- Asteroseismic determination of stellar angular momentum
- 1 Introduction
- 2 The relationship between oscillation frequencies and rotation
- 3 Seismic determination of the moment of inertia
- 4 Conclusions
- References

A&A 402, 683-692 (2003)

DOI: 10.1051/0004-6361:20030239

**F. P. Pijpers ^{1} **

Theoretical Astrophysics Center (TAC), Department of Physics and Astronomy, Aarhus University, Ny Munkegade, 8000 Århus C, Denmark

Received 29 November 2002 / Accepted 14 February 2003

**Abstract**

The total angular momentum of stars is an important
diagnostic for testing theories of formation of
stars. Angular momentum can also play a decisive role in
the evolution of stars, in particular towards the higher
masses. Asteroseismology can play an important role
in providing measurement of this parameter. The spectrum
of stellar oscillations is affected by rotation in a way
similar to the splitting of spectral lines through the
magnetic Zeeman effect. It is shown here that this can be
used to determine accurately the total angular momentum
of stars, taking full differential rotation into account,
and without any dependence of inclination angle of rotation axis
with respect to the line of sight. All aspects of this
technique are addressed, including the issue of mode
identification, determination of the moment of inertia,
and the influence of the reference model on the seismic
inference. It is a realistic prospect to do
asteroseismogyrometry with planned space satellite
missions such as MOST, MONS, COROT, and Eddington.

**Key words: **stars: oscillations -
stars: evolution -
stars: rotation

Previous asteroseismic work on rotation has concentrated on the
possibility of measuring differential rotation in Scuti stars
(Goupil et al. 1996) and in white dwarf stars (Kawaler et al. 1999). Scuti stars are multimode pulsators. In principle two separate driving
mechanisms could operate in these stars (cf. Samadi et al. 2002); the
opacity driving mechanism that also operates in large amplitude
pulsators such as Cepheids and RR Lyrae variables, and the turbulent
driving by convection that is known from the Sun. Because of
this Goupil et al. (1996) assumed that modes with *l* up to 4 or 6 might
be detectable in these stars, where *l* is the number of node lines
of the pulsation mode across the stellar surface. In stars more
similar to the Sun, where only the turbulent driving mechanism
is expected to operate, the oscillation amplitudes are likely to
be smaller because of stronger damping effects. This means that the
cancellation in the signal, due to averaging over the stellar surface,
for the higher *l* values may limit the range in *l* for which a
signal is measurable to
or even .
In the best
studied oscillating white dwarf so far, PG 1159 (Kawaler et al. 1999), only
*l*=1 modes could be used for the determination of an internal
rotation rate, with a marginal detection of radially differential
rotation. It is therefore of interest to investigate what measures of
internal rotation can be determined given more pessimistic assumptions
than those of Goupil et al. (1996).

The angular momentum of stars is one important example of a parameter that is of astrophysical interest and that can be determined with reasonable accuracy even from relatively little data. This paper presents a method for the measurement of the total angular momentum of stars from asteroseismology: asteroseismogyrometry.

Section 2 of this paper gives a brief overview of the relationship between oscillation frequencies and rotation, shows how the angular momentum can be inferred, and presents tests of this method using solar data. Section 3 presents a method for determining the moment of inertia, and also discusses issues concerning mode identification and the choice of reference model.

where is known as the large separation and the small separation is defined as . The large and small separation can be used as measures of mass and evolutionary age of the star (cf. Christensen-Dalsgaard 1993b). The rotational term is responsible for the splitting up of multiplets: modes with different

with respect to spherical polar coordinates : here and are calculable functions, given a stellar model; ; and

(3) |

where

Here and in the following, the radial variable

where

being the density and , the components of the displacement eigenfunction in the non-rotating star, and

and

G^{lm}_{1} |
|||

G^{lm}_{2} |
(8) |

with

(9) |

A different but equivalent way of describing the effect of rotation on the oscillation frequencies is through so-called a-coefficients. The a-coefficients are constructed by fitting polynomials in

(10) |

Here the polynomials have degree

(11) |

specify the polynomials completely. As noted in Appendix A of Schou et al. (1994) the polynomials are usually constructed in an iterative process. However explicit expressions exist as shown by Pijpers (1997), and the equivalent expression of Eq. (4) is:

where the "mode kernels'' are now given by:

in which the

The factors

(15) |

For

(16) |

with the moment of inertia kernel :

(17) |

where

The Subtractive Optimally Localised Averages (SOLA) algorithm
(Pijpers & Thompson 1992; Pijpers & Thompson 1994) provides measures of the rotation
rate by constructing appropriate linear combinations of the kernels,
and the data, from expression (5) or from expression (13).
For resolving the rotation rate as a function of *r*and
one would construct a linear combination of kernels that
would resemble as closely as possible for instance a two-dimensional
Gaussian. For measuring the total angular momentum one instead
constructs a linear combination of kernels, that resembles as closely
as possible the kernel .

