Precession and Nutation in the eta Carinae binary system: Evidences from the X-ray light curve
aa r X i v : . [ a s t r o - ph . S R ] S e p Mon. Not. R. Astron. Soc. , 1– ?? (2005) Printed 1 November 2018 (MN L A TEX style file v2.2)
Precession and Nutation in the η Carinae binarysystem: Evidences from the X-ray light curve
Z. Abraham ⋆ , D. Falceta-Gon¸calves Instituto de Astronomia, Geof´ısica e Ciˆencias Atmosf´ericas, Universidade de S˜ao Paulo, Rua do Mat˜ao 1226,Cidade Universit´aria 05508-090, S˜ao Paulo, Brazil N´ucleo de Astrof´ısica Te´orica, CETEC, Universidade Cruzeiro do Sul, Rua Galv˜ao Bueno 868, CEP 01506-000 S˜ao Paulo, Brazil
ABSTRACT
It is believed that η Carinae is actually a massive binary system, with thewind-wind interaction responsible for the strong X-ray emission. Although theoverall shape of the X-ray light curve can be explained by the high eccentricityof the binary orbit, other features like the asymmetry near periastron passageand the short quasi-periodic oscillations seen at those epochs, have not yetbeen accounted for. In this paper we explain these features assuming thatthe rotation axis of η Carinae is not perpendicular to the orbital plane ofthe binary system. As a consequence, the companion star will face η Carinaeon the orbital plane at different latitudes for different orbital phases and,since both the mass loss rate and the wind velocity are latitude dependent,they would produce the observed asymmetries in the X-ray flux. We wereable to reproduce the main features of the X-ray light curve assuming thatthe rotation axis of η Carinae forms an angle of 29 ◦ ± ◦ with the axis ofthe binary orbit. We also explained the short quasi-periodic oscillations byassuming nutation of the rotation axis, with amplitude of about 5 ◦ and periodof about 22 days. The nutation parameters, as well as the precession of theapsis, with a period of about 274 years, are consistent with what is expectedfrom the torques induced by the companion star. Key words: stars: individual: η Carinae – stars: binaries: general – stars:winds c (cid:13) Z. Abraham and D. Falceta-Gon¸calves
The intensity and spectrum of the high energy X-ray flux, and its strict periodicity, areprobably the strongest evidence of the binary nature of the η Carinae system. The 2-10keV X-ray emission of η Carinae is monitored by the
Rossi X-Ray Timing Explorer RXTE since 1996, and the published results cover two cycles in the 5.52 year periodic light curve(Corcoran 2005). The duration of the shallow minima, as well as the general qualitativebehavior of the light curve, were similar in the two cycles. The long lasting intervals ofalmost stationary intensity were modulated by low amplitude quasi-periodic flares, and thelarge flux increase that occurred before the minima was enhanced by strong short durationflares (Ishibashi et al. 1999). Although the X-ray light curve was successfully reproduced byanalytical approximations involving wind-wind collisions (Ishibashi et al. 1999; Corcoran etal. 2001) and by numerical simulations (Pittard et al. 1998; Okazaki et al. 2008), which alsoreproduced the high resolution spectra obtained with
Chandra (Pittard & Corcoran 2002),some features are still controversial, like the asymmetry near periastron passage, the shortquasi-periodic oscillations seen at those epochs, and the difference in the phases of theseoscillations between the two cycles (Okazaki et al. 2008; Parkin et al. 2009).Besides X-rays, other observational features can be related to wind-wind collision. Abra-ham & Falceta-Gon¸calves (2007) were able to reproduce the HeII λ η Carinae probably coincides with the axis of the Homunculus; theshape of the nebula and the measured latitude dependent stellar