainty is typically only a few percent of the absolute value), the luminosity of stars without a parallax measured by Hipparcos is usually subject to considerable inaccuracy (which often makes up a few 10 percent of the absolute value). The radii determined with the help of the Stefan-Boltzmann law are therefore at best accurate to about 10 percent of the absolute value, i. H. this method lags far behind interferometry in terms of accuracy. As already shown in the second section of the article, the temperature-based radius deviates significantly from the other radius definitions, especially for stars with extended photosphere. For comparison, the radiation energy radii with the interferometric again use model atmospheres have to.

### Radius values

Finally, typical values ​​for stars of different spectral and luminosity classes according to Scheffler and Elsässer (1990) are compiled. The solar radius is used as the unit.

Star radius depending on the spectral type and luminosity class
Spectral type Main series Giants Supergiant
O5 12.0 - -
B0 07.5 15th 030th
A0 02.5 05 060
F0 01.5 05 080
G0 01.1 06th 100
K0 00.9 16 200
M0 00.6 40 500

## literature

• Abraham Z., I. Iben: More Solar Models and Neutrino Fluxes . In: American Astronomical Society (Ed.): Astrophysical Journal . tape 170 , 1971, p. 157 .
• B. Baschek, M. Scholz, R. Wehrse: The parameters R and in stellar models and observations${\ displaystyle T_ {eff}}$ . In: Astronomy and Astrophysics . tape 246 , 1991, pp. 374 .
• MS Bessell, JM Brett, M. Scholz, PR Wood: The effects of photospheric extension upon the spectra of M type Mira variables . In: Astronomy and Astrophysics . tape 213 , 1989, pp. 209 .
• RL Gilliland, AK Dupree: HST Imaging of Betelgeuse . In: International Astronomical Union Symposium . tape 176 . Kluwer Academic Publishers, Dordrecht, 1996, p. 165 .
• X. Haubois, G. Perrin, S. Lacour, T. Verhoelst, S. Meimon, L. Mugnier, E. Thiébaut, JP Berger, ST Ridgway, JD Monnier, R. Millan-Gabet, W. Traub: Imaging the spotty surface of Betelgeuse in the H band . In: Astronomy and Astrophysics . tape 505 , 2009, p. 923 ff .
• V. Malyuto, MO Oestreicher, Th. Schmidt-Kaler : Quantitative Spectral Classification of Galactic Disk KM Stars from Spectrophotometric Measurements . In: Monthly Notices of the Royal Astronomical Society . tape 286 , 1997, pp. 500 .
• K. Ohnaka, K.-H. Hofmann, M. Benisty, A. Chelli, T. Driebe, F. Millour, R. Petrov, D. Schert, Ph. Stee, F. Vakili, G. Weigelt: Spatially resolving the inhomogeneous structure of the dynamical atmosphere of Betelgeuse with VLTI / AMBER . In: Astronomy and Astrophysics . tape 503 , 2009, p. 183 .
• Scheffler H., Elsässer H .: Physics of the stars and the sun . Ed .: BI Wissenschaftsverlag. 2nd Edition. 1990, ISBN 3-411-14172-7 , pp. 122, also pp. 88-92 .