Period-luminosity relationship

The period-luminosity relationship is the relationship between luminosity and the period with which the absolute brightness of radially pulsating stars changes. This relationship is an important means of measuring distances in astronomy .

A period-luminosity relationship was first established in 1912 on Cepheids in the Magellanic Cloud. Since then, other types of giant stars with a similar connection have also been discovered. The luminosity (absolute luminosity) of the star results from the period of the brightness fluctuation , from which its distance can be calculated with the measured brightness .

Cepheids

The first period-luminosity relationship was discovered by Henrietta Swan Leavitt in 1912 . For a classical Cepheid or Type I - Cepheid of the mean absolute brightness M and the period duration P in days ( d ) results for the V-band :

${\displaystyle M_{V}=-2{,}78\cdot \log _{10}{\frac {P}{d}}-1{,}32}$

with error consideration:

${\displaystyle M_{V}=(-2{,}78\pm 0{,}09)\cdot \log _{10}{\frac {P}{d}}-1{,}32\pm 0{,}04}$

and for the near-infrared K-band:

${\displaystyle M_{K}=-3{,}26\cdot \log _{10}{\frac {P}{d}}-1{,}84}$

with error consideration:

${\displaystyle M_{K}=(-3{,}26\pm 0{,}03)\cdot \log _{10}{\frac {P}{d}}-1{,}84\pm 0{,}03}$

The formula shows that Cepheids shine more brightly (larger p, smaller M ), the longer their pulsation period is. Therefore, the average absolute brightness can be determined from the period and the distance r of the Cepheid can be calculated from this by using the spectroscopic parallax to calculate the absolute brightness M with the apparent brightness m in the sky ${\displaystyle \rightarrow }$

${\displaystyle m-M=-5+5\cdot \log _{10}{\frac {r}{pc}}}$

compares. Type I and Type II Cepheids , which have a different period-luminosity relationship, can be differentiated on the basis of their light curves . Precondition for an exact determination of the distance is to take into account the influence of the extinction in the line of sight, for which the entity function is often used.

This type of distance determination enables distances to be determined far beyond the local group . One of the main tasks of the Hubble Space Telescope was the search and measurement of a large number of Cepheids in the neighboring galaxy clusters (e.g. Virgo clusters ) for precise distance determination. The Hubble telescope thus continued the work of Edwin Hubble , who first applied the Cepheid method to neighboring galaxies in the 1920s and whose discoveries led to the introduction of Hubble's law and the Hubble constant .

Since Cepheids are very luminous stars, they can still be examined photometrically even at relatively great distances . For this reason, the method of determining the distance using Cepheids is suitable for relatively large distances. Here, as well as with densely populated spiral arms, however, the blending effect sets in, whereby, due to the lack of resolution, the light does not come exclusively from the variable star, but the average brightness is falsified by other unresolved stars in the distant galaxy. Since the amplitude of the light change also depends on the metallicity of the stars, the effect can only be compensated for by spectral analyzes, the limit brightness of which, however, is considerably lower than with photometric measurements.

RR Lyrae Stars

Also, RR-Lyrae stars are periodically radially pulsating variable, the mean absolute brightness is a function of the period length. In contrast to the Cepheids, there is a period-luminosity relationship only in the near infrared

${\displaystyle M_{K}=-2{,}33\cdot \log _{10}{\frac {P}{d}}-0{,}88}$.

Since RR Lyrae stars can also pulsate in the first harmonic , there is a separate period-luminosity relationship for these RRc or RR1 stars.

RR Lyrae stars are fainter than the Cepheids, but more numerous and belong to Population II . They are therefore better suited to investigate the structure and distance of both the halo and dwarf galaxies of the local group, as only low-metal stars of population II occur in them.

Delta Scuti Stars

Delta Scuti stars , also called dwarf cepheids, are pulsating variable stars with low amplitudes and periods of less than 0.3 days. They are multiperiodic and vibrate in both radial and non-radial modes. The period-luminosity relationship

${\displaystyle M_{V}=-3{,}725\cdot \log _{10}{\frac {P}{d}}-1{,}969}$

the Delta Scuti star is only valid for the period of the fundamental oscillation and only achieves an accuracy of 5 percent. Delta Scuti stars are suitable for determining distances in open star clusters and star associations .