wind velocity are strongindications that the rotational velocity is close to its critical value (Smith 2002; Dwarkadas& Owocki 2002). A consequence of the inclination of the rotation axis relative to the axis ofthe orbital plane is that the secondary star faces η Carinae at different latitudes as it movesalong the orbit, and therefore, the latitude dependent velocity and mass loss rate of theprimary’s wind will affect the intensity of the X-rays produced in the wind-wind collisionregion. ⋆ E-mail:[email protected] c (cid:13) , 1– ?? recession and Nutation in η Carinae η Carinae will affect its internal mass distribution,which will depart from spherical symmetry. The torque induced by the companion starwill result in apsidal motions, as seen in other massive binary systems (e.g. see referencesin Claret & Gim´enez 1993). Finally, the inclination of the rotation axis of η Carinae willproduce nodding motions, which will further affect the strength of the wind-wind collisionand the consequent X-ray intensity.In this paper we will take all these effects into account and calculate the X-ray light curveof η Carinae using the analytical approximation derived by Usov (1992) and the orbitalparameters found by Abraham et al. (2005) and Abraham & Falceta-Gon¸calves (2007). Wewill show that for reasonable values of the precession and nutation periods and amplitudes,it is possible to reproduce the asymmetries in the light curve close to periastron passage andthe amplitudes and phases of the short quasi-periodic oscillations for the two binary cyclesobserved by
RXT E . We will use the model derived by Usov (1992) to calculate, at each point of the orbit, theX-ray luminosity originated in the shock heated gas at both sides of the contact surface,which depends on the mass loss rates ( ˙ M p and ˙ M s ) and wind velocities ( V p and V s ) of theprimary and secondary stars, respectively, and on the distance D between them. Pittard &Stevens (2002) showed that for the η Carinae binary system, the major contribution to theX-ray flux comes from the interaction surface of the secondary wind, because of its highertemperature so that the expression derived by Usov (1992) and valid for adiabatic shocksbecomes: F X ( θ s ) = 1 . × πd D (cid:18) ˙ M s V s (cid:19) / ( ˙ M p V p ) / e − τ ( θ s ) , (1)where d is the distance to η Carinae, taken as 2.3 kpc, and τ ( θ s ) the optical depth for X-rayabsorption; ˙ M p and ˙ M s are expressed in units of 10 − and 10 − M ⊙ yr − respectively, V p and V s in units of 10 km s − , and D in units of 10 cm; θ s is the true anomaly, with θ s = 0at periastron. We will not take into account any possible cooling of the very dense shockedgas very close to periastron passage (Parkin et al. 2009).We will assume that ˙ M s and V s have constant values, although Parkin et al. (2009)proposed a reduced secondary wind velocity near periastron to explain the observed changein the X-ray hardness ratio. On the other hand, we will assume that ˙ M p and V p depend on c (cid:13) , 1– ?? Z. Abraham and D. Falceta-Gon¸calves
Figure 1.
Left : geometrical description of the intersection of the binary system orbital plane with the surface of η Carinae atlatitude λ , Φ is the angle between the rotation axis and the perpendicular to the orbital plane; Right : definition of the differentangles and coordinate systems involved in the precession of the line of apsis and nutation of the rotation axis of η Carinae the latitude λ ( θ s ) at which the orbital plane intercepts the side of η Carinae that faces thesecondary star, and can be expressed as (Dwarkadas & Owocki 2002):˙ M ( λ ) = ˙ M (90 ◦ )[1 − Ω cos λ ] , (2) V ( λ ) = V (90 ◦ )[1 − Ω cos λ ] / , (3)with Ω = ω/ω c ; ω is the rotation