Impact of the stellar oblation effect on estimation of the magnetic dipole strength
The surface magnetic field structure of an ellipsoidal star is modelled in the frame of the magnetic charge description (MCD) approach. It is shown that the stellar oblation effect can lead to the essential overestimation of the magnetic dipole strength value obtained from the mean crossover effect...
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Головна астрономічна обсерваторія НАН України
2005
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| Назва видання: | Кинематика и физика небесных тел |
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| Цитувати: | Impact of the stellar oblation effect on estimation of the magnetic dipole strength / V. Khalack // Кинематика и физика небесных тел. — 2005. — Т. 21, № 5-додаток. — С. 299-302. — Бібліогр.: 8 назв. — англ. |
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irk-123456789-796632015-04-04T03:01:53Z Impact of the stellar oblation effect on estimation of the magnetic dipole strength Khalack, V. MS3: Physics of Stars and Galaxies The surface magnetic field structure of an ellipsoidal star is modelled in the frame of the magnetic charge description (MCD) approach. It is shown that the stellar oblation effect can lead to the essential overestimation of the magnetic dipole strength value obtained from the mean crossover effect (up to 12%) and quadratic magnetic field (up to 8%) in comparison with its theoretical value obtained for the case of the spherically symmetric star. Taking into account the gravity-darkening phenomenon is argued that this overestimation increases with the growth of the effective gravity difference at the equator and poles of the star. The data of the mean longitudinal magnetic field provide the most correct estimation of the magnetic dipole strength value in the ellipsoidal star. 2005 Article Impact of the stellar oblation effect on estimation of the magnetic dipole strength / V. Khalack // Кинематика и физика небесных тел. — 2005. — Т. 21, № 5-додаток. — С. 299-302. — Бібліогр.: 8 назв. — англ. 0233-7665 http://dspace.nbuv.gov.ua/handle/123456789/79663 en Кинематика и физика небесных тел Головна астрономічна обсерваторія НАН України |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine |
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English |
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MS3: Physics of Stars and Galaxies MS3: Physics of Stars and Galaxies |
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MS3: Physics of Stars and Galaxies MS3: Physics of Stars and Galaxies Khalack, V. Impact of the stellar oblation effect on estimation of the magnetic dipole strength Кинематика и физика небесных тел |
| description |
The surface magnetic field structure of an ellipsoidal star is modelled in the frame of the magnetic charge description (MCD) approach. It is shown that the stellar oblation effect can lead to the essential overestimation of the magnetic dipole strength value obtained from the mean crossover effect (up to 12%) and quadratic magnetic field (up to 8%) in comparison with its theoretical value obtained for the case of the spherically symmetric star. Taking into account the gravity-darkening phenomenon is argued that this overestimation increases with the growth of the effective gravity difference at the equator and poles of the star. The data of the mean longitudinal magnetic field provide the most correct estimation of the magnetic dipole strength value in the ellipsoidal star. |
| format |
Article |
| author |
Khalack, V. |
| author_facet |
Khalack, V. |
| author_sort |
Khalack, V. |
| title |
Impact of the stellar oblation effect on estimation of the magnetic dipole strength |
| title_short |
Impact of the stellar oblation effect on estimation of the magnetic dipole strength |
| title_full |
Impact of the stellar oblation effect on estimation of the magnetic dipole strength |
| title_fullStr |
Impact of the stellar oblation effect on estimation of the magnetic dipole strength |
| title_full_unstemmed |
Impact of the stellar oblation effect on estimation of the magnetic dipole strength |
| title_sort |
impact of the stellar oblation effect on estimation of the magnetic dipole strength |
| publisher |
Головна астрономічна обсерваторія НАН України |
| publishDate |
2005 |
| topic_facet |
MS3: Physics of Stars and Galaxies |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/79663 |
| citation_txt |
Impact of the stellar oblation effect on estimation of the magnetic dipole strength / V. Khalack // Кинематика и физика небесных тел. — 2005. — Т. 21, № 5-додаток. — С. 299-302. — Бібліогр.: 8 назв. — англ. |
| series |
Кинематика и физика небесных тел |
| work_keys_str_mv |
AT khalackv impactofthestellaroblationeffectonestimationofthemagneticdipolestrength |
| first_indexed |
2025-07-06T03:41:10Z |
| last_indexed |
2025-07-06T03:41:10Z |
| _version_ |
1836867409465573376 |
| fulltext |
IMPACT OF THE STELLAR OBLATION EFFECT
ON ESTIMATION OF THE MAGNETIC DIPOLE STRENGTH
V. Khalack
Main Astronomical Observatory, NAS of Ukraine
27 Akademika Zabolotnoho Str., 03680 Kyiv, Ukraine
e-mail: khalack@mao.kiev.ua
The surface magnetic field structure of an ellipsoidal star is modelled in the frame of the magnetic
charge description (MCD) approach. It is shown that the stellar oblation effect can lead to the es-
sential overestimation of the magnetic dipole strength value obtained from the mean crossover
effect (up to 12%) and quadratic magnetic field (up to 8%) in comparison with its theoretical value
obtained for the case of the spherically symmetric star. Taking into account the gravity-darkening
phenomenon is argued that this overestimation increases with the growth of the effective gravity
difference at the equator and poles of the star. The data of the mean longitudinal magnetic field
provide the most correct estimation of the magnetic dipole strength value in the ellipsoidal star.
