Barnes-Evans relation

The Barnes-Evans relation relates the luminosity or, indirectly, the surface brightness of a star to its VR color index in the UBVRI system , which characterizes stars based on their luminous color.

It was published in 1976 by the American astrophysicists Thomas G. Barnes and David S. Evans . Although formalisms for a color index-dependent calculation of the luminosity of stars existed before the relation was published, Barnes and Evans succeeded for the first time in a much more extensive confirmation of this relationship, also for cooler spectral classes up to M8, since, in contrast to earlier researchers, they rely on extensive data on a lunar eclipse s program at the McDonald Observatory , which provided the angular diameters of various stellar objects necessary to prove the theory .

overview

If you first consider the general relationship that the luminosity of a star is proportional to the 2nd power of its linear diameter and the 4th power of its effective temperature , then mathematically: ${\displaystyle L}$ ${\displaystyle D}$ ${\displaystyle T}$

${\displaystyle L\thicksim D^{2}\cdot T^{4}}$

If we now use the sun as the calibration point , one can easily derive the equation for the luminosity of a star with the help of the bolometric correction , the angular diameter and the apparent brightness in the visual spectrum of the star under consideration . It is: ${\displaystyle C}$ ${\displaystyle \phi }$ ${\displaystyle V}$

${\displaystyle \log T-0{,}1\cdot C=4{,}2207-0{,}1\cdot V-0{,}5\cdot \log \phi }$

One of the main objectives of the Barnes-Evans relation was to show that the calculation methods based on the bolometric correction or the effective temperature deliver correct results, but the same results can also be determined from more easily measurable quantities such as the apparent brightness and the angular diameter of a star can, which brings with it a significantly increased, practical applicability.

Here, the value of the right side of the equation is referred to below as . So it applies to : ${\displaystyle Fv}$ ${\displaystyle Fv}$

${\displaystyle Fv=4{,}2207-0{,}1\cdot V-0{,}5\cdot \log \phi }$

As you can now see, the value is now linearly linked to the apparent brightness in the visual spectrum and can be calculated specifically with a known angular diameter . ${\displaystyle Fv}$ ${\displaystyle V}$ ${\displaystyle \phi }$

This luminosity parameter is related to the surface flux density (unit: energy per area and time) as follows: ${\displaystyle Fv}$ ${\displaystyle {\mathcal {Fv}}}$

${\displaystyle \log {\mathcal {Fv}}=4\cdot Fv\ +\ c}$, where denotes any constant that depends on the length unit under consideration. ${\displaystyle c}$

It is also important that the value for is calculated without taking into account interfering effects such as interstellar extinction , since the deviations that would result from this interference source, according to previous studies by Warner (1972) and Dunham et al. (1973) can be neglected, because Barnes and Evans also found that none of the stars they examined would fluctuate by more than a few tenths of a magnitude if the absorbance were taken into account. ${\displaystyle Fv}$ ${\displaystyle Fv}$

In addition, Barnes and Evans found that the reddening lines that occur with the reflection and refraction effects of starlight in the atmospheric layers of the earth ( extinction ) can be recognized in different diagrams if the size is optionally selected depending on the VR, BV or RI color index in the UBVRI system . However, relatively complex procedures must be used for stars that are further away in order to exclude interference effects in the measured values. Such stars were initially deliberately left out of the calculations by Barnes and Evans, in order to first prove the general correlation between color index and luminosity with reliable measured values. ${\displaystyle Fv}$

Likewise, in order to make the various values ​​for angular diameters of various stars comparable, only those values ​​from external sources could be used that were approximately in the same wavelength range as the angular diameters determined by the McDonald observation, because both Bonneau and Labeyrie (1973) and Currie, Knapp and Liewer (1974) had already suspected that the angular diameter of stars in the very later spectral classes could be dependent on the wavelength that was selected during the measurement, and this effect had to be minimized as far as possible. In the specific case of Barnes and Evans, it was decided to use a wavelength for measuring the angular diameter of about 7000 Å . ${\displaystyle \phi }$

Since variable stars were also examined, it was essential to try to obtain photometric data (i.e. values ​​of different UBVRI color indices) at about the same time as the measurement of the angular diameter of the star in question, in order to prevent temporal fluctuations in the spectrum . In addition, some transformations between the UBVRI system according to Johnson (1966) and the narrow band system by Eggens, who himself developed such a transformation in 1969, which is now also used.