velocity and ω c = ( GM p /R p ) / its critical value; G isthe gravitational constant, M p and R p are the mass and radius of η Carinae, respectively.These expressions are valid when Ω is close to unity; they were already used to reproducethe observed wind velocity as a function of latitude in η Carinae, as well as the shape of theHomunculus nebula (Smith 2002; Dwarkadas & Owocki 2002).By replacing eq. (2) and (3) in (1) we obtain: F X ( θ s ) = G ( t ) D (cid:20) ˙ M p ( λ )˙ M p (90 ◦ ) (cid:21) / (cid:20) V p ( λ ) V p (90 ◦ ) (cid:21) / e − τ p ( θ s ) , (4)with G ( t ) = 1 . × πd (cid:18) ˙ M s V s (cid:19) / (cid:20) ˙ M p (90 ◦ ) V p (90 ◦ ) (cid:21) / e τ , (5)where we have defined τ ( θ s ) = τ p ( θ s ) + τ ( t ); τ p ( θ s ) represents the absorption produced bythe wind of η Carinae intercepting the line of sight, and τ ( t ) is constant or a slowly varying,phase independent function of time, representing all other sources of absorption. λ ( θ s ) η Carinae must be highly distorted, both by rotation and by the presence of the companionstar in a highly eccentric orbit. Although the total angular momentum in a detached binarysystem is conserved (except for a small amount lost by the stellar winds), energy will bedissipated by the tidal forces, until a minimum energy equilibrium configuration is reached, c (cid:13) , 1– ?? recession and Nutation in η Carinae λ ( θ s ).We will assume that η Carinae rotates with angular velocity ~ω around an axis that forms anangle Φ with the perpendicular to the orbital plane ( z axis in Figure 1), and its projectionon the orbital plane forms an angle Θ with the line of apsis ( x axis), so that:sin λ ( θ s , t ) = sin Θ( θ s , t ) sin Φ( t ) , (6)wheretan Θ ( θ s , t ) = ω y ω x , (7)sin Φ( t ) = q ω x + ω y ω . (8)As a consequence of nutation, the rotation angular velocity vector ~ω will describe a cone ofamplitude ∆ ϕ and period P n around its non-perturbed direction, which forms an angle ϕ with the polar axis z (Figure 1).In a coordinate system ( x ′′ , y ′′ , z ′′ ), in which z ′′ is directed along the unperturbed rotationaxis, the components of ~ω will be ( ω sin ∆ ϕ sin ω n ( t ), ω sin ∆ ϕ cos ω n ( t ), ω cos ∆ ϕ ), with ω n ( t ) = (2 π/P n )∆ t + θ n ; θ n is a constant phase and ∆ t = t − t , with t = 2 , , ~ω in the( x, y, z ) coordinate system, in which z coincides with the orbital axis, the ( x ′′ , y ′′ , z ′′ ) axismust be rotated around y ′′ by an angle ϕ , so that the new z ′ axis coincides with z , and thenaround z ′ by and angle θ ′ s = θ s + 2 π/P p + θ p , to take into account the orbital motion and theprecession of the line of apsis; θ p is a constant phase. These rotations can be representedby the matrix M : M = cos θ ′ s sin θ ′ s − sin θ ′ s cos θ ′ s
00 0 1 cos ϕ ϕ − sin ϕ ϕ , (9)so that ( x, y, z ) T = M ( x ′′ , y ′′ , z ′′ ) T c (cid:13) , 1– ?? Z. Abraham and D. Falceta-Gon¸calves - k e V X -r a y F l u x - k e V X -r a y F l u x - k e V X -r a y F l u x JD - 2.400.000 - k e V X -r a y F l u x Figure 2.
Observed 2-10 keV X-ray flux, from Corcoran(2005), shown as open circles, and model, shown as a continuous line,obtained with the parameters listed in Table 1, but with ∆ ϕ = 0 ◦ (no nutation) in the two upper graphs and ∆ ϕ = 4 . ◦ (cid:13) , 1– ?? recession and Nutation in η Carinae I column density As mentioned before, the optical depth for X-ray absorption was divided into two parts: τ ( θ s ) = τ ( t ) + τ p ( θ s ), where τ p ( θ s ) = σ ph N H ( θ s ) represents the photoelectric absorptionproduced by the unshocked wind of η Carinae intercepting the line of sight to the vertex ofthe X-ray emitting cone, with column density N H , and τ ( t ) that represents all other sourcesof absorption, excluding