**1. Historic Background and Naming**

Studying the brightest stars in M 31 and M 33 Hubble and Sandage [1] found irregular variable stars that defined a new object class: Var 19 in M 31 and Var 2, Var A, Var B and Vary C in M 33. The variability of Var 2 has been recognized already 1922 by Duncan [2] and 1923 by Wolf [3]. All irregular variable stars Hubble and Sandage found showed the three common characteristics: high luminosity, blue color indice and at the date of observation an intermediate F-type spectrum. Objects of this class became known as *Hubble-Sandage Variables*. In 1974 Sandage and Tammann [4] observed bright stars in NGC 2366, NGC 4236, IC 2574, Ho I, Ho II, and NGC 2403 originally to further constrain the Hubble constant using Cepheids. In some of these galaxies however they identified stars they designated as *Irregular Luminous Blue Variables*. At the same time Humphreys [5] published additional spectral analysis on the M 31 and M 33 Variables and put them into context to the *η* Carina-like objects. Few years later Humphreys and Davidson [6] studied our galaxy and the LMC and identified the most luminous and massive stars. In that work it became more and more obvious that a certain region in the HRD is not populated: very luminous cool stars seems to not exist or more likely stay for only for a very short time in this region. The boundary to that area was defined by the authors and has been referred to as the Humphreys-Davidson limit. Shortly after in his publication entitled "The stability limit of hypergiant photospheres" de Jager [7] was the first to addressed the presence of such a limit and related possible instabilities from a more theoretical perspective. His argumentation was based on turbulent pressure initiating an instability. Lamers & Fitzpatrick [8] however showed in a 1986 publication that —as still accepted now—radiation and not turbulent pressure is the driver. This linked Humphreys and Davidson observations to the fact that stars will become unstable in this cool and luminous state.

The variability of S Doradus in the LMC was first notices by Pickering in 1897 [9], he also found the star to be bright in <sup>H</sup>*β*, <sup>H</sup>*γ* and H*δ*. Later further studies of its variability [10] showed that S Dor characteristics are very similar to those of the Hubble-Sandage Variables. In our own galaxy, a class known as the P Cygni type stars also showed the same behavior. Note in that context that not all stars that show P Cygni line profiles were automatically members of this historically defined class. Humphreys noted already in her 1975 paper [5] that: "The spectral and photometric properties of these extragalactic variables sugges<sup>t</sup> that they may all be related to stars like *η* Car in our Galaxy and S Dor in the Large Magellanic Cloud.". This hints to the fact that all are only samples of one larger class of variable stars.

In 1984 Peter Conti [11] used the term Luminous Blue Variable during a talk at the IAU Symposium 105 on Observational Tests of the Stellar Evolution Theory. Herewith he finally united—as Humphreys already suggested in her 1975 paper—the earlier defined stellar subgroups of Hubble-Sandage Variables, S Dor Variables, P Cygni and *η* Car type stars, and explicitly excluded Wolf-Rayet stars and normal blue supergiants from LBVs.

### **2. Characteristic of Luminous Blue Variables**

The name already suggests that features that LBVs seem to have in common are being blue and luminous stars that are variable. This however is a rather weak constraint and not even true for a LBV all the time.

It is not simple to disentangle a LBV from a blue O B supergiant and even cooler supergiant of spectral type A of F. A significant number of LBVs have at least temporarily an Of/WN type spectrum [12,13], indicating the presence of emission line and in particular a larger amount of nitrogen in their photosphere. Others were detected with a Be or B[e] spectrum.

It is not possible to identify and classify an LBV by its spectrum or analog its color. It is the a specific variability or an eruption that distinguishes LBVs from "normal stars". The variability of LBVs is a combination of a photometric brightness and color change, caused and accompanied by changes in the stellar spectrum. During such a S Dor variability or S Dor cycle which lasts years or decades [14,15] the star varies from a optically fainter to a brighter star and back. This variability is therefore caused by the star changing from an early (hot) to a late (cool) spectral type, it implies also that not only brightens up but also goes from a blue to a redder color. Historically the brightening of a LBV in the bright (cool) phase during an cycle has also been called an eruption (or S Dor eruption). As we will see later this term is confusing.

