Scaling of Observable Properties in Rapidly Rotating Stars
SScaling of Observable Properties in Rotating Stars
D. Castañeda, R. G. Deupree
Institute for Computational Astrophysics and Department of Astronomy and PhysicsSt. Mary’s University, Halifax, NS B3H 3C3, Canada [email protected]
Abstract
The spectral energy distribution as a function of inclination is computed using2D rotating stellar models and NLTE plane parallel stellar atmospheres. Thesemodels cover the range from . M (cid:12) to . M (cid:12) . The deduced effective tem-perature is determined by B-V computed from the spectral energy distribution,and the deduced luminosity is computed as the integral of the spectral energydistribution over all frequencies, assuming the distance and reddening are known.These deduced quantities are obtained from the observed spectral energy distri-bution assuming the objects are spherically symmetric, and thus the results aredependent on the inclination. Previous work has shown that the surface proper-ties between two rotating stellar models with the same surface shape scale, andthis is also true for the deduced effective temperature and luminosity over thislimited mass range. Subject headings: stars: atmospheres — stars: fundamental parameters — stars:rotation
1. INTRODUCTION
Advances in the general study of rotating stars have been limited by both theoreticaland observational difficulties. A key example is that fundamental properties such as theeffective temperature ( T eff ) and luminosity ( L ) that one would deduce from observationsnow depend significantly on the angle of inclination ( i ) between the line of sight and thestar’s rotation axis for sufficiently rapidly rotating stars (e.g., Collins & Harrington 1966;Hardorp & Strittmatter 1968; Maeder & Peytremann 1970). This greatly complicates thedetermination of the star’s position on the HR diagram and hence nearly all other usefulinformation unless its inclination can be determined. a r X i v : . [ a s t r o - ph . S R ] O c t L , and theapplication of nonrotating color – effective relations to the SED produces what we will denoteas deduced T eff Both these quantities will be strongly inclination dependent for sufficientlyrapid rotation. There is also no straightforward relationship with the actual L (i.e. the totalamount of energy coming out of the star per unit time) or the actual T eff , which we define as ( L/ ( Aσ )) / where A is the surface area of the star and σ is the Stefan–Boltzmann constant.Naturally the comparison between a computed SED and one observed has much prospectof success only if the inclination and the oblateness of the star are known. Fortunatelyimportant advances in interferometric instrumentation over approximately the last decadehave permitted resolved observations of some nearby rapid rotators (e.g., van Belle et al.2001; Domiciano de Souza et al. 2003; Aufdenberg et al. 2006; Monnier et al. 2007; Zhaoet al. 2009; Che et al. 2011). The direct process requires performing the calculations withdifferent models until the observed SED properties are matched to the extent possible. Thiscan be laborious and it would be far preferable to be able to start with the observed SEDproperties and work backward to what the luminosity and the latitudinal variation of theeffective temperature must be. A crucial first step in this process is to demonstrate thatthere is a well defined relation between the deduced quantities and their physically significantcounterparts, even though this relation depends on both the inclination and the amount ofrotation. We explore this relationship in this work.Recently Deupree (2011) showed that a number of properties of rotating models, par-ticularly the surface effective temperature as a function of latitude, is proportional betweenmodels as long as the surface shape remains the same. Of course the surface radius as afunction of latitude scales by definition. For the surface shapes to be exactly the same,the two models must have the same rotation law to within a multiplicative constant. Theindependence of latitude for the actual effective temperature and radius ratios suggests thatobservable properties such as the deduced luminosity and deduced effective temperature asfunctions of inclination may scale as well. If true, one might be able to at least place con-straints on models and parameters which could produce the observed properties. It wouldalso allow stepping backwards from the observed properties to the actual luminosity andeffective temperature in a straightforward way for cases in which both the inclination andsurface shape are known. Being able to reduce the uncertainty in a star’s actual propertiesis important in determining whether differences between observed and computed oscillation 3 –frequencies are due to not having a model with the correct actual properties or not havingthe interior model structure right. This is very timely because the discovery of multiple os-cillation frequencies in rapidly rotating stars such as α Oph (Monnier et al. 2010) for whichinterferometric observations (Zhao et al. 2009) have been made provides our best opportu-nity at the moment to study in detail the interior structure of rotating stars (e.g. Deupreeet al. 2012; Mirouh et al. 2013).We develop the model scaling relationships and discuss how the deduced propertiesscale in the next section. The following sections provide an example of how these scalingproperties can be used and what are the limitations of the method.