the stellar wind; σ ph is the cross section of photoelectric absorption,multiplied by the heavy element abundance relative to H. N H ( θ s ) is calculated from: N H ( θ s ) = 14 πµm H (cid:18) ˙ M p V p (cid:19) Z ∞ s sh dss + r , (10)where µ is the molecular weight and m H the mass of the hydrogen atom, ˙ M p and V p arethe mean values of the mass loss rate and wind velocity of the primary star; s is measuredalong the line of sight to the apex of the X-ray source; r = b sin Ψ and s sh = b cos Ψ, where b is the distance from η Carinae to the shock, measured in the orbital plane; Ψ is calculatedfrom:sin Ψ = sin( θ s − θ ) sin i, (11)where i is the inclination of the orbit and θ is the true anomaly at conjunction. Whilethe inclination is one of our model parameters, the value of the input parameter θ is stillcontroversial (Pittard et al. 1998; Corcoran et al. 2001; Falceta-Gon¸calves, Jatenco-Pereira &Abraham 2005; Kashi & Soker 2007; Hamaguchi et al. 2007; Abraham & Falceta-Gon¸calves2007; Okazaki et al. 2008; Falceta-Gon¸calves & Abraham 2009; Parkin et al. 2009). As wewill see later, it affects mostly the opacity near periastron, where the model anyway fails toreproduce the duration of the shallow minima.By solving the integral of equation (10), we can write: τ p ( θ s ) = C τ b sin Ψ (cid:18) π − arctan 1tan Ψ (cid:19) , (12)where C τ = σ ph ˙ M p / πµm H V p .The contribution to the opacity of the unshocked wind of the secondary secondary star ismuch smaller than that of the wind of η Carinae, because of the much smaller value ˙ M s /V s .However, depending on the position of the secondary star on the orbit near periastron,the absorption due to the shocked gas intercepting the line of sight could be large (Falceta-Gon¸calves, Jatenco-Pereira & Abraham 2005; Parkin et al. 2009) and can affect the duration c (cid:13) , 1– ?? Z. Abraham and D. Falceta-Gon¸calves
Table 1.
Input and model parametersInput Model e = 0 . P p = 274 ±
15 years θ = − ◦ θ p = 5 . ◦ ± ◦ P = 2024 days P n = 22 . ± .
04 days A = 15 A.U. θ n = 90 ◦ ± ◦ t = 2 , ,
795 JD Ω = 0 . ± . ϕ = 29 ◦ ± ◦ ∆ ϕ = 4 . ◦ ± . ◦ C x = (7 ± × − C τ = 7 . ± . G = (2200 ± × − erg cm − s − i = 60 ◦ Table 2.
Inclination of the binary system orbit i and of η Carinae rotation axis i ∗ i i ∗ ( η = 0 . i ∗ ( η = 0 . of the minima in the light curve; this issue will not be addressed here since it requiresnumerical simulations. We will use the orbital parameters derived by Abraham et al. (2005) from the observed7-mm light curve of η Carinae during the 2003.5 minimum and listed in the left columnof Table 1. They were successfully used to reproduce the He II λ ◦ (Davidson et al. 2001; Smith 2006), for each inclination i of theorbit relative to the observer, an orientation for the Homunculus axis i ∗ was found, for whichthe reflected line profiles and velocities could be reproduced (Abraham & Falceta-Gon¸calves2007). In Table 2 we present the values of these angles for two values of η = ˙ M s V s / ˙ M p V p .In the next section we will use the value of i ∗ = ϕ + ∆ ϕ , obtained from the model thatreproduced the observed X-ray light curve, to constrain the value of the orbital inclination i . c (cid:13) , 1– ?? recession and Nutation in η Carinae - k e V X -r a y F l u x . JD - 2.400.000 - k e V X -r a y F l u x . Figure 3.
Details of the X-ray emission model without precession and nutation, close to the epochs of the two shallow minima,for three different values for the precession phase θ p : − ◦ , ◦ and 13 ◦ . All other model parameters are shown in Table 1. -0.50-0.30-0.100.100.300.500.70-180 -135 -90 -45 0 45 90 135 180Orbital Phase (degrees) -40-20020406080100120140Density x 100Mass-loss rateVelocityLatitude Figure 4.