Recalling the expression for the associated Legendre polynomial
(cf. Gradshteyn & Ryzhik 1994):

(18) |

it can be seen that the expressions (14) for

which means that they have the same latitudinal (

It is convenient at this point to normalise the density by the mean
density of the star which defines

(20) |

where

(21) |

where is defined as

and defined as

The normalisation of these kernels refers to both and giving unity when integrated over their respective domains. For a homogeneous gas sphere , for real stars . Because the two

in which the are new, normalised kernels, that are functions of

(25) |

and the factors are constants of order unity, defined by:

(26) |

The problem of finding the SOLA estimate of the total angular momentum

is minimised. The term is added in (27) because there are measurement errors on the a-coefficients, for which

(28) |

In order to estimate the error for

(29) |

because of the normalisation of the kernels. In this case there is no need to attempt to construct a linear combination of kernels that matches and instead one can use weights resulting in the minimal variance for

(30) |

In this case the contribution to the error estimate for

(31) |

It is usually not known a-priori whether a star rotates as a solid body. However if there is no trend or correlation of the

Application of the method to helioseismic data from the Global
Oscillations Network Group (GONG) network and also data from the Solar
and Heliospheric Observatory (SoHO) satellite yields a solar total angular
momentum of:
(Pijpers 1998). In Fig. 1 are plotted the SOLA
coefficients *c*_{nl} for the latter dataset as a function of the turning
point radius
of the mode in the Sun defined by:

(32) |

where is the internal solar sound speed. Although this set included modes with

The same solar data set after excluding all modes for which *l*>4contains 39 modes. The value of *H* obtained is
,
where the error is
just the statistical error propagated from the data. There is an extra
contribution, not included in this error estimate, which comes from
the mismatch between the averaging kernel
and the target kernel
which is of
course rather larger here than it is for the full solar set.

If instead all modes for which *l*>2 are excluded, only 14 modes
remain and
.
The mismatch between the averaging kernel and the target kernel is
substantial, and the close correspondence between this *H* and the
value obtained for the full set is in part through fortuitous cancellation:
the averaging kernel oscillates around zero with a large amplitude and
this happens to cancel out in the integration.

Evidently it is possible to obtain an accurate estimate of the solar
angular momentum even from small sets of data covering a limited range
in *l*. It is realistic to expect to observe in stars with the upcoming
satellite projects as many multiplets or more over the same limited
range of *l*, with a comparable precision of the frequencies and *a*_{1}
coefficients. Thus stellar angular momenta can be measured with a
precision comparable to the one for the Sun shown here.

(33) |

where is the vector composed of all the coefficients

(34) |

where

(35) |

Adding the constraint that the sum of the coefficients be unity is implemented by augmenting with a row and a column filled with 1's except for the diagonal element which is 0. Further the vector is augmented with a single element 1.

For a small set of stellar models computed using the stellar evolution code of Christensen-Dalsgaard (cf. Christensen-Dalsgaard 1993a) the scaled angular momentum is calculated and shown in Fig. 3 and the appropriate evolutionary tracks in Fig. 2. Every model has been evolved from Zero Age Main Sequence (ZAMS) until a time that is roughly its main sequence life time. The same He and heavy element abundance is used for all models, as well as the same properties for the convection which is parameterised using a standard mixing length recipe.

Figure 3:
The scaled moment of inertia
normalised to the
solar value
versus the mean stellar
density in solar units, for the stellar models of
Fig. 2. For all models the mean density
and
decrease as a function of time while on the
main sequence. |

Beyond the main sequence, as these stars evolve towards and up the giant branch, their mean density will tend to continue to decrease. However, the moment of inertia at some point increases quite sharply. Stellar models with masses below have a convective envelope, models with masses higher than that value have little or no convective envelope but instead a convective core that increases in fractional mass with the total mass. In Fig. 2 the evolution for the lower mass stars first is directed towards higher before turning around whereas the higher mass stars evolve towards lower throughout this evolutionary stage. For each evolutionary track in Fig. 2 the corresponding track is shown in Fig. 3: the lower mass tracks appear to correspond to a single relation between mean density and whereas the higher mass tracks lie roughly parallel to each other. For the purposes of this paper the important point to notice is that the value of between stars of different masses, and also for the same star during its life time, changes by as much as 50%.