The BV color index as an argument for the luminosity function ${\displaystyle Fv}$

It was possible to obtain a linear curve for the value as a function of the BV color index on the basis of various observation data from stars, but only in the interval . After a BV value of 1.5 mag , disruptive effects increase, because the titanium oxide contained in the atmosphere of cool stars actually leads to a strong obscuration in the spectrum of starlight. Therefore, the linearity collapses more and more from this value. These typical deviations can also be clearly seen in the figure opposite for the stars of the latest spectral class R Leo and o Cet . ${\displaystyle Fv}$ ${\displaystyle -0{,}88\leq (B\ V)>\,\thickapprox 1{,}5}$

The luminosity parameter Fv depending on the BV color index (selection of a few stars)

Since the complex correction function developed by Nather and Wild (1973) in a study to filter out the titanium oxide veils from the star spectra entails considerable uncertainties, Barnes and Evans decided to define the function only within the BV interval mentioned above which, similar to Warner (1972), who gave a linear coefficient of 0.365, is based on the following equation: ${\displaystyle Fv}$

${\displaystyle Fv=3{,}964-0{,}333\cdot (B\ V)}$

At 0.333, the linear coefficient determined by Barnes and Evans is very close to Warner from 1972.

Since the linear relationship between the value and BV color index as an argument was only consistent for a small interval, Barnes and Evans looked for another color index from the UBVRI system that had a broader linearity too . ${\displaystyle Fv}$ ${\displaystyle Fv}$

The UB color index as an argument for the luminosity function ${\displaystyle Fv}$

The UB color index could be ruled out as a possible argument for a functional rule for fast compared to the previously examined BV index, since Smak (1964), among others, had shown that Mira stars in particular , for which the relation should also be shown by Barnes and Evans , show strong and irregular fluctuations in the UB color index due to the Balmer emission lines. ${\displaystyle Fv}$

The VR color index as an argument for the luminosity function ${\displaystyle Fv}$

One of the main results produced by the study by Barnes and Evans is the confirmation of a linear relationship between the value and the VR color index for a wide range of stars in a wide range, with the following mathematical relationships based on numerous measurements of angular diameters, apparent brightness and color indices were calculated: ${\displaystyle Fv}$

The luminosity parameter Fv depending on the VR color index (selection of individual stars)

${\displaystyle Fv=3{,}977-0{,}429\cdot (V\ R)}$ for the interval ${\displaystyle 0{,}00\leq (V\ R)\leq \,1{,}26}$

${\displaystyle Fv=3{,}837-0{,}320\cdot (V\ R)}$ for the interval ${\displaystyle 1{,}26\leq (V\ R)\leq \,4{,}20}$

as well as beyond:

${\displaystyle Fv=3{,}977-1{,}390\cdot (V\ R)}$ for the interval ${\displaystyle -0{,}17\leq (V\ R)\leq 0{,}00}$

Due to the large number of star types, the values ​​of which were investigated by Barnes and Evans, it can be reasonably assumed that these relationships are valid for S stars , red (super) giants , Mira stars and carbon stars and thus have a broad applicability in the Own astronomy.