Variations with orbital phase of the wind velocity, mass loss rate, wind density at the contact surface, and latitudeat the intersection of η Carinae with the orbital plane facing the companion star. The first two quantities, given by the left axis,are relative to their values at λ = 90 ◦ , the latitude values, in degrees, are given in the right axis. The wind density, relative toits value at λ = 90 ◦ , in units of 1 . × cm − is also displayed at the right axis. We used equations (1) to (12) to model the X-ray light curve of η Carinae, with the orbitalparameters listed in the first column of Table 1. No formal fitting was attempted; instead,the model parameters were changed until they reproduced the general shape of the lightcurve (except the shallow minima), and the amplitude and period of the oscillations that c (cid:13) , 1– ?? Z. Abraham and D. Falceta-Gon¸calves occurred just before them. The parameters that fulfilled these criteria are listed in the lastcolumn of Table 1, and the model superposed to the observed light curve is presented inFigure 2; the two upper graphs represent the model without nodding motions (∆ ϕ = 0), andthe last two graphs include nodding. The multiplicative term G ( t ) in equation (5) was fittedby a function G ( t ) = G [1 + C x ( t − t )], with G and C x having constant values. The quotederrors were obtained by changing the values of each parameter while keeping the othersconstant until the model was no longer acceptable. As mentioned before, no formal fittingwas attempted, but the combination of model parameters could not be changed arbitrarily,since each of them affect some particular feature of the X-ray light curve, as discussed bellow.The parameters Ω and θ p are responsible for the asymmetry in the light curve at bothsides of the shallow minima, as can be seen in the first two graphs in Figure 2. To illustratethis dependence, we plotted in Figure 3 the model light curves close to the minima forΩ = 0 .
975 and several values of θ p , when no precession or nutation are present. We foundthat θ p = 5 . ◦ θ p ∼ ◦ was necessary to reproduce the minimum of 2003.5.The difference in angles was attributed to the motion of the apsis, resulting in the precessionperiod listed in Table 1. A large value of Ω was necessary to get the observed discontinuity;it indicates that η Carinae is rotating at almost its critical velocity, as expected from theepisodes of large mass loss. Of course, if other causes were responsible for the asymmetry inthe X-ray light curve, like changes in opacity or stellar wind parameters, the value of theseparameters should be revised.The nutation parameters P n , θ n and ∆ ϕ are responsible for the amplitude and period ofthe large oscillations that occur before the minima, as can be seen in the two lower graphs inFigure 2; it is important to notice that, near periastron, the oscillations remained in phasein the two cycles for the same initial phase θ n . However, the model does not reproduce theamplitude and the period of the oscillations far from periastron, which could be expectedfrom the variation of the torque of the secondary star acting on η Carinae.The opacity parameter C τ determines the shape of the light curve before the minima,and has no influence at other phases. The linear dependence of G ( t ) with time implies anincrease of about 30% in the X-ray intensity between the two cycles, which coincides withan increase in the optical flux during the same time interval (Martin et al. 2004) and couldmaybe attributed to an overall decrease in opacity.No assumptions were made on the values of the mass loss rates and wind velocities of the c (cid:13) , 1– ?? recession and Nutation in η Carinae G and the comparisonbetween the model hydrogen column density N H and that inferred from the XMM
X-rayspectra observed far from periastron passage (Hamaguchi et al. 2007) allowed us to put someconstrains on their magnitudes. From eq. (5) and assuming τ = 0 we can write:˙ M s V s = G . × − η / (90 ◦ ) (13)where η (90 ◦ ) = ˙ M s V s ˙ M p (90 ◦ ) V p (90 ◦ ) = ˙ M p ( λ ) V p ( λ )˙ M p (90 ◦ ) V p (90 ◦ ) η ( λ ) (14)From our model η (90 ◦ ) = 0 . η (30 ◦ ), resulting that for η (30 ◦ ) = 0 . V s = 3000 kms − , ˙ M s = 8 × − M ⊙ y − , well within the values of the mass loss rate of the secondarystar used in the literature.The hydrogen column density inferred from the X-ray spectra in January 2003 was N H = 9 × cm − (Hamaguchi et al. 2007). Using this value in equation (10) we obtain˙ M p /V p = 4 . × g cm − , where we have used: Z ∞ s sh dss + r = 1 b sin Ψ (cid:18) π − arctan 1tan Ψ (cid:19) = 0 .
062 AU − (15)Assuming a wind velocity of 500 km s − for η Carinae, we find ˙ M p = 3 . × − M ⊙ y − ,also consistent with the values found in the literature.Finally, the model opacity for January 2003 was τ p = 0 .