2. SCALING PROPERTIES
We wish to examine the surface relationships between models which have the samesurface shape. We assume that the surface is an equipotential, requiring the rotation lawto be conservative. Some of the assumptions we make in the analysis hold for the laws weconsider, and it is not clear what deviations from a conservative law are allowed before noneof the scaling results hold. We shall return to this point in the final section.
We first consider what scaling in this context means. Essentially, it means that for anysurface variable of two models ( y , y ) with the same shape: y ( θ j ) y ( θ j ) = c y ∀ j (1)where θ j is the co-latitude and c y is a constant. This is true for the surface radius bydefinition.Having the surface shape of two different models the same also imposes a number ofconditions on those models. First, models that have the same surface shape have the samerotation law form, except for an overall scale factor: Ω ( X R , θ j ) = c Ω × Ω ( X R , θ j ) ∀ j (2)where Ω is the rotation rate, c Ω is a constant and X R = r R /R = r R /R . The scaling of the rotation rate and the surface radius between two models with thesame surface shape means the rotational velocities will scale as well. The scale factor can be 4 –determined from the ratio of the surface equatorial velocities of the two models. With boththe rotational velocity and radius scaling, the centrifugal force scales as well.Letting Φ denote the gravitational potential, we can write the total equipotential ( Ψ )at the surface in the following way: Ψ = Φ ( θ ) + Ω ( θ ) R ( θ ) sin ( θ )2 = Ψ c Ψ = 1 c Ψ (cid:20) Φ ( θ ) + Ω ( θ ) R ( θ ) sin ( θ )2 (cid:21) = 1 c Ψ (cid:20) Φ ( θ ) + c c R Ω ( θ ) R ( θ ) sin ( θ )2 (cid:21) (3)Given that the gravitational potential will have only a small nonradial component, we seethat this equation can be solved only if both terms are true individually. Thus, Φ = c Ψ Φ and c Ψ = c c R (4)We see that the gravitational potentials scale as well. To the extent one can approximatethe gravity at the surface by the Roche potential (which one can do quite well for all butthe most rapid uniformly rotating models), one can obtain an estimate for the mass once weknow the appropriate scaling constants.Deupree (2011) has shown that scaling as defined in equation (1) is true for the effectivetemperature for ZAMS models from . to M (cid:12) with twenty different shapes. Theseeffective temperature ratios appear to show a maximum variation of . over all latitudes.A part of the variation could be due to the fact that radii are discretized in the 2D finitedifference mesh - a zone is either inside the model or it is not. The surface at each latitudeis taken to be the outer radial boundary of the zone which the equipotential that describesthe surface passes through.We can see why the the effective temperatures might scale by application of von Zeipel’s(1924) law to two rotating homologous models which have the same shape. This means thatthe equipotential surfaces will have the same shape at the same fractional radius. Becausethe flux, assumed to be radiative, is perpendicular to the equipotential surface, one has (cid:20) F θ ( X R , θ j ) F r ( X R , θ j ) (cid:21) = (cid:20) F θ ( X R , θ j ) F r ( X R , θ j ) (cid:21) (5)where the subscripts 1 and 2 refer to the two models. This means that the same fractionof the flux is being diverted from the radial direction for each of the two models. If this istrue at all locations inside the model, then the distribution of the flux emerging from themodel outer boundary must have the same relative distribution with latitude in both models.Because the effective temperatures are defined in terms of the flux emitted from a surfacezone, the effective temperatures of the two models must scale, satisfying equation (1). 5 – We can also obtain this result from the work of Espinosa Lara & Rieutord(2011) , which finds that one may write T eff = a g β eff , where β decreases slightly asthe model becomes more oblate. Because g eff scales, then T eff scales as long asone is comparing models that have the same shape as we are here. We thankthe referee for this insight. Given sufficient information, one could expect to compute the latitude independentratios of the surface radius, effective temperature, surface rotation velocity, and effectivegravity. While these results may be of some theoretical interest, they would be more beneficialif properties obtained from observations, i.e. the deduced T eff and deduced L , also scaled. Two key variables one wishes to obtain from a star are the actual T eff as defined aboveand L . This remains true for rotating stars, with the complication that neither the tem-perature nor the luminosity one would deduce from observations of a rapidly rotating stardirectly relate to intrinsic stellar properties because the deduced properties are strongly in-clination dependent (e.g.Collins & Harrington 1966; Hardorp & Strittmatter 1968; Maeder &Peytremann 1970; Gillich et al. 2008; Dall & Sbordone 2011). To obtain a deduced effectivetemperature and luminosity from a spectral energy distribution for a rotating star requiresthe same knowledge about