Here and are the square of the sound speed and the density, and refers to the difference in these quantities between the star and the reference model, which are functions of

In the same spirit of linearisation around a reference model the aim
is now to measure the difference
between the moment of
inertia of the star and the reference model. Using Eq. (22)
an expression for the relative difference in moment of inertia is:

(37) |

where the kernel is the same as before, given by Eq. (23). Using the SOLA method the task is now to find coefficients

with the additional constraint of unimodularity:

The in Eq. (38) are weights that must be chosen to balance the various contributions to the function to be minimised. This method has been used for inversions for resolving solar structure (cf. Dziembowski et al. 1994; Basu et al. 1996; Rabello-Soares et al. 1999; Di Mauro et al. 2002). It should be noted also that the minimization is usually done under the additional constraint that the total mass for the star and the reference model be identical:

It can be seen that Eq. (40) is of the same form as Eq. (36) but instead of a frequency difference, there is a single datum 0, and the kernels are:

= | |

= 0 . | (41) |

In this case the measurement error for the datum 0 is the relative error on the stellar mass. For the Sun this relative error is 10

There are various ways of carrying out the surface term
filtering. One can add constraints in the same way as is done for
Eqs. (39) and (40) using for instance a set of orthogonal
polynomials in frequency
of low order. Thus one adds
constraints of the following form:

(42) |

Equivalently one can apply an appropriate linear transform to both the data and the kernels (cf. Basu et al. 1996) in Eq. (36), so that a new linear inverse relation is generated very similar to Eq. (36) except that the surface contribution disappears.

It is known from helioseismology that, given enough data, it is
possible to accommodate all of these constraints, even if instead of
a Gaussian function with a small width is chosen
(cf. Basu et al. 1997). Generally the uncertainty in the localised
estimate of the density difference or sound speed difference increases
as the width of the Gaussian target kernel is reduced. Since the
function
is quite broad, the smaller expected
signal-to-noise and number of *nl*-multiplets obtainable for stars
compared to the Sun, should not prevent a seismic determination of
.
However, for each additional surface term to fit, the
number of degrees of freedom is reduced. Since there is less data
available for other stars it is likely to be more difficult than it is
for the Sun to accommodate all constraints including extensive surface term
filtering. Because of this it is worth noting that the particular
combination of
and
as variables for which to invert may
not be optimal. If the equation of state is assumed to be known, then
it is possible to reformulate Eq. (36) in terms of
differences in
and in Helium abundance *Y*. The kernels for *Y*have large amplitudes only near the surface and the influence of *Y*can therefore be considered as a surface effect to be filtered out.
In that case the term analogous to the term with
can be
removed from Eq. (38). Since the number of constraints on
the inversion is reduced by removing this term, one will obtain a
better match to the kernel
and/or a smaller propagated
statistical error. Detailed testing is deferred at this stage.

Once the coefficients *f*_{nl} are obtained, by solving a set of
linear equations similar to those in Sect. 2.5, the angular momentum
is then estimated as follows:

(43) |

where the subscript refers to the reference model. The issue of choosing the most appropriate reference model is closely linked with the issue of mode identification which is discussed in the next section.

One aid in assigning the appropriate *l* value to a multiplet for a
rotating star is the structure of the multiplet itself. Since *m*cannot be higher than *l*, the number of peaks in the multiplet
determines the minimum possible value of *l*. The oscillatory
signal corresponding to some *m* values within a multiplet can be
suppressed, or too weak to measure given the noise, due to for
instance projection effects because of the angle of inclination of
the rotation axis with the line of sight. This means that for a
given oscillation spectrum multiplet the value of *l* is higher than
indicated by the number of peaks present in the multiplet.