Since, in particular, the data on the variable (Mira) stars, regardless of their cyclical development phase, still led to a detectable correlation of the value with the VR color index, it can be assumed that the Barnes-Evas relation is largely independent of variable star phases . Furthermore, the linearity between and the VR color index for the entire range of spectral classes from O5 (concrete star: ζ Pup ) to M8 (concrete star: R Leo ) was proven. Exclusively o Cet forms an outlier in the above diagram of some of the stars examined. ${\displaystyle Fv}$ ${\displaystyle Fv}$

A later study by Barnes et al. (1977) showed a corrected correlation especially for stars of the spectral classes A to G. It is:

${\displaystyle Fv=[3{,}966\pm 0{,}005]-[0{,}390\pm 0{,}019]}$ for the interval ${\displaystyle 0{,}00\leq (V\ R)\leq 0{,}60}$

The RI color index as an argument for the luminosity function ${\displaystyle Fv}$

There is also a linear assignment for the RI color index as an argument of size , but not in such a large range of definition, because according to Barnes and Evans it reads: ${\displaystyle Fv}$

The luminosity parameter Fv depending on the RI color index (selection of a few stars)

${\displaystyle Fv=3{,}824-0{,}386\cdot (R\ I)}$ for the interval ${\displaystyle 0{,}42\leq (R\ I)\leq \,3{,}25}$

The sometimes strong deviations from the linear assignment rule for values ​​for the RI index outside the determined interval cannot be attributed exclusively to measurement inaccuracies, so that interference effects of molecular veils on the light spectrum of the examined stars are probably responsible. See also various example stars in the figure on the right.

History of origin

As early as 1969 the Dutch astronomer Adriaan Wesselink was able to observe a certain linear correlation between a value similar to the luminosity parameter later used by Barnes and Evans and the BV color index. For small (approx. Up to 1.5 mag) values ​​of the BV color index, Wesselink was able to calculate a good linear approximation, although only 4 of 19 of the stars he examined had a spectral class below F5, so that the correlation only for stars of earlier spectral classes could be reasonably credibly supported by evidence. ${\displaystyle Fv}$

The American astronomer Brian Warner took these findings in 1972 as an opportunity to test this linearity for four other stars with known angular diameters and thus came to the justified assumption that the linearity discovered by Wesselink would also be valid for stars up to and including spectral class M2. However, Warner had to assume that the stars μ Gem and α Her were outliers that could not confirm the otherwise valid correlation, so he excluded them from his evaluations.

However, Harwood et al. Warner's assumption in 1975 on the basis of various measured values ​​of angular diameters of different stars with the occultation technique , which were mainly developed at the South African Astronomical Observatory , confirm.

From the beginning of the 1970s, Barnes and Evans were able to access an extensive collection of measured values ​​for the first time, i.e. H. Angular diameters and photometric data, of stars, later spectral classes ("red stars"), can be accessed from a program at the McDonald Observatory . With the help of this data, it was not only possible to confirm the linearity between the luminosity parameter within a certain interval (see overview ) with greater certainty, but also to exclude the UB and RI color index as unsuitable as an argument for a function . Mainly, however, Barnes and Evans found that the most stable correlation results for the previously neglected VR color index. ${\displaystyle Fv}$ ${\displaystyle Fv}$

Due to the large number of measured values ​​of the angular diameters of different stars, which were determined either during the occultation or by other methods, Barnes and Evans were also able to show that if one simply unifies the wavelength at which the measurement is effectively carried out, the different measuring methods are surprising deliver identical values ​​with minor deviations, so that Gold's hypothesis, which described the occultation measurement as inaccurate in 1954, could finally be refuted.

Applications

Calculation of the radii of stars and binary stars

One possible application of the Barnes-Evans relation is the estimation of the radii of near-Earth single and double stars, of which reliable data are available about the distance to Earth, which enables considerable insights into star development and evolution. Since the Barnes-Evans relation when considering the VR color index within the luminosity classes Ia to V and within the spectral classes B to G is free of effects that are related to the gravitation on the surface of the stars, such stars are mainly used for used to calculate their radii.