reddening and distance as for a spherical star, so we assume thatthis transformation can be performed to some degree and will address obtaining the deducedluminosity and effective temperature from dereddened SEDs with a known absolute flux.The computation of the deduced effective temperature and luminosity as a function ofinclination requires several steps. First, we must have the surface properties of the model,which here we take from the suite of ROTORC (Deupree 1990, 1995) ZAMS models computedby Deupree (2011). We note that these models force a relationship between the local effectivetemperature and the local surface temperature, unlike von Zeipel’s law which assumes thatthe surface is an equipotential (and hence constant temperature) surface while the effectivetemperature can vary significantly from pole to equator. The net effect is that the ROTORCmodels have a flatter relationship between the effective temperature and effective gravity,closer to 0.2 instead of the 0.25 of von Zeipel’s law. We note that the behavior and values ofour exponent with increasing rotation are quite similar to those of Espinosa Lara & Rieutord(2011). Previous studies consistently find a lower value preferable (Monnier et al. 2007; Cheet al. 2011; Claret 2012). Second, we must also have the intensities emerging from thesurface for each member of a grid of stellar atmospheres. For the intermediate mass mainsequence models we wish to explore, plane parallel model atmospheres are satisfactory. The 6 –model atmospheres are computed with the PHOENIX code (Hauschildt & Baron 1999),and the grid covers the range in effective temperature from 7500K to 11000K in steps of250K and in log g from 3.333 to 4.333 in steps of 0.333. The spectrum was computed fromthe far ultraviolet to 20000Å, with minimum wavelength (wavelength interval) maximumwavelength = 600Å (0.005Å) 1500Å (0.01Å) 3000Å (0.015Å) 4000Å (0.02Å) 6000Å (0.03Å)8000Å (0.04Å) 12000Å (0.06Å) 16000Å (0.08Å) 20000Å. These intervals were chosen to keepthe resolution greater than 250000 below 8000Å and about 200000 above 8000Å. Lines inthe four lowest ionization stages of Al, S, and Fe; in the three lowest of C, N, O, Mg, K,and Ca; in the two lowest of He, Li, and Na; and in the lowest of H and Ne are computed inNLTE. More specific details are given by Gillich et al. (2008) and Deupree et al. (2012). Atlower temperatures than included we would need to include more species in NLTE, and athigher temperatures photometric temperature indicators in the visible region of the spectrumbecome harder to find. The net result of these two steps is that at any place on the surfaceof the rotating model, one can interpolate through the grid of model atmospheres in log T eff and log g to obtain the emergent intensity in any direction with respect to the local vertical.The third step calculates the direction to the observer, and thus the angle of the observerwith respect to the local vertical, at every point on the surface, and performs the weightedintegral over all the contributions of the intensities from every point on the surface visible tothe observer to obtain the flux the observer would see. This approach for the third step israther frequently used (e.g.,Slettebak et al. 1980; Linnell & Hubeny 1994; Frémat et al. 2005;Gillich et al. 2008; Aufdenberg et al. 2006; Yoon et al. 2008; Dall & Sbordone 2011), andthe specific details in our calculations are outlined by Lovekin et al. (2006). The final SEDswere obtained by using a 50Å wide boxcar filter. Because of this filtering and the fact thatrotation does not affect the equivalent width, the Doppler shifts were not included in theflux integrals, making the flux calculation computationally “embarrassingly parallel”. Thereis an option to include the Doppler shift when one wishes to compute specific line profileswith no filtering.SEDs were obtained for uniformly rotating models for five masses ( . , , . , . and M (cid:12) ) and six different rotation rates characterized by flatness ( − R p /R eq ) values of0.112, 0.134, 0.156, 0.180, 0.207 and 0.234. To give an idea of how much rotation this isin more conventional terms, we note that the surface equatorial velocities range from about230 km s − to about 360 km s − . The most rapidly rotating model was chosen to keepthe minimum effective temperature above 7500K, below which we would need to includeother low ionization potential metals in NLTE. Models with slower rotation rates were notincluded because the pole to equator temperature variation was less than about 1000K. Somecomputed effective temperatures for the . M (cid:12) models for the two most oblate calculationswere below this lower temperature limit, and those models were not included. The SEDs 7 –were computed at ten equally spaced inclinations from pole on to equator on.Both the deduced effective temperature and deduced luminosity were obtained from thecomputed SEDs. As expected, we found that B-V provided a good indicator of the deducedeffective temperatures, using a NLTE PHOENIX model of Vega with the parameters ofCastelli & Kurucz (1994) to calibrate the color indices. The deduced effective temperatureswere obtained from the