INTRODUCTION
The upper layer of stellar gas is stressed by the mutual action of radiation and gas pressure, centrifugal and gravi-
tational forces and forms the respective (often symmetrically spherical) shape of stellar surface. For a rapidly
rotating star the rotational deformation of the stellar surface could be essential and leads to the observable
stellar oblation on the rotational poles. Furthermore, the essentially high stellar rotation leads to the surface
brightness distribution, as it was originally shown in [8]. Nowadays, a common model of a differentially rotating
star is characterized by the highly anisotropic turbulence which has a strong horizontal component [7]. This
horizontal turbulence strongly reduces the horizontal differential rotation, so that rotation varies only radially.
This means that the angular velocity is uniform at the surface of isobaric shells and, consequently, the rotation
is usually said to be “shellular”. The presence of a magnetic field generated, for example, by the Tayler–
Spruit dynamo mechanism does not affect the shape of the equipotentials, but strongly amplifies the horizontal
coupling.
The structure of a surface magnetic field for an ellipsoidal star can be sufficiently well described in the frame
of the magnetic charge description (MCD) method [1, 2, 4]. The aim of this research is to consider the stellar
oblation influence on the results of the surface magnetic field modelling.
MAGNETIC FIELD AT THE SURFACE OF AN ELLIPSOIDAL STARS
The surface of a rapidly rotating star has a shape that can be well described with the help of the model of
a rotating Maclaurin ellipsoid in which a small semi-axis Rp is directed along the rotational axis, while a large
semi-axis Re lies in the equatorial plane. For such a model the distance of an arbitrary surface point M from
the stellar centre in the Rp units is:
ρ =
1√
1 − ε2 cos2 δ
, (1)
where eccentricity is defined as ε =
√
1 − R2
p/R2
e and δ specifies point’s latitude in the Spherical reference
frame related to the Star (SS). According to the main relations obtained in [5] the integrated radiation flux
from a visual surface of the ellipsoidal star with a weak oblation (ε2 � 1) can be expressed as
I0 � π
30
[
10 (3 −u) + ε2(15 (1 +cos2 i) −u (7 −cos2 i))
]
, (2)
where i specifies the inclination of the stellar rotational axis to the line of sight and u is the limb darkening
coefficient.
c© V. Khalack, 2004
299
The corresponding equations for the Cartesian components of the magnetic field vector at the point M of
ellipsoid’s surface can be deduced from [2] and applied for calculation of the surface magnetic field characteristics.
Supposing the presence of a centered symmetric magnetic dipole inside the ellipsoidal star, we can obtain
the expressions for the magnetic field characteristics according to [5, 6] for the star with the weak oblation
effect. The corresponding equations have the following form: for the mean longitudinal magnetic field
<Bz >� πBd
120I0
{
2Z0(15+u) +3ε2X0 sin i cos i(35−11u)− ε2Z0[15(3 −7 cos2i)−u(5−17 cos2i)]
}
, (3)
for the mean crossover effect
< Bc > � −πBdveY0 sin i
2240 I0
{ 56 (8− 3u) − b [ 64 (1 + 2 cos2 i) − u (29 + 23 cos2 i)]
−4ε2[ 8 (13 − 16 cos2 i) − u (34 − 23 cos2 i)]}, (4)
and for the mean quadratic magnetic field
〈Bmq〉2 � πB2
d
4I0
{
525 − 193u
210
+ Z2
0
105 + 13u
105
+ ε2
(
105−247u
140
+ Z0X0 sin i cos i (21−5u)
+ sin2 i
[
X2
0
245 − 117u
40
− 2205− 157u
240
]
+ Z2
0
[
1365− 181u
140
− sin2 i
651− 11u
48
])}
, (5)
where ⎧⎪⎪⎨
⎪⎪⎩
X0 = ρ1[cosβ sin i −(1−ε2)sinβ cos i cos(λ1+ϕ)],
Y0 = ρ1[1 − ε2] sinβ sin (λ1 + ϕ),
Z0 = ρ1[cosβ cos i +(1−ε2)sinβ sin i cos(λ1+ϕ)]
(6)
and
ρ1 =
1√
(1 − ε2)2 sin2 β + cos2 β
. (7)
Here variable β defines the angle between the magnetic dipole axis and the axis of axial stellar rotation while
variables λ1 and ϕ specify the longitude of the probe magnetic charge in the SS reference frame and rotational
phase of the star with the equatorial velocity ve, respectively. The magnetic dipole strength is defined as
Bd = 4aQ/R2
p (see, e.g., [3]).