69, which together with theobserved hydrogen column density gives a value for σ ph = τ p /N H = 4 . g − , consistentwith the opacity to 3 keV photons of a 10 -10 K gas (Parkin & Pittard 2008).From the inclination of the rotation axis of η Carinae we estimated that, relative to theobserver, the orbit has an inclination i ∼ ◦ − ◦ , for η varying between 0.2 and 0.1, ascan be seen in Table 2. The variation of the wind velocity, mass loss rate and wind densityalong the orbital period, as well as the variation in λ are shown in Figure 4 for the firstorbital period.Although the eccentricity and period of the binary orbit used in the model agree withthose used by other authors (Pittard et al. 1998; Ishibashi et al. 1999; Corcoran et al. 2001;Pittard & Corcoran 2002; Okazaki et al. 2008), the value of θ is still controversial. The modelpresented in this paper was calculated for θ = − ◦ ; changing its value to θ = +45 ◦ will nothave any effect on the X-ray emission but affect the absorption, mostly close to periastrompassage, where the model does not anyway reproduce the X-ray light curve; however, using c (cid:13) , 1– ?? Z. Abraham and D. Falceta-Gon¸calves -3-2-112 0.5 1.5 2.5 3.5 4.5 p /A) l og k M S /M P k n = 0123 Figure 5.
Asymmetry parameter k of the internal mass distribution of η Carinae , produced by rotation and gravitationaltorque induced by the companion star, versus the ratio between the primary star radius and the orbital semi-major axis, forthree values of the primary to secondary mass ratio M s /M p . Horizontal lines are the expected values of k for polytropes ofindex n = 0, 1, 2 and 3. θ = +45 ◦ will not reproduce the shape and central velocity of the He II λ As mentioned before, the model does not adjust the amplitude and period of the oscillationsin the X-ray light curve far from periastrom passage. From Figure 2 we can also see thatit does not reproduce the duration of the shallow minima. In fact, neither the analyticalmodels nor the numerical simulations developed up to the present time were able to accountfor the extended minima as a result of X-ray photoelectric absorption by the dense wind of η Carinae intercepting the line of sight (Pittard et al. 1998; Ishibashi et al. 1999; Corcoranet al. 2001; Pittard & Corcoran 2002; Hamaguchi et al. 2007). Possible explanation are theincrease in the H column density to ∼ − cm − due to additional material providedeither by a slowly expanding shell of shocked material formed during periastron passage(Falceta-Gon¸calves, Jatenco-Pereira & Abraham 2005), or by the primary wind itself, which”engulfs” the secondary star (Okazaki et al. 2008), or simply by the suppression of thesecondary wind due to accretion of matter from the close primary star (Soker 2005; Akashi,Socker & Behar 2006) . c (cid:13) , 1– ?? recession and Nutation in η Carinae p /A) D j M S /M P Figure 6.
Angle ∆ ϕ between the total and orbital angular momenta as a function of R p /A for several values of the mass ratio M s /M p . The horizontal line shows the value of ∆ ϕ obtained from the model. p /A) l og t t alig t circ t TF t spin Figure 7.
Time-scales for tidal friction t TF , alignment t alig , circularization t circ , and synchronization t spin for ( M s /M p = 0 . The precession and nutation parameters are related to the torques of the secondary acting onthe fast rotating non-spherical primary star and can provide constrains on the stellar masses,internal structure and orbital stability. Hut (1982) derived expressions for the evolution ofthe orbital and spin parameters in highly eccentric binary system, assuming that the tidalbulges lag by a constant angle from the line that joins the stars. Eggleton et al. (1998)obtained an expression for this angle, considering that the dissipative force is proportionalto the rate of change of the quadrupole tensor of the stars, as seen by an observer thatrotates with them.The precession rate of the apsis, is independent of dissipation and, neglecting the torques c (cid:13) , 1– ?? Z. Abraham and D. Falceta-Gon¸calves M s /M p ( R p / A ) D j = 4.5° j = 29° D j = 4.0° j = 24° D j = 5.0° j = 34° Figure 8.