If we now consider the triangle that arises between an observer on earth and a star with the diameter at a distance (in pc ), the following formula can be established using the definition of the angular diameter (in milli-arcseconds ): ${\displaystyle 2R}$ ${\displaystyle d}$ ${\displaystyle \phi }$

${\displaystyle \phi ={\frac {2R}{d}}}$

Or using the solar radius : ${\displaystyle R\odot }$

${\displaystyle \log {\frac {R}{R\odot }}=0{,}1074\cdot \phi \cdot d}$

If one uses the definition of the luminosity parameter already explained in detail in the section Overview , which reads:, then this leads to the extended equation: ${\displaystyle Fv}$ ${\displaystyle Fv=4{,}2207-0{,}1\cdot V-0{,}5\cdot \log \phi }$

${\displaystyle \log {\frac {R}{R\odot }}=7{,}4724-0{,}2\cdot V-2Fv+\log d}$

For suitable (observable) double stars, in which the individual stars both have visible spectra and periodically cover / darken each other, all quantities of the above equation can be determined from spectroscopic and photometric measurements except for the distance to the observation point, so that it can be easily calculated. However , if the distance is known , the star radius of such double stars can only be determined on the basis of photometric measurements. ${\displaystyle d}$ ${\displaystyle d}$ ${\displaystyle R}$ ${\displaystyle d}$

Of course, the deviations from the star radii calculated with other methods, namely both the random and the system-related deviations, of the values ​​are of particular interest. Claud H. Lacy recognized in a study published in 1977 that there was actually only a standard deviation of 18% compared to the comparison values. This could be expected, however, since the value of the angular diameter was only available with an average accuracy of 18% and a systematic error of around 2% for the calibration lines had to be taken into account, whereby it should be mentioned that the individual deviations are sometimes smaller or greater than this average value. However, another source of error in calculating the star radius using the Barnes-Evans relation is that the techniques used to measure the angular diameters of different stars differ. For late spectral classes, occultation measurements are preferred, in which the complete darkening of the stars is assumed, while the results of the intensity interferometer for stars of earlier spectral classes are mostly based on an edge darkening law, which is based on atmospheric models in order to determine the angular diameter as precisely as possible. Lacy, for example, estimated that this again meant deviations in the ratio of about 5% for stars of later spectral classes and of about 2% for early spectral classes. In addition, the distances between the stars and the measuring point can also be considered as a possible source of error, which is why Lacy only selected stars that have a reliably determined trigonometric parallax. In order to obtain comparable photometric data, not only Eggen's narrow-band system often has to be integrated into the UBVRI system , as already mentioned in the overview section , but also the photometric system according to Kron et al. (1957) based on the theory of Johnson et al. (1966) were transferred to the UBVRI system and was as follows: ${\displaystyle \phi }$ ${\displaystyle \phi }$ ${\displaystyle \phi }$ ${\displaystyle {\frac {R}{R\odot }}}$

${\displaystyle R_{\mathrm {J} }=R_{\mathrm {K} }-0{,}37}$ for the interval ${\displaystyle (R-I)_{\mathrm {K} }\geq 0{,}4}$

${\displaystyle R_{\mathrm {J} }=R_{\mathrm {K} }-0{,}20-0{,}39\cdot (R-I)_{\mathrm {K} }}$ for the interval ${\displaystyle (R-I)_{\mathrm {K} }<0{,}4}$

Here referred to Johnson, and the R-color index the R color index after Kron et al. (1957) ${\displaystyle R_{\mathrm {J} }}$ ${\displaystyle R_{\mathrm {K} }}$

To further refine the data points for the radius ratio , Lacy suggests calculating the standard deviation as follows: ${\displaystyle {\frac {R}{R\odot }}}$ ${\displaystyle \operatorname {var} \left({\frac {R}{R\odot }}\right)}$

${\displaystyle \operatorname {var} \left({\frac {R}{R\odot }}\right)=4\cdot \sigma _{\mathrm {F} }^{2}+0{,}41\cdot \sigma _{\mathrm {V} }^{2}+\sigma _{\mathrm {V-R} }^{2}+0{,}19\cdot \left({\frac {\sigma }{\pi }}\right)^{2}}$