simulated (B-V) color using the (B-V) - effective temperature relationfor the plane parallel model atmospheres with the Vega calibration. The gravity used for the(B-V) - effective temperature relationship for the rotating models was the effective gravityat the co-latitude which corresponds to the inclination angle. However, the variation ineffective gravity between the equator and pole is only a little larger than a factor of two,which would lead to a maximum error in the effective temperature of about ± K basedon a comparison of the change in the color – effective temperature relations with gravityfor the plane parallel model atmospheres. The scaling should still be successful because themodels at the same shape have the same effective gravity distribution.The deduced luminosities were computed by integrating the computed flux over allwavelengths, including a Rayleigh-Jeans tail from the end of the calculated wavelengths toinfinite wavelength, and multiplying the result by πd , where d is an assumed distanceto the model from the observer. Because the SED is inclination dependent, the deducedluminosity and effective temperature will be also. We also note that, because determiningthe gravity becomes part of the scaling algorithm if the inclination and shape are known,one can iterate the process to make the deduced gravity and the assumed gravity consistent.We first turn to the deduced luminosities to determine how well they scale from onemodel with the same shape to another. For each model we divide the deduced luminosity ateach inclination by the actual luminosity. The results are presented in Figure 1 for modelswith the ratio of the polar to equatorial radius of 0.82, the most rapidly rotating case forwhich the temperatures of all five masses fall within the range allowed. We see that thecurves all have the same shape, but that the variation from pole to equator increases slightlyas the mass increases, particularly at small inclination. While not perfect, the results inFigure 1 are sufficient to indicate that a reasonable determination of the intrinsic luminositycould be made given an observed luminosity, inclination, and polar to equatorial radius ratio(assuming uniform rotation), at least in the mass range covered.For the deduced effective temperatures, we proceed in a manner similar to that used forthe luminosity. Here the actual effective temperature, defined as the effective temperatureobtained from the flux given by the actual luminosity divided by the total surface areaof the model, plays the role that the actual luminosity played in the previous discussion.We take the ratio of the deduced effective temperature at each inclination divided by the 8 –actual effective temperature for each mass at a given shape. The results are shown in Figure2. Again we see that the curves for different masses show the same form. Interestingly,the largest differences are shown for models seen equator on instead of pole on, except forthe M (cid:12) model, whose ratio at low inclination is noticeably larger than that for all theother masses. This variation with mass for both the deduced effective temperature anddeduced luminosity suggests that these might be analogous to homology transforms forrealistic models of stars – it works well over a restricted mass range, but is not universaland progressively degrades as the physical properties of the models become less similar. Itis worth mentioning that the ratio of the model effective temperatures at a given latitudefor these masses does not show any significant latitudinal variation, so that the variation inthe deduced effective temperatures must originate in the conversion from physical effectivetemperatures to observed ones.Finally we consider whether the bolometric corrections deduced from these simulatedSEDs are affected by any substantial changes rotation may introduce into the SED. Thiswould be important if a full SED was not available. We computed the visual magnitudeof our models and calculated M V using the assumed distance. The visual magnitude wascalibrated by scaling the flux of our spherical model for Vega to match the observed valueabove the earth’s atmosphere at Å (Hayes & Latham 1975) and then integrating theflux in the V filter and requiring V = 0 . mag (Bessell et al. 1998). The absolute bolometricmagnitude comes directly from the model luminosity with the bolometric magnitude of thesun set to 4.74. The bolometric corrections have been computed at inclinations between 0and 90 degrees in ten degree intervals for all models. The results are shown in Figure 3, aplot of the bolometric corrections as a function of (B-V) for all inclinations of all models atall rotation speeds. Also shown in Figure 3 are the bolometric corrections for sphericallysymmetric models, the stars for log g = 4 . and the triangles for log g = 4 . . Figure 3indicates that the temperature is the key to the determination of the bolometric correction,but also that the effective gravity also plays a role (to about 0.07 magnitudes). Exceptthrough the effective gravity, rotation by itself does not appear to produce any particularmodifications to the bolometric corrections for these models.