Relation between the mass ratio M s /M p and R p /A , for the combination of model parameters ( ϕ, ∆ ϕ ) = (34 ◦ , . ◦ ◦ , . ◦ ◦ , . ◦ of η Carinae on the secondary star, can be expressed as: PP p = 15 kf ( e ) (cid:18) M s M p (cid:19)(cid:18) R p A (cid:19) + k Ω g ( e ) (3 cos ϕ − (cid:18) R p A (cid:19) (16)with f ( e ) = (1 − e ) − (1 + 32 e + 18 e ) (17) g ( e ) = (1 − e ) − , (18)where k is the constant part of the quadrupole moment, which depends on the internal massdistribution of the primary star, A is the semi-major axis of the orbit, and R p is the radiusof the primary star.In massive binary systems, the measured precession period P p , together with the orbitalparameters, stellar masses and radii, are used to calculate k and improve the stellar structuremodels (e.g. Claret & Gim´enez 1993). Since for η Carinae the orbital and stellar parametersare unknown, we will take M s /M p and R p /A as free parameters.In Figure 5 we show the values of k that satisfy equation (13) for P/P p = 0 .
02 (obtainedfrom our model) as a function of R p /A , for several values of M s /M p . The maximum valueallowed for R p /A corresponds to the separation between the stars at periastron: R p /A =(1 − e ) = 0 .
05. The horizontal lines represent the values of k for rotating polytropes withindices n = 0 , , n = 0 corresponds to a star with constantdensity while for n = 1 the radius is independent of the central density.Another constrain for the ratios M s /M p and R p /A can be obtained from the amplitude c (cid:13) , 1– ?? recession and Nutation in η Carinae ϕ , which represents the angle between the orbital and total angular momenta,and can be expressed as:cot ∆ ϕ = cot ϕ + hIω sin ϕ , (19)where h is the orbital angular momentum, I = M p r g R p the momentum of inertia of theprimary star and r g a parameter that depends on its internal structure. We can also write h/Iω in terms of M s /M p and R p /A : hIω = (1 + M s /M p ) / M s /M p (cid:18) R p A (cid:19) / r g Ω(1 − e ) / , (20)In Figure 6 we present the relation between ∆ ϕ and R p /A for ϕ = 29 ◦ (obtained fromour model), and M s /M p = 0 .
1, 0.4, and 1.0. In the calculation, we used an interpolatedrelation between r g and k for rotating polytropes obtained from Motz (1952):log r g = 0 .
453 log k − .
307 (21)The value of ∆ ϕ = 4 . M s /M p ∼ . r g , andshould be consider only in the context of a consistency test for the parameters derived fromthe X-ray light curve.The last parameter derived from the observations is the nutation period, which accordingto Eggleton et al. (1998) depends on both the conservative torques and those produced bytidal dissipation and can be written as: PP n = k r g M s M p (cid:18) M s M p (cid:19) − / (cid:18) R p A (cid:19) / cos ϕ (1 − e ) / (22)+ 38 r g M s M p (cid:18) M s M p (cid:19) − (cid:18) AR p (cid:19) e (1 + 1 / e )(1 − e ) / P πt T F , where t T F is the tidal friction time scale.When only non-dissipative torques, represented by the first tem in eq. (19), are con-sidered, the nutation period turns out to be several orders of magnitude larger than thatobtained from our model ( P n ∼ = P/
92) for any combination of k , ( R p /A ) and ( M s /M p )given by equation (13); therefore, the observed nutation must be produced by the dissi-pative torques, i.e. the second term in eq. (19), and its value can be used to estimate thedissipative time-scale t T F : c (cid:13) , 1– ?? Z. Abraham and D. Falceta-Gon¸calves t T F = 38 r g M s M p (cid:18) M s M p (cid:19) − (cid:18) AR p (cid:19) e (1 + 1 / e )(1 − e ) / P n π . (23)In Figure 7 we display t T F as a function of R p /A for M s /M p = 0 . yt y = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) dydt (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , (24)where t y = ( t alig , t spin , t circ ) for y = ( ϕ, ω, e ), respectively.For the case of a highly eccentric