With S Doradus in the Large Magellanic Cloud as the first to show this and therefore the prototype, this alternation from hot to cool and back was accordingly named a S Dor cycle and is observed in LBVs only!. The S Dor variability is the one and only clear distinction of LBVs from other massive evolved stars. An example of a long term lightcurve is given for the LBV Var B in M 33 in Figure 1, the analog version for Var C was published by Burggraf [16]. Also plotted here are the changes of the spectral type for the star, that mark an S Dor cycle.

Bernhard Wolf [17] noticed that the change of the spectrum (or equivalent Teff) within an S Dor cycle from a hot to a cool type is larger for more luminous LBVs. This became known as the *amplitude-luminosity-relation*. His plot as well as a new version we made to visualize this relation is given in Figure 2. Instead of a classical HRD we plotted the change of Temperature (ΔTeff) versus the Luminosity L by using the LBVs given in the HRD in Figure 3. The new plot visualizes nicely how tight this relation really is.

**Figure 1.** This figure taken from Burggraf (2015) [18] shows a lightcurve spanning more than 100 years of the LBV and original Hubble Sandage Variable Var B in M 33. In addition to the B magnitudes upper section the spectral type if know for the same date is plotted in the lower section. Note the for S Dor cycles typical changes in the spectral type.

**Figure 2.** This figure shows the classical plot by Wolf [17] (**top**) and in a new version (**bottom**) we plotted the luminosity L and change in Temperature ΔTeff for a new way to visualize the amplitudeluminosity-relation.

A more elaborate photometric classification, based on the duration of the S Dor cycle was made by van Genderen in 2001 [19]. He subdivided the phase and thereby objects into long S Dor (L-SD), here the cycle lasts ≤ 20 years and the short S Dor (S-SD) with the cycle being less than 10 yrs. Beside that he added a group he designated as ex-/dormant for those that currently (within the last 100 years) showed only a weak or no activity at all. Note that these variation are much larger as the microvariability which is common for supergiants in general [20].

In contrast to the more ordered S Dor variability (or eruption) LBVs can undergo more energetic events. In spontaneous *giant eruptions* the visible brightness increases spontaneously by several magnitudes [21]. The best known and well documented event is the giant eruption of the LBV *η* Carinae around 1843. During the eruption the star (or rather outburst) was with −1m the second brightest star in the sky, surpassed only by Sirius with −1.46m [21,22]. Other known and documented historic and present giant eruptions of LBVs are those of P Cygni around ∼1600 [23], SN1954J (=V12) in NGC 2403 ([21,24], and SN1961V in NGC 1058 ([25,26]). It is really important to distinguish between the "S Dor eruption" and a giant eruption. The latter being much more energetic and have changes of Δ ∼ 5mag. With that different strength of the "eruptions" both are most likely caused by very different physical mechanism.

LBVs that showed a giant eruption are referred to as *giant eruption LBVs* or *η Car Variables*, to distinct them from LBVs that show only S Dor variations. Or more precisely for which we at least do not know if they have had a giant eruption, since we are limited to historic records of the last centuries, several giant eruption could have passed unnoticed. See the contribution by Kris Davidson in this volume for more details on giant eruption LBVs and there important distinction from LBVs with S Dor variability only. Concerning these two very different variabilities it has so far not been observed and therefore is not clear if the S Dor variability and the giant eruptions occur separately or a LBV can show both variations.

Beside their variability LBVs stand out by having a rather high mass loss rate. In 1997 Leitherer [27] gave a first list for the mass loss rates of LBVs. They range from 7 10−<sup>7</sup> to 6.6 10−<sup>4</sup> with a typical values around 10−<sup>5</sup> M /yr−1. Stahl et al. [28] used the H*α* line to determined the mass loss rate the during one complete S Dor cycle of AG Car. They find that the derive mass-loss rates in the visual maximum is about a factor five higher as in the visual minimum. More recent studies of the same object by Groh et al. [29] support and extend this study. The authors associate the changes with the bistability limit. Lamers et al. [30] first discussed that while evolving from hot to cool temperatures stars will pass the bistability limit at roughly 21000 K. At this limit a change in the stellar wind occurs. On the hot side the wind velocities are higher and the mass loss rates lower (see also [31]). The cool side of the bistability limit matches in the HRD to the region of LBVs in their cool state and causes a high mass loss in that phase. The closeness to the Eddington limit [32,33] of LBV in their cool phase also favors a high mass loss. This is even more so if the stars rotate fast and the modified Eddington limit the ΩΓ limit applies [33] lowering the gravitational force even further. And indeed AG Car [34] and HR Car [35] are fast rotating LBVs.

