| 1. |
- Heiter, Ulrike, et al.
(author)
-
Cool stars in the Gaia photometric system
- 2005
-
In: 13th Cambridge Workshop on Cool Stars, Stellar Systems and the Sun. ; , s. 635-
-
Conference paper (other academic/artistic)
|
|
| 2. |
- Heiter, Ulrike, et al.
(author)
-
VALD — an atomic and molecular database for astrophysics
- 2008
-
In: Journal of Physics, Conference Series. - : IOP Publishing. - 1742-6588 .- 1742-6596. ; 130, s. 012011-
-
Journal article (peer-reviewed)abstract
- The VALD database of atomic and molecular data aims to ensure a robust and consistent analysis of astrophysical spectra. We offer a convenient e-mail and web-based user interface to a vast collection of spectral line parameters for all chemical elements and in the future also for molecules. An international team is working on the following tasks: collecting line parameters from relevant theoretical and experimental publications, computing line parameters, evaluating the data quality by comparison of similar data from different sources and by comparison with astrophysical observations, and incorporating the data into VALD. A unique feature of VALD is its capability to provide the most comprehensive spectral line lists for specific astrophysical plasma conditions defined by the user.
|
|
| 3. |
- Howard, Andrew W., et al.
(author)
-
THE NASA-UC ETA-EARTH PROGRAM : I. A SUPER-EARTH ORBITING HD 7924
- 2009
-
In: Astrophysical Journal. - 0004-637X .- 1538-4357. ; 696:1, s. 75-83
-
Journal article (peer-reviewed)abstract
- We report the discovery of the first low-mass planet to emerge from the NASA-UC Eta-Earth Program, a super-Earth orbiting the K0 dwarf HD 7924. Keplerian modeling of precise Doppler radial velocities reveals a planet with minimum mass M-P sin i = 9.26M(circle plus) in a P = 5.398 d orbit. Based on Keck-HIRES measurements from 2001 to 2008, the planet is robustly detected with an estimated false alarm probability of less than 0.001. Photometric observations using the Automated Photometric Telescopes at Fairborn Observatory show that HD 7924 is photometrically constant over the radial velocity period to 0.19 mmag, supporting the existence of the planetary companion. No transits were detected down to a photometric limit of similar to 0.5 mmag, eliminating transiting planets with a variety of compositions. HD 7924b is one of only eight planets detected by the radial velocity technique with M-P sini < 10 M-circle plus and as such is a member of an emerging family of low-mass planets that together constrain theories of planet formation.
|
|
| 4. |
|
|
| 5. |
|
|
| 6. |
- Kochukhov, Oleg, et al.
(author)
-
Magnetic Doppler Imaging of Active Stars
- 2006
-
In: Solar and Stellar Activity Cycles, 26th meeting of the IAU, Joint Discussion 8.
-
Conference paper (other academic/artistic)
|
|
| 7. |
- Kochukhov, Oleg, et al.
(author)
-
Magnetic fields in M dwarf stars from high-resolution infrared spectra
- 2009
-
In: COOL STARS, STELLAR SYSTEMS AND THE SUN: Proceedings of the 15th Cambridge Workshop on Cool Stars, Stellar Systems and the Sun. - : AIP. - 9780735406278 ; , s. 124-129
-
Conference paper (peer-reviewed)abstract
- Magnetic fields play a central role in the atmospheric properties and variability of active M dwarfs. Information on the strength and structure of magnetic fields in these objects is vital for understanding dynamo mechanisms and magnetically-driven activity of low-mass stars, and for constraining theories of star formation and evolution. We have initiated the first systematic high-resolution survey of magnetically sensitive infrared spectral lines in M dwarf stars using the CRIRES instrument at the ESO VLT. We have completed observations for a sample of 35 active and inactive M dwarfs. Here we report first results of our project, demonstrating a clear detection of magnetic splitting of lines in the spectra of several M dwarfs. We assess diagnostic potential of different Zeeman-sensitive lines in the observed spectral region and apply spectrum synthesis modelling to infer magnetic field properties of selected M dwarfs.
|
|
| 8. |
- Kochukhov, Oleg, et al.