orbit, they can be obtained from Hut (1982): t alig = t T F (cid:20) kr g (cid:18) M s M p (cid:19) (cid:18) R p A (cid:19) H (1 − e ) (cid:21) − , (25) t spin = t T F (cid:20) kr g (cid:18) M s M p (cid:19) (cid:18) R p A (cid:19) H (1 − e ) (cid:21) − , (26) t circ = t T F (cid:20) k (cid:18) M s M p (cid:19)(cid:18) R p A (cid:19) H (1 − e ) / (cid:21) − , (27)with H = h ( ε )Ω (cid:18) M s M p (cid:19) / (cid:18) R p A (cid:19) / − − e ) / (1 + e ) (1 − η ) h ( ǫ ) , (28) H = h ( ε )Ω (cid:18) M s M p (cid:19)(cid:18) R p A (cid:19) / − − e ) / (1 + e ) h ( ǫ ) , (29) H = h ( ε ) (30) − h ( ε )Ω 112117 (1 − e ) / (1 + e ) (cid:18) M s M p (cid:19) − / (cid:18) R p A (cid:19) − / η = Iωh , (31) ǫ = 1 − e (32) h ( ε ) = 1 − ε + 3511 ε − ε + 57 ε − ε ε , (33) h ( ε ) = 1 − ε + 443140 ε − ε + 5170 ε − ε ε . (34) h ( ε ) = 1 − ε + 345143 ε − ε + 511 ε − ε ε , (35) h ( ε ) = 1 − ε + 20584 ε − ε + 1942 ε − ε ε . (36) c (cid:13) , 1– ?? recession and Nutation in η Carinae t alig , t spin , and t circ as a function of R p /A for M s /M p = 0 . T T F is of the order of 10 years, the timescales for alignment,synchronization and circularization are larger than 10 years, which means that the binarysystem did not reached yet its equilibrium configuration, considering an evolution timescaleof 10 − years for the massive stars.We should notice in eq. (27) that H can be negative, implying that the eccentricity canincrease with time, which corresponds to an unstable orbit. This occurs when the spin ofthe primary star is larger than the orbital angular velocity at periastron, or: R p A < − e (1 + e ) / M s /M p ) / (37)For M s /M p = 0 . R p /A = 0 . ϕ = 29 ◦ , ∆ ϕ = 4 . ◦ ϕ = 34 ◦ , ∆ ϕ = 5 . ◦
0) and ( ϕ = 24 ◦ , ∆ ϕ =4 . ◦ M s /M p and R p /A for the binary system should lie between these twoextreme lines. We were able to reproduce the general features of the 2-10 keV X-ray light curve of η Carinaeobtained by
RXT E (Corcoran 2005) during two cycles, including the amplitudes and phasesof the short period oscillations that occurred prior to the shallow minima, assuming thatthe star rotates around an axis that is not perpendicular to the orbital plane, so that thesecondary star faces η Carinae at different latitudes as it moves along the orbit, using thefact that both the mass loss rate and the terminal wind velocity of η Carinae are latitudedependent. According to the model, the star should be rotating with a fraction Ω = 0 . ◦ with the axis of theorbital plane, nutates with an amplitude of about 5 ◦ and a period of 22.5 days. We alsofound that the line of apsis precesses with a period of about 274 years. According to theresults of Abraham & Falceta-Gon¸calves (2007), the inclination obtained for the rotationaxis of η Carinae implies that the inclination of the binary orbit relative to the observermust be i ∼ ◦ − ◦ , depending on the ratio of the wind momenta ( η ∼ . − . c (cid:13) , 1– ?? Z. Abraham and D. Falceta-Gon¸calves dependent opacity due to the wind of η Carinae intercepting the line of sight, and introduceda phase independent absorption, which decreased linearly with time and explained the overallincrease in the X-ray flux between the two cycles. We used the precession period and nutationamplitude and rate to constrain the mass ratio of the binary system and the radius of theprimary star relative to the semi-major orbital axis. We found that the orbit is stable if theradius of η Carinae is larger than 0.035 times the orbit major axis, and the mass of thecompanion star at least half the mass of η Carinae. Finally we found that for stable orbits,the time scale for orbit circularization, spin alignment and synchronization is much largerthan the lifetime of the stars.
ACKNOWLEDGMENTS
This work was partially supported by the Brazilian agencies FAPESP and CNPq.
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