The values denote the standard deviation of the indexed sizes and the unindicated stands for the standard deviation in parallax. Finally, in 1977 Lacy was even able to show that there was a broad agreement between the values ​​for the star radii, which were calculated using the Barnes-Evans relation and those determined by other methods such as, among others, Gray (1967, 1968). Gray used a theoretically as well as practically based method for his calculation of various star radii by measuring a luminous flux density distribution and calculating the surface flux density distribution on the basis of an adapted model of a solar atmosphere for stars with a known distance , so that the following applies to the radius : ${\displaystyle \sigma }$ ${\displaystyle \sigma }$ ${\displaystyle \pi }$ ${\displaystyle Fv}$ ${\displaystyle {\mathcal {Fv}}}$ ${\displaystyle d}$ ${\displaystyle R}$

${\displaystyle R=d\cdot {\sqrt {\frac {Fv}{\mathcal {Fv}}}}}$

Actually arises when the logarithmic values and against each other applies in a coordinate system, a remarkable correlation that a perfect first bisecting approaches (with exactly equal values), although Grays values are on average 10% too large, but very likely must be due to inaccuracies in Gray's calculation, otherwise the results of Veeder (1974) would even be 35-40% too small, although they show an excellent correlation with other research work. ${\displaystyle \log {\frac {R}{R\odot }}(Lacy)}$ ${\displaystyle \log {\frac {R}{R\odot }}(Gray)}$

Calculating the distances from Cepheid stars

The uncertainty in the calculation of various variable stars , including Cepheid stars, is in the extragalactic order of magnitude, i.e. outside of our Milky Way, is probably very considerable, so that it is an important goal for astrophysics to have a model for calculation that is independent of the previous methods to develop this distance in order to actually recognize the possible systematic deviations. In mid-1992, the calibration of individual Cepheids in open star clusters was used to determine the distances of such Cepheids by developing techniques that were to be adapted to the main sequence of stars.

This prompted the Canadian astronomer Douglas L. Welch to use the Barnes-Evans relation to develop a method in 1992 that is based on more precise photometric data and more precisely determined radial velocities of Cepheids in the Milky Way and the magellanic clouds , as well as with the help of various angular diameters and surface brightnesses different stars, which have been determined to a high degree of accuracy, which makes the distance from Cepheid stars possible in an alternative way and possibly with smaller deviations. In fact, it is possible for Welch to determine the distance of Cepheids from the Magellanic clouds to within a few percent.

For the later calculations, however, it must first be clarified which color index from the UBVRI system , which has been extended to include the longer-wave indices J, H and K, is best suited, because the selected color index must be used to reliably determine the individual Cepheids strongly influenced by the temperature of the star. Although the BV, as well as the UB and UV color indexes are very temperature-dependent, they all show too great deviations due to fog in the spectrum of the stars and the gravitation on the surface of the stars, which is why they are unsuitable for comparable data. Welth found out that the VK color index should deliver the best values ​​because it covers a large wavelength range (from V ~ 545 nm to K ~ 2190 nm), which causes a high temperature dependence and since this index is only very slightly affected by obscurations in the Spectrum is influenced. In addition, even with simple interferometers, the V and K phases of visible light can be regularly determined to an accuracy of approx. 1%, which does not apply to other long-wave phases such as L.

After numerous Fourier models for the periodic fluctuations in the spectrum of the Cepheids, Welth obtained a simple mathematical relationship between the radius of a star in astronomical units and the angular diameter calculated using the model of the edge darkening , which is in the wavelength range of the K phase (approx. 2200 nm ) was determined in order to be able to determine the distance of the Cepheid in parsec with an uncertainty in the radius of : ${\displaystyle R_{\mathrm {AU} }}$ ${\displaystyle \theta _{\mathrm {K} }}$ ${\displaystyle d_{\mathrm {pc} }}$ ${\displaystyle r_{\mathrm {AU} }}$

${\displaystyle 2\cdot r_{\mathrm {AU} }+2\cdot R_{\mathrm {AU} }=d_{\mathrm {pc} }\cdot \theta _{\mathrm {K} }}$