3. APPLICATION TO MODELS NOT ON THE ZAMS
The scaling relationship for deduced luminosities and deduced T eff described above forthe case of ZAMS models can be extended to models that are not in the same evolutionarystate. To demonstrate this we consider a model of α Ophiuchus, a rapidly rotating A-typestar. Interferometric observations of α Oph imply that it has a polar to equatorial radius 9 –ratio of 0.836 (Monnier et al. 2010) and Vsin i in the range of 210 - 240 km s − (e.g., Bernacca& Perinotto 1970; Abt & Morrell 1995; Royer et al. 2002). We compared this model withtwo ZAMS models which have the same shape as α Oph: one has a similar mass but differentactual T eff and the other with similar actual T eff to α Oph but different mass. A summaryof the properties of each model is given in Table 1. Using as input the deduced luminosityand effective temperature at a specific inclination from the α Oph model and applying thescaling relations to each ZAMS model allows the determination of the deduced effectivetemperatures and luminosities at all inclinations, the effective temperature as a function oflatitude, and finally the actual effective temperature and luminosity for the α Oph model.We can then compare the results predicted by the two ZAMS models with those for the α Oph model itself.The comparisons of the deduced effective temperatures and actual effective tempera-tures are shown in Figure 4. We show the deduced temperatures as functions of inclinationdetermined from the α Oph model (crosses), from the . M (cid:12) model (circles), and from the . M (cid:12) model (squares). Somewhat in keeping with the interferometric results for α Oph,we have chosen the deduced effective temperature and luminosity to be at an inclination of ◦ . The latitudinal variation of the effective temperatures of the model is represented bythe dotted line, and the variation of the deduced effective temperatues with inclination isshown by the dashed line. The same quantities are presented for the scaled . M (cid:12) and . M (cid:12) models. We note that in both cases each of the scaled two ZAMS models agree wellwith the actual model for α Oph.
These results suggest that the precise details ofthe comparison model are not too important as long as the interior structuresare sufficiently homologous. While “sufficiently homologous” is somewhat looselydefined, clearly these two models fit the requirements. On the other hand, onewould not expect a M (cid:12) main sequence star to be an appropriate model foreither α Oph or for a M (cid:12) red giant. The result is the same for the deduced luminosity, as shown in Figure 5. Both ZAMSmodels scale well to the deduced luminosity for the α Oph model at all inclinations, althoughTable 1. Model properties
Model Mass V eq Actual T eff Actual L (M (cid:12) ) (km/s) (K) (K)ZAMS 1 .
25 287 9474 . . ZAMS 2 .
85 237 8187 . . α Oph .
19 229 8122 . .
10 –the . M (cid:12) ZAMS model agrees with the α Oph model a little better. As one might expectfrom the agreement of these features, the actual luminosity and the actual effective temper-ature for each ZAMS model agree to within . L (cid:12) and 40K with the value of the α Ophmodel.