### **3. The Evolutionary Status of LBVs**

LBVs are massive evolved stars. The LBV phase is in comparison to other phases massive stars will pass with roughly 25,000 years rather short [36]. Originally, in the classical Conti scenario [37,38], only stars above roughly 50 M were thought to turn into LBVs. Observations however identified LBVs that have a significantly lower mass. The position of LBVs in the HRD, see Figure 3 is associated with bright and generally blue stars (like AG Car, R 127, S 61, P Cyg, WRA 751), but an additional area is populated with LBVs that are fainter and somewhat cooler (HR Car, R 71, HD 160529). These maybe indeed hint for two subclasses the first group being massive LBVs and the latter less massive LBVs. Figure 3 shows the position of galactic and LMC LBVs and LBV candidates in the HRD. If known the position is given for both the cool (open circles) and the hot phase (filled circles).

**Figure 3.** HRD with Galactic and LMC LBVs and LBV candidates. Circles are used for LBVs with an emission line (optical/NIR) nebulae, squares for all others. If an S Dor cycle has been observed both the cool (open symbol) and the hot phase (filled symbol) are marked. Otherwise an open grayish symbol is used. In color evolutionary tracks for different masses are added. The tracks are based on the data from the Geneva code for Z = 0.02 and v*rot* = 300 km/s, colors code the generally three different evolutionary scenarios, see text for details.

Stellar evolution models by the Geneva group [39] that include rotation also shows the position of stars with lower mass matching both location of LBVs in the HRD Figure 3. Also plotted in this figure are tracks of the Geneva group, the Humphreys Davidson limit as well as the LBV/S Dor instability strip, the area of LBV in the hot phase.

The Geneva models [39] yield the following evolutionary scenarios:

least massive stars (red color code in Figure 3):

M < MWR : O – BSG/RSG

:

M >

intermediate massive stars (blue color code in Figure 3):

MWR < M < MOWR : O – LBV

O

–

*or alternatively:* O – RSG – eWNL – eWNE – WC/WO

eWNE

–

WC/WO

–

most massive stars (green color code in Figure 3): eWNL

MOWR The authors define that mass limits as follows: "MOWR is the minimum initial mass of a single star entering the WR phase during the MS phase...MWR is the minimum initial mass of a single star entering the WR phase at any point in the course of its lifetime." Both limits MWR and MOWR depend on the rotation rate and metallicity. For a rotation rate of 300 km/s and solar metallicity MWR = 22 M and MOWR = 45 M. Both values are higher for lower metallicity and lower for higher metallicity. This leads to a mass as low as 21 M for LBVs at Z = 0.04. Depending on the mass and mass loss LBVs either evolve into Wolf-Rayet or directly turn supernovae. Figure 3 with the tracks and LBV positions also yield a clue to Wolfs amplitude-luminosity-relation: In their evolution the point in temperature (open circle) the stars start to turned back around towards hotter temperatures is relatively independent of the stars mass. The more massive, luminous stars start with a hotter temperature, so for them the

crossing in the HRD (or change in Teff) to the turning point is larger. This cool limit is caused by the stars forming an "extended envelope" or pseudo-photosphere in an opaque stellar wind [40,41]. An analysis of this concept using NLTE expanding atmosphere models showed that the formation of a pseudo-photosphere due to strongly increased mass-loss alone does not explain large brightness excursions [42,43]. A later discussion in context of the bi-stability jump implied that the formation of a pseudo-photosphere might work for rotating, relatively low mass LBVs [44,45]. Still, the idea of pseudo-photospheres may explain the power-law shape of the variability spectrum at higher frequencies [46]. An promising alternative idea to explain S Dor variability is envelope inflation [47] potentially induced by changes of the stars rotation. A similar idea based on an instability induced by the lowering of the effective stellar mass by rotation was also suggested [35,48].