Individual evidence

1. ↑ ^{a b c} Thomas G. Barnes, David S. Evans: Stellar angular diameters and visual surface brightness-I . Ed .: Monthly Notices of the Royal Astronomical Society. No. 174 . Oxford University Press, London 1976, pp. 489 ( oup.com ).
2. ^ Thomas G. Barnes, David S. Evans: Stellar angular diameters and visual surface brightness-I . Ed .: Monthly Notices of the Royal Astronomical Society. No. 174 . Oxford University Press, London 1976, pp. 491 .
3. ^ Claud H. Lacy: Radii of nearby stars: an application of the Barnes-Evans relation . Ed .: Astrophysical Journal. No. 34 . Institute of Physics Publishing (USA), Washington DC August 1977, p. 479, line 38 .
4. ^ ^{A b c} John Faulkner, Brian P. Flannery, Brian Warner: Ultrashort-period binaries. II. HZ 29 (= AM CVn): A double-white-dwarf semidetached postcataclysmic nova? Ed .: The Astrophysical Journal. No. 175 . Institute of Physics Publishing (USA), Washington DC July 1972, p. 79-83 , bibcode : 1972ApJ ... 175L..79F .
5. ↑ David W. Dunham et al .: The angular diameter of Upsilon Capricorni and an occultation of SAO 118655 . Ed .: Astronomical Journal. No. 78 . University of Chicago Press, Chicago March 1973, pp. 199–201 , bibcode : 1973AJ ..... 78..199D .
6. ^ Thomas G. Barnes, David S. Evans: Stellar angular diameters and visual surface brightness-I . Ed .: Monthly Notices of the Royal Astronomical Society. No. 174 . Oxford University Press, London 1976, pp. 495-496, lines 12 ff .
7. ^ Thomas G. Barnes, David S. Evans: Stellar angular diameters and visual surface brightness-I . Ed .: Monthly Notices of the Royal Astronomical Society. No. 174 . Oxford University Press, London 1976, pp. 496, line 2 ff .
8. ^ D. Bonneau, A. Labeyrie: Speckle Interferometry: Color-Dependent Limb Darkening Evidenced on Alpha Orionis and Omicron Ceti . Ed .: The Astrophysical Journal. No. 181 . Institute of Physics Publishing (USA), Washington DC April 1973, p. 1-4 , bibcode : 1973ApJ ... 181L ... 1B .
9. ↑ SL Knapp, DG Currie, KM Liewer: On the effective temperature of Alpha Herculis A . Ed .: University of Maryland. July 1974 ( dtic.mil [PDF]).
10. ^ Thomas G. Barnes, David S. Evans: Stellar angular diameters and visual surface brightness-I . Ed .: Monthly Notices of the Royal Astronomical Society. No. 174 . Oxford University Press, London 1976, pp. 491-493 .
11. ↑ ^{a b} Harold L. Johnson: Astronomical measurements in the infrared . Ed .: Annual Review of Astronomy and Astrophysics. No. 4 . Annual Reviews (USA), April 1966, p. 193-206 .
12. ^ ^{A b} O. J. Eggens: Narrow-and Broad-Band Photometry of Red Stars, IV. Population Separation in Giant Stars . Ed .: Astrophysical Journal. No. 158 . Institute of Physics Publishing (USA), Washington DC October 1969, p. 225-242 .
13. ^ Thomas G. Barnes, David S. Evans: Stellar angular diameters and visual surface brightness-I . Ed .: Monthly Notices of the Royal Astronomical Society. No. 174 . Oxford University Press, London 1976, pp. 493, line 10 ff .
14. ^ Thomas G. Barnes, David S. Evans: Stellar angular diameters and visual surface brightness-I . Ed .: Monthly Notices of the Royal Astronomical Society. No. 174 . Oxford University Press, London 1976, pp. 496, line 18 .
15. ↑ Thomas G. Barnes, David S. Evans: Stellar angular diameters and visual surface brightness-I . Ed .: Monthly Notices of the Royal Astronomical Society. No. 174 . Oxford University Press, London 1976, pp. 496, line 8 ff .