(author)
-
Weather in stellar atmosphere revealed by the dynamics of mercury clouds in alpha Andromedae
- 2007
-
In: Nature Physics. - : Springer Science and Business Media LLC. - 1745-2473 .- 1745-2481. ; 3:8, s. 526-529
-
Journal article (peer-reviewed)abstract
- The formation of long-lasting structures at the surfaces of stars is commonly ascribed to the action of strong magnetic fields. This paradigm is supported by observations of evolving cool spots in the Sun and active late-type stars, and stationary chemical spots in the early-type magnetic stars. However, results of our seven-year monitoring of mercury spots in non-magnetic early-type star αAndromedae show that the picture of magnetically driven structure formation is fundamentally incomplete. Using an indirect stellar-surface mapping technique, we construct a series of two-dimensional images of starspots and discover a secular evolution of the mercury cloud cover in this star. This remarkable structure-formation process, observed for the first time in any star, is plausibly attributed to a non-equilibrium, dynamical evolution of the heavy-element clouds created by atomic diffusion, and may have the same underlying physics as the weather patterns on terrestrial and giant planets.
|
|
| 9. |
- Korn, Andreas, et al.
(author)
-
A probable stellar solution to the cosmological lithium discrepancy
- 2006
-
In: Nature. - : Springer Science and Business Media LLC. - 0028-0836 .- 1476-4687. ; 442:7103, s. 657-659
-
Journal article (peer-reviewed)abstract
- The measurement of the cosmic microwave background has strongly constrained the cosmological parameters of the Universe. When the measured density of baryons (ordinary matter) is combined with standard Big Bang nucleosynthesis calculations, the amounts of hydrogen, helium and lithium produced shortly after the Big Bang can be predicted with unprecedented precision. The predicted primordial lithium abundance is a factor of two to three higher than the value measured in the atmospheres of old stars. With estimated errors of 10 to 25%, this cosmological lithium discrepancy seriously challenges our understanding of stellar physics, Big Bang nucleosynthesis or both. Certain modifications to nucleosynthesis have been proposed, but found experimentally not to be viable. Diffusion theory, however, predicts atmospheric abundances of stars to vary with time, which offers a possible explanation of the discrepancy. Here we report spectroscopic observations of stars in the metal-poor globular cluster NGC6397 that reveal trends of atmospheric abundance with evolutionary stage for various elements. These element-specific trends are reproduced by stellar-evolution models with diffusion and turbulent mixing. We thus conclude that diffusion is predominantly responsible for the low apparent stellar lithium abundance in the atmospheres of old stars by transporting the lithium deep into the star.
|
|
| 10. |
- Korn, Andreas, et al.
(author)
-
Atomic Diffusion and Mixing in Old Stars : I. Very Large Telescope FLAMES-UVES Observations of Stars in NGC 6397
- 2007
-
In: Astrophysical Journal. - : American Astronomical Society. - 0004-637X .- 1538-4357. ; 671:1, s. 402-419
-
Journal article (peer-reviewed)abstract
- We present a homogeneous photometric and spectroscopic analysis of 18 stars along the evolutionary sequence of the metal-poor globular cluster NGC 6397 ([Fe/H] -2), from the main-sequence turnoff point to red giants below the bump. The spectroscopic stellar parameters, in particular stellar parameter differences between groups of stars, are in good agreement with broadband and Strömgren photometry calibrated on the infrared flux method. The spectroscopic abundance analysis reveals, for the first time, systematic trends of iron abundance with evolutionary stage. Iron is found to be 30% less abundant in the turnoff point stars than in the red giants. An abundance difference in lithium is seen between the turnoff point and warm subgiant stars. The impact of potential systematic errors on these abundance trends (stellar parameters, the hydrostatic and LTE approximations) is quantitatively evaluated and found not to alter our conclusions significantly. Trends for various elements (Li, Mg, Ca, Ti, and Fe) are compared with stellar structure models including the effects of atomic diffusion and radiative acceleration. Such models are found to describe the observed element-specific trends well, if extra (turbulent) mixing just below the convection zone is introduced. It is concluded that atomic diffusion and turbulent mixing are largely responsible for the subprimordial stellar lithium abundances of warm halo stars. Other consequences of atomic diffusion in old metal-poor stars are also discussed.
|
|
| 11. |
- Korn, Andreas J., et al.