A second aid in assigning *l* is the use of colour information in the
oscillations. The number of nodelines over the surface of a star
increases with *l*. Because of averaging over regions which have
alternating excess or deficit local brightness (or velocity) the
signal that remains for unresolved observations of the stellar disk
reduces quickly with increasing *l* (cf. Christensen-Dalsgaard & Gough 1982). In this
averaging procedure the limb darkening of the stellar surface has to
be taken into account, which is wavelength dependent. By comparing the
amplitude of a signal in (at least) two different wavelength bands it
is possible to identify *l*. The limb darkening effectively functions as a
(non-optimal) spatial filter for *l* (cf. Heynderickx et al. 1994; Bedding et al. 1996).
The factor by which the amplitude of intensity variations has to be
multiplied in order to obtain the signal for unresolved observations
in a spectral band *X* of an oscillation with degree *l* and for
*m*=0 is:

where

The spectral response of the filter for the band *X* is
assumed to be accounted for in
.
For observations in
velocity or equivalent width of spectral lines,
similar relations hold (cf. Bedding et al. 1996). If one has time series
in several bands, or in a combination of intensity, velocity, and
equivalent width, construction of a filter to select a given degree *l*_{0} can be considered as an inverse problem (cf. Christensen-Dalsgaard 1984;
Toutain & Kosovichev 2000). Ideally one would combine the observed relative
amplitudes
with weights *g*_{X}(*l*_{0}) so that:

where

where is a weighting term analogous to those used in Eq. (38), and

If it were possible to do this and if the entire stellar surface were to have been observed, the orthogonality of Legendre polynomials would ensure that Eq. (45) would be satisfied exactly. This can be seen by multiplying in Eq. (44) left- and right-hand sides with the

where the function selects the even element of the pair

(49) |

In practice this minimization procedure usually will not achieve the exact equality of Eq. (47) and thus not achieve an exactly 0 cross-talk between modes for which . Better results can be achieved for increasing numbers of bands (or other independent observables)

(50) |

An integral relation equivalent to Eq. (36) can be obtained by subtracting Eq. (36) for

where the appropriate kernels are defined as:

By doing this one achieves elimination of the lowest order surface term compared to Eq. (36). The other surface terms are still functions of frequency only. If necessary, further filtering can be carried out using the same procedure as outlined in Sect. 3.1. However, it is expected, based on experience in helioseismology, that the remaining surface terms are negligible up to some frequency . At low frequencies the asymptotic relation Eq. (1) loses accuracy, but there is an intermediate frequency range where the are constant and equal to the large separation from asymptotic theory. is determined by the structure of the star through the crossing time for sound waves :

(53) |

If in the intermediate frequency range the individual relative differences of separations between model and star are not consistent with 0, this implies that the mean sound crossing time of the star is different from the reference model. In this case it would be beneficial to use a different reference model for which the sound crossing time is identical to that of the star. As soon as the appropriate value of

for the two variables

At this point it should be noted that there may be a range of
models that have the same sound crossing time as the star, and thus
in the minimum of Eq. (54), but for
which *n*' is not necessarily the same. The minimum value of
for all of these different models should then
indicate the most appropriate reference model and the associated mode
identification. The implication is that mode identification performed
in this way is model dependent. In practice one would perform
inversions using reference models spanning the allowed range, with *n*' also being allowed to vary within an allowed range. Linear
inversion methods, of which SOLA is one, are fast. Therefore no
significant computational effort is required in estimating the
influence of these uncertainties on the uncertainty in the moment of
inertia.

It may be necessary to repeat the same inversion using several different reference models, all matching within the uncertainties the classical parameters, and also the sound crossing time of the star as discussed in Sect. 3.2.2. That model for which the differences between observed and measured frequencies are least is the model most likely to be an adequate reference model. By doing this one can account for effects from propagation of measurement errors on the oscillation frequencies, as well as minimising systematic effects due to linearisation around a reference model that may be too dissimilar from the star.

A method is outlined as well for identifying oscillation modes, i.e.
assigning appropriate values of *n* and *l* to measured frequencies.
In part this is achieved through solving an inverse problem as well.
It is shown that an essential tool in this process are accurate wavelength
dependent limb-darkening profiles for the target stars of
asteroseismic campaigns. In order to obtain these it is necessary to
construct appropriate 3D stellar atmosphere models and/or measure
stellar limb darkening using interferometric techniques. In order
to prepare for data to be gathered with upcoming satellite missions
it is important to commence such work now.

The author thanks Jørgen Christensen-Dalsgaard and Teresa Teixeira for useful comments and discussions when writing the paper. The author also thanks the Theoretical Astrophysics Center, a collaborative centre between Copenhagen University and Aarhus University funded by the Danish Research Foundation, for support of this work. GONG is managed by NSO, a division of NOAO that is operated by the AURA under co-operative agreement with NSF. The GONG data were acquired by instruments operated by the BBSO, HAO, Learmonth, Udaipur, IAC and CTIO. The MDI project operating the SOI/MDI experiment on board the SoHO spacecraft is supported by NASA contract NAG5-3077 at Stanford University.

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