In Eqs. (3)–(5) the influence of the stellar oblation effect on the magnetic field characteristics is taken into
account by the terms proportional to ε2. In order to estimate its contribution the following function is calculated
for some magnetic field characteristics
Λm(ε, u, i, β, ϕ) =
(
1 − < Bm(ε) >
< Bm(0) >
)
× 100%, (8)
where < Bm(ε) > denotes the field characteristic estimation (the mean crossover effect or the mean quadratic
magnetic field) for the case of non-zero stellar oblation, while < Bm(0) > is the field estimation obtained by
neglecting the stellar oblation effect (ε = 0). For the mean longitudinal magnetic field the other expression of
the Λ-function is applied as
Λz(ε, u, i, β, ϕ) =
<Bz(0)> − <Bz(ε)>
<Bz(0)>max
× 100%, (9)
in order to avoid the division by zero for the certain sets of the variables i, β and λ1 (see Eqs. (3), (8)).
Dependence of these functions on the variables ε, i, β, ϕ is shown in Fig. 1.
300
0
0.1
0.2
0.3
ε
04080120160 ϕ
–2
–1
0
1
2
Λ
, %
0
0.1
0.2
0.3
ε
04080120160 ι
0
2
4
6
8
10
12
Λ
,%
a) b)
0
0.1
0.2
0.3
ε
04080120160 ι
0
2
4
6
Λ
, %
04080120160 β
0 40 80 120 160 ι
2
4
6
Λ
, %
c) d)
Figure 1. The shapes of the Λm-functions: (a) for the mean longitudinal magnetic field with the values of u = 0.6,
i = 90◦, and β = 90◦; (b) for the mean crossover effect with b = 0.2 (solar differential rotation); (c) for the mean
quadratic magnetic field with u = 0.6, β = 90◦, and ϕ = 0; (d) for the mean quadratic magnetic field with ε = 0.3,
u = 0.6, and ϕ = 0
DISCUSSION
It is widely accepted to consider a spherically symmetric star in the existing models of the magnetic field
structure at the stellar surface. An estimate of the strength of the magnetic dipole is based on the main surface
magnetic field characteristics such as the mean longitudinal magnetic field, crossover effect and quadratic field
is a function of the geometrical characteristics of the stellar surface. In this situation the main question is: How
strong should the rotational deformation be in order to essentially distort our estimation of the magnetic dipole
strength?
From the analysis of Eqs. (3)–(5) it follows that the aforementioned magnetic field characteristics are not
equally sensitive to the stellar oblation effect. For the mean longitudinal magnetic field we can ignore the con-
tribution of the oblation effect because it does not exceed 2% (see Fig. 1a). Nevertheless, for the mean crossover
effect and the mean quadratic field the effect of stellar oblation increases their theoretical value (under ε = 0.3)
up to 12% (see Fig. 1b) and up to 8% (see Figs. 1c and 1d), respectively. That could lead to the essential overes-
timation of the magnetic dipole strength value obtained from the mean crossover effect and quadratic magnetic
field observations if we neglected the effect of stellar oblation.
For an ellipsoidal star, its polar zones distorted by rapid rotation are hotter than its equatorial zones, because
T 4
eff ∼ (geff)β1 [8] and the effective gravity is higher at the poles of the star. This phenomenon is known as gravity-
darkening. In order to account for the this phenomenon, we construct a simplified approach supposing that
the effective gravity at the stellar surface is inversely proportional to the square of its distance from the center of
the star. The ordinary limb darkening law is combined with the brightness distribution over the stellar surface
ρ−2γ (see Eq. (1)). The new expressions for the integrated radiation flux as well as for the mean longitudinal
magnetic field, the crossover effect and the quadratic field are obtained and the respective Λ-functions are
recalculated. Choosing the values of the rest of the parameters in Eqs. (8), (9) in the way that they provide
the maximum of Λ-function (see Figs. 1a, 1b, 1c), we analyse its dependence on the new parameter γ. It is
obtained for all the surface magnetic field characteristics that their Λ-functions grow with the increasing of
γ parameter. By increasing the value of parameter γ we can change the brightness distribution over the surface
301
of the star and, consequently, the difference of Teff at the stellar pole and equator. The Λ-function obtained
for the mean longitudinal magnetic field is almost insensitive to the increase of γ, while for the mean crossover
effect the Λ-function grows up to 2%, when we increase the parameter γ from 0 to 1. Therefore, the given
in Fig. 1 values of the magnetic dipole strength overestimation due to the effect of stellar oblation should be
considered as the bottom limit of the real error.
For the star with an essential axial rotation the local flux depends on the gravity according to the von Zeipel’s
theorem and a resulting polarized line profile can differ from the line profile that is usual for the spherically
symmetric star. This means that correction of the magnetic field measurements for the stellar oblation effect
and the corresponding gravity-darkening should be performed on the stage of line profile analysis or later, during
the modelling of stellar magnetic field structure.
Acknowledgements. I am kindly grateful to Dr. J. Zverko and Dr. J. Žižňovský for their useful comments.
I would also like to express my gratitude to the Local Organizing Committee of MAO–2004 for the partial
financial support.
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