4. SCALING ALGORITHM
We have argued that the deduced luminosity and effective temperature scale for rotatingmodels with the same shape over some limited range of conditions, and that this allows usto determine reasonable values for the luminosity and actual effective temperature of theunknown star from the models. Here we develop an algorithm to use the deduced and modelscaling relations to obtain some intrinsic properties of a rotating star (for convenience, weshall refer to the rotating stellar models with known properties as the “model” and theunknown object whose properties we wish to obtain as the “star”) . We start by assumingthat we have a deduced effective temperature, deduced luminosity, and a measurement ofVsin i . For the moment we assume that we also have the shape and inclination for the staras well. The reference model must have the surface radius, effective temperature, surfacerotational velocity, and surface effective gravity as functions of latitude, the deduced effectivetemperatures and luminosities as functions of inclination, and the mass and luminosity ofthe model.The algorithm proceeds as follows: from Vsin i and the inclination of the star, computethe surface equatorial velocity, V eq . Compute the ratio of the surface equatorial velocityof the model and the star. Because equation (1) is true for the surface velocity, one cancompute the star’s surface rotation velocity at all latitudes. Because both the deducedeffective temperatures and the latitudinal effective temperatures scale as shown in Figure 4,we have T eff , ( θ j ) T eff , ( θ j ) = T d , ( i j ) T d , ( i j ) = T eff ,a, T eff ,a, ∀ j (6)here 1 refers to the model, 2 refers to the star, d refers to the deduced temperature, and a refers to the actual T eff previously defined. Because the deduced temperatures are known atthe inclination of the star for both the model (by interpolation) and the star (by observation),we can obtain the effective temperature at all latitudes and the actual effective temperaturefor the star. The actual luminosity of the star can be computed from the deduced luminosityat the assumed inclination, the deduced luminosity as a function of inclination for the modeland the actual luminosity for the model. With the actual luminosity and actual effectivetemperatures known for both the star and the model, one can compute the ratio of theradii because the only difference between the surface areas of the model and the star is the 11 –difference in the radius. Hence, the radius of the star at every latitude of the star followsfrom the radius profile of the model.The steps so far have depended only on the model, its deduced properties, and thescaling relations for both. These steps have resulted in the surface properties of the staras a function of latitude. The next step requires that the surface be an equipotential. Asshown in Section 2.1, we use the fact that the gravitational part of the total potential andcentrifugal potential must scale the same way for a given shape (i.e., at a given latitude,the ratio between the centrifugal potentials of the model and star and the ratio betweenthe gravitational potentials of the model and the star must be the same). Because wehave computed both the surface rotational velocity and the surface radius as functions oflatitude for the star and have them for the model, we can obtain the ratio of the centrifugalpotentials and hence of the gravitational potentials. Figure 6 presents a diagram ofthe information required from both the model and the star, as well as how thevarious unknowns of the star are determined from the scaling.
If we assume that the gravitational potential at the surface is given by that of a sphericalstar, at least at the equator, we can compute an estimate for the mass of the star. Thisassumption is generally good unless there is significant differential rotation where the materialclose to the rotation axis rotates much faster than the material farther away from the axis.It is certainly excellent for uniformly rotating models except those very near critical rotation(e.g., Ostriker & Mark 1968; Faulkner et al. 1968; Jackson et al. 2004; Deupree 2011).The reason for performing these last steps was to get an estimate of the mass which couldbe used with the actual luminosity to provide a check on the results. All the models utilizedhere are core hydrogen burning objects for which the main sequence mass - luminosity lawshould hold. A check on the reasonableness of the assumed inclination and shape can bemade through how well the derived mass and actual luminosity fit the mass-luminosity law.
5. LIMITATIONS OF THE ALGORITHM
The scaling we have described relies on certain assumptions, and it is reasonable to seeto what extent they can be relaxed. We have assumed that the surface is an equipotential,which only exists if the rotation law is conservative. Even for conservative rotation laws,it remains an assumption that the surface is an equipotential. This likely matters for thepart of the solution that makes an estimate of the mass, but it need not affect the scalingof the observable properties as long as whatever mechanism determines the surface shapedetermines it in the same way for both the unknown and comparison objects. Our very 12 –limited knowledge of the surfaces of rotating stars does not allow an answer to this question.We also assume that we know both the inclination and the surface shape of the unknownobject. This in general is not true, and it turns out that there are combinations of inclinationand shape which produce reasonable results, including masses which fit the mass-luminosityrelation. The general trend is that more rapid rotation (i.e., more oblate shapes) can be offsetby smaller inclinations. One might also add that determining the surface shape accuratelypotentially pays dividends by possibly placing constraints of the rotation law.We have also used a single composition for all our models. A different compositionwould make a difference by producing a different color - effective temperature relation. Wecan obtain a crude estimate of how this might affect results by comparing the color-effectivetemperature relations of several spherical models at different temperatures with two differentcompositions. We calculated NLTE plane parallel model atmospheres with temperatures of8000, 9000, and 10000K using half the metallicity of our previously computed models. Thededuced temperatures for spherical models at these temperatures were all within 50K ofthe actual temperature when using the color-effective temperature relation from the fullmetallicity models.These results lead us to believe that scaling of observables can be a useful technique tomake the bridge between what one observes for rotating stars and physically useful informa-tion under appropriate conditions. We should caution that these results cover only a limitedrange in gravity and effective temperature and that extension far outside this range may notbe warranted.The authors thank Compute Canada and ACEnet for the computational resources usedin this research.