16. ↑ ^{a b c} Thomas G. Barnes, David S. Evans: Stellar angular diameters and visual surface brightness-I . Ed .: Monthly Notices of the Royal Astronomical Society. No. 174 . Oxford University Press, London 1976, pp. 490 and 492 .
17. ^ R. Edward Nather, PAT Wild: The angular diameter of R Leonis . Ed .: Astronomical Journal. No. 78 . University of Chicago Press, Chicago September 1973, pp. 628-631 , bibcode : 1973AJ ..... 78..628N .
18. ^ Thomas G. Barnes, David S. Evans: Stellar angular diameters and visual surface brightness-I . Ed .: Monthly Notices of the Royal Astronomical Society. No. 174 . London 1976, p. 496, line 19 .
19. ^ J. Smak: Photometry and Spectrophotometry of Long-Period Variables . Ed .: The Astrophysical Journal supplements. No. 9 . Institute of Physics Publishing (USA), Washington DC 1964, p. 141-184 , bibcode : 1964ApJS .... 9..141S .
20. ^ Thomas G. Barnes, David S. Evans: Stellar angular diameters and visual surface brightness-I . Ed .: Monthly Notices of the Royal Astronomical Society. No. 174 . Oxford University Press, London 1976, pp. 493, line 1 ff .
21. ^ Thomas G. Barnes, David S. Evans: Stellar angular diameters and visual surface brightness-I . Ed .: Monthly Notices of the Royal Astronomical Society. No. 174 . Oxford University Press, London 1976, pp. 497, line 5 ff .
22. ^ Claud H. Lacy: Radii of nearby stars: an application of the Barnes-Evans relation . Ed .: Astrophysical Journal. No. 34 . Institute of Physics Publishing (USA), Washington DC August 1977, p. 479, line 52 f .
23. ^ Thomas G. Barnes, David S. Evans: Stellar angular diameters and visual surface brightness-I . Ed .: Monthly Notices of the Royal Astronomical Society. No. 174 . Oxford University Press, London 1976, pp. 498, line 1 ff .
24. ^ Thomas G. Barnes, David S. Evans: Stellar angular diameters and visual surface brightness-I . Ed .: Monthly Notices of the Royal Astronomical Society. No. 174 . Oxford University Press, London 1976, pp. 498, line 1 ff .
25. ↑ Thomas G. Barnes, James F. Dominy, David S. Evans, Phillip W. Kelton, SB Parsons, Richard J. Stover: The distances of cepheid variables . Ed .: Monthly Notices of the Royal Astronomical Society. No. 178 . Oxford University Press, London April 1977, pp. 661, line 12 , bibcode : 1977MNRAS.178..661B .
26. ^ Thomas G. Barnes, David S. Evans: Stellar angular diameters and visual surface brightness-I . Ed .: Monthly Notices of the Royal Astronomical Society. No. 174 . Oxford University Press, London 1976, pp. 496, line 14 .
27. ^ Thomas G. Barnes, David S. Evans: Stellar angular diameters and visual surface brightness-I . Ed .: Monthly Notices of the Royal Astronomical Society. No. 174 . Oxford University Press, London 1976, pp. 498, line 15 ff .
28. ^ Adriaan J. Wesselink: Surface Brightnesses in the U, B, V System with Applications of Mυ and Dimensions of Stars . Ed .: Monthly Notices of the Royal Astronomical Society. No. 144 . Oxford University Press, London June 1969, pp. 297-311 , bibcode : 1969MNRAS.144..297W .
29. ^ Thomas G. Barnes, David S. Evans: Stellar angular diameters and visual surface brightness-I . Ed .: Monthly Notices of the Royal Astronomical Society. No. 174 . Oxford University Press, London 1976, pp. 489, line 32 ff .
30. ^ Thomas G. Barnes, David S. Evans: Stellar angular diameters and visual surface brightness-I . Ed .: Monthly Notices of the Royal Astronomical Society. No. 174 . Oxford University Press, London 1976, pp. 489, line 37 ff .