Radial pulsing red giants

Radial pulsing red giants and stars on the asymptotic giant branch follow a multitude of period-luminosity relationships in the far infrared. These long-period variables have been intensively investigated during surveys in the Magellanic Clouds and each seem to have a period-luminosity relationship for their respective vibration mode (fundamental or first harmonic), their chemical composition (oxygen-rich stars or carbon stars ) and the cause of the variability. In addition to radial oscillations due to the Kappa mechanism, there is a period-luminosity relationship for an ellipsoidal change of light , which is caused by the orbit of a companion star in a binary star system .

Mira stars

Mira stars are subject to a period-luminosity relationship in the infrared. They are mostly observed in the K band, because there the extinction due to the circumstellar matter is very low.

${\displaystyle M_{K}=-3{,}56\cdot \log _{10}{\frac {P}{d}}+1{,}14}$.

A small group of AGB stars of higher mass in the state of hot bottom burning deviate from this period-luminosity relationship. The advantage of determining the distance through a period-luminosity relationship with Mira stars compared to the Cepheids is:

• Mira stars are more luminous in the far infrared than Cepheids and can therefore be observed over a greater distance
• Mira stars are also found in galaxy types such as dwarf galaxies that do not contain Cepheids
• Mira stars can also be seen in the halos of spiral galaxies . In contrast to the spiral arms, these are not densely populated and the problem of the superposition of several stars can be avoided.

Individual evidence

1. ^ Cuno Hoffmeister , G. Richter, W. Wenzel: Veränderliche Sterne . JA Barth Verlag, Leipzig 1990, ISBN 3-335-00224-5 .
2. ↑ ^{a b} G. Benedict, Fritz; McArthur, Barbara E .; Feast, Michael W .; Barnes, Thomas G .; Harrison, Thomas E .; Patterson, Richard J .; Menzies, John W .; Bean, Jacob L .; Freedman, Wendy L .: Hubble Space Telescope Fine Guidance Sensor Parallaxes of Galactic Cepheid Variable Stars: Period-Luminosity Relations . In: the Astronomical Journal . 133, No. 4, 2007. bibcode : 2007AJ .... 133.1810B . doi : 10.1086 / 511980 .
3. ↑ John R. Percy: Understanding Variable Stars . Cambridge University Press, Cambridge 2007, ISBN 978-0-521-23253-1 .
4. ↑ Joy M. Chavez, Lucas M. Macri, Anne Pellerin: Blending of Cepheids in M33 . In: Astrophysics. Solar and Stellar Astrophysics . 2012, arxiv : 1208.1048 .
5. ^ HA Smith: RR Lyrae Stars . Cambridge University Press, Cambridge 2003, ISBN 0-521-54817-9 .
6. ↑ Laura Watkins: Galactic substructure traced by RR Lyraes in SDSS Stripe 82 . In: Astrophysics. Solar and Stellar Astrophysics . 2011, arxiv : 1111.4390v1 .
7. ↑ C. Ulusoy et al .: Mode identification in the high-amplitude delta Scuti star V2367Cyg . In: Astrophysics. Solar and Stellar Astrophysics . 2012, arxiv : 1210.7147 .
8. ↑ I. Soszynski, P. Wood: SEMI REGULAR VARIABLES WITH PERIODS LYING BETWEEN THE PERIOD LUMINOSITY SEQUENCES C ', C AND D . In: Astrophysics. Solar and Stellar Astrophysics . 2012, arxiv : 1212.0549 .
9. ^ Patricia A. Whitelock: Asymptotic Giant Branch Variables as Extragalactic Distance IndicatorsI . In: Proceedings of the International Astronomical Union . tape 289 , 2012, p. 209-216 , doi : 10.1017 / S1743921312021400 , arxiv : 1210.7307 .