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This preprint was prepared with the AAS L A TEX macros v5.2.
15 –Fig. 1.— Ratio of the deduced luminosity to the luminosity as a function of inclination forZAMS uniformly rotating models with masses of . M (cid:12) (circles), M (cid:12) (crosses), . M (cid:12) (stars), . M (cid:12) (triangles) and M (cid:12) (squares). The ratio of the polar radius to equatorialradius is 0.82, corresponding to a surface equatorial velocity of about 300 km s − .Fig. 2.— Ratio of the deduced effective temperature to the actual effective temperature(defined as the luminosity divided by the total surface area) as a function of inclination forthe same models as in 1. Note the general agreement, although the results for the M (cid:12) suggest that there are limits to the applicability of the scaling.Fig. 3.— The dots show bolometric corrections for the SEDs of all models and inclinations.The stars denote plane parallel model atmospheres with log g = 4 . and the trianglesdenote those with 4.000. Note that the bolometric corrections are not significantly differentbetween the rotating models and the spherical models.Fig. 4.— Plot of the effective temperature as a function of latitude (symbols on the dottedline) and the deduced effective temperature as a function of inclination (symbols on thedashed line). The diamonds, triangles, and crosses for the effective temperature refer tothe results scaled from the . M (cid:12) ZAMS model, the results scaled from the . M (cid:12) ZAMSmodel, and the actual effective temperatures for the evolved model. The circles, squares, andcrosses for the deduced effective temperatures refer to the results scaled from the . M (cid:12) ZAMS model, the results scaled from the . M (cid:12) ZAMS model, and the actual effectivetemperatures for the evolved model. A temperature range of K is also indicated and itis clear that all three temperatures agree with each other for all cases considered.Fig. 5.— The deduced luminosity as a function of inclination for the evolved model basedon the scaled . M (cid:12) ZAMS model (circles), the scaled . M (cid:12) ZAMS model (squares), andfor the actual evolved model itself (crosses).Fig. 6.— Diagram showing representation of the scaling algorithm (see section 4). It showsthe information required from both the model and the star, as well as how the variousunknowns of the star are determined from the scaling are presented. 16 –
Inclination [deg]
Luminosity ratio
Fig. 1.— 17 –
Inclination [deg]
Temperature ratio
Fig. 2.— 18 –
B-V [mag] BC V Fig. 3.— 19 –
Inclination or Colatitute (Degrees) E ff e c t i v e T e m pe r a t u r e and D edu c ed T e m pe r a t u r e ( K ) Fig. 4.— 20 –
Inclination (Degrees) D edu c ed Lu m i no s i t y ( L s un ) Fig. 5.— 21 – • From reference model: – Same R p /R eq as star – As function of colatitude: ∗ Surface radius ∗ Surface T eff ∗ log g eff ∗ Rotational velocity( V ) – Known actual L ( L ac )and actual T eff ( T eff , ac ) – Deduced T eff for variousinclinations • From star: – Deduced T eff – Inclination ( i ) – Observed V sin i – R p /R eq – R eq • New surface parameters for star: – V eq from V sin i and i ∗ From eq. 1: V ( θ ) for any θ – L ac , star = (cid:0) L d , star /L d , model (cid:1) obs i × L ac , model – With L ac and T eff , ac for both the star and the model, theratio of the radii can be computed because both stars havethe same shape – g eff ( θ ) can be found with R ( θ ) and V ( θ ) using eq. 3 andsubsequent discussionObservables can be calculated:(+) SED(+) Line profiles(+) Synthetic interferometry1