31. ^ Thomas G. Barnes, David S. Evans: Stellar angular diameters and visual surface brightness-I . Ed .: Monthly Notices of the Royal Astronomical Society. No. 174 . Oxford University Press, London 1976, pp. 489, line 39 f .
32. ↑ JM Harwood, RE Nather, AR Walker, B. Warner, PAT Wild: Photoelectric Observations of Lunar Occultations . Ed .: Monthly Notices of the Royal Astronomical Society. No. 170 . Oxford University Press, London January 1975, pp. 229-236 , bibcode : 1975MNRAS.170..229H .
33. ^ Thomas G. Barnes, David S. Evans: Stellar angular diameters and visual surface brightness-I . Ed .: Monthly Notices of the Royal Astronomical Society. No. 174 . Oxford University Press, London 1976, pp. 491, line 28 ff .
34. ↑ T. Gold: Occultations of Antares . Ed .: The Observatory. No. 74 , 1954, pp. 38-40 , bibcode : 1954Obs .... 74 ... 38G .
35. ^ Claud H. Lacy: Radii of nearby stars: an application of the Barnes-Evans relation . Ed .: Astrophysical Journal. No. 34 . Institute of Physics Publishing (USA), Washington DC August 1977, p. 479, line 18 ff .
36. ^ Claud H. Lacy: Radii of nearby stars: an application of the Barnes-Evans relation . Ed .: Astrophysical Journal. No. 34 . Institute of Physics Publishing (USA), Washington DC August 1977, p. 480, line 16 ff .
37. ^ Claud H. Lacy: Radii of nearby stars: an application of the Barnes-Evans relation . Ed .: Astrophysical Journal. No. 34 . Institute of Physics Publishing (USA), Washington DC August 1977, p. 480, line 34 ff .
38. ↑ Kron, GE, Gascoigne, SCB, & White, H. S: Red and infrared magnitudes for 282 stars with known trigonometric parallaxes . Ed .: Astronomical Journal. No. 62 . Chicago University Press, Chicago September 1957, pp. 205–220 , bibcode : 1957AJ ..... 62..205K .
39. ^ Claud H. Lacy: Radii of nearby stars: an application of the Barnes-Evans relation . Ed .: Astrophysical Journal. No. 34 . Institute o Physics Publishing (USA), Washington DC August 1977, p. 480, line 90 ff .
40. ^ David F. Gray: A list of photometric stellar radii . Ed .: Astronomical Journal. No. 73 . University of Chicago Press, Chicago November 1968, pp. 769-771 , bibcode : 1968AJ ..... 73..769G .
41. ^ ^{A b} Claud H. Lacy: Radii of nearby stars: an application of the Barnes-Evans relation . Ed .: Astrophysical Journal. No. 34 . Institute of Physics Publishing (USA), Washington DC August 1977, p. 490, line 47 ff .
42. ↑ Glenn J. Veeder: Luminosities and temperatures of M dwarf stars from infrared photometry . Ed .: Astronomical Journal. No. 79 . University of Chicago Press, Chicago October 1974, p. 1056-1072 , bibcode : 1977ApJS ... 34..479L .
43. ^ Jacoby, GH, Branch, D., Ciardullo, R., Davies, RL, Harris, WE, Pierce, MJ: A critical review of selected techniques for measuring extragalactic distances . Ed .: Astronomical Society of the Pacific. No. 104 . Publications of the Astronomical Society of the Pacific, San Francisco August 1992, pp. 599-662 , bibcode : 1992PASP..104..599J .
44. ^ ^{A b} Douglas L. Welch: A Near-Infrared Variant of the Barnes-Evans Method For Finding Cepheid Distances Calibrated with High-Precision Angular Diameters . Ed .: Department of Physics and Astronomy, McMaster University, Hamilton. Ontario August 1994, p. 3 ( arxiv.org [PDF]).
45. ↑ Douglas L. Welth: A Near-Infrared Variant of the Barnes-Evans Method For Finding Cepheid Distances Calibrated with high-precision Angular Diameters . Ed .: Department of Physics and Astronomy, McMaster University, Hamilton. Ontario August 1992, p. 11 .