(author)
-
New Abundances for Old Stars – Atomic Diffusion at Work in NGC 6397
- 2006
-
In: The Messenger. - 0722-6691. ; 125, s. 6-10
-
Research review (pop. science, debate, etc.)abstract
- A homogeneous spectroscopic analysis of unevolved and evolved stars in the metal-poor globular cluster NGC 6397 with FLAMES-UVES reveals systematic trends of stellar surface abundances that are likely caused by atomic diffu-sion. This finding helps to understand, among other issues, why the lithium abundances of old halo stars are sig-nificantly lower than the abundance found to be produced shortly after the Big Bang.
|
|
| 12. |
- Korn, Andreas, et al.
(author)
-
Pinning Down Gravitational Settling
- 2006
-
In: Chemical Abundances and Mixing in Stars in the Milky Way and its Satellites. - Berlin, Heidelberg : Springer Berlin Heidelberg. - 9783540341352 ; , s. 294-
-
Conference paper (other academic/artistic)
|
|
| 13. |
- Lueftinger, T., et al.
(author)
-
3D atmospheric structure of the prototypical roAp star HD 24712 (HR1217)
- 2008
-
In: Contributions of the Astronomical Observatory Skalnate Pleso. - 1335-1842. ; 38:2, s. 335-340
-
Journal article (peer-reviewed)abstract
- The first analysis of the structure of the surface magnetic field of a rapidly oscillating Ap (roAp) star is presented. We obtain information about abundance distributions of a number of chemical elements on the surface of the prototypical roAp star HD 24712 and about magnetic field geometry. Inverting rotationally modulated spectra in Stokes parameters I and V obtained with the SOFIN spectropolarimeter attached to the NOT, we recover surface abundance structures of sixteen different chemical elements, including Mg, Ca, Sc, Ti, Cr, Fe, Co, Ni, Y, La, Ce, Pr, Nd, Gd, Tb, and Dy. Our analysis reveal a pure dipolar structure of the stellar magnetic field and surprising and unexpected correlations of the various elemental surface abundance structures to this field geometry. Stratification analysis at phases of both magnetic extrema enable us to obtain the vertical dimension in the atmosphere of HD 24712. High time resolved spectroscopic data and observations obtained with the MOST space photometer give us the possibility to compare (Luftinger, 2007) our results to detailed pulsational analysis.
|
|
| 14. |
- Lyra, Wladimir, et al.
(author)
-
Embryos grown in the dead zone : Assembling the first protoplanetary cores in low mass self-gravitating circumstellar disks of gas and solids
- 2008
-
In: Astronomy and Astrophysics. - : EDP Sciences. - 0004-6361 .- 1432-0746. ; 491:3, s. L41-L44
-
Journal article (peer-reviewed)abstract
- Context: In the borders of the dead zones of protoplanetary disks, the inflow of gas produces a local density maximum that triggers the Rossby wave instability. The vortices that form are efficient in trapping solids. Aims: We aim to assess the possibility of gravitational collapse of the solids within the Rossby vortices. Methods: We perform global simulations of the dynamics of gas and solids in a low mass non-magnetized self-gravitating thin protoplanetary disk with the Pencil Code. We use multiple particle species of radius 1, 10, 30, and 100 cm. The dead zone is modelled as a region of low viscosity. Results: The Rossby vortices excited in the edges of the dead zone are efficient particle traps. Within 5 orbits after their appearance, the solids achieve critical density and undergo gravitational collapse into Mars sized objects. The velocity dispersions are of the order of 10 m s-1 for newly formed embryos, later lowering to less than 1 m s-1 by drag force cooling. After 200 orbits, over 300 gravitationally bound embryos were formed, 20 of them being more massive than Mars. Their mass spectrum follows a power law of index -2.3 ± 0.2.
|
|
| 15. |
- Lyra, Wladimir, et al.
(author)
-
Global magnetohydrodynamical models of turbulence in protoplanetary disks : I. A cylindrical potential on a Cartesian grid and transport of solids
- 2008
-
In: Astronomy and Astrophysics. - : EDP Sciences. - 0004-6361 .- 1432-0746. ; 479, s. 883-901
-
Journal article (peer-reviewed)abstract
- Aims.We present global 3D MHD simulations of disks of gas and solids, aiming at developing models that can be used to study various scenarios of planet formation and planet-disk interaction in turbulent accretion disks. A second goal is to demonstrate that Cartesian codes are comparable to cylindrical and spherical ones in handling the magnetohydrodynamics of the disk simulations while offering advantages, such as the absence of a grid singularity, for certain applications, e.g., circumbinary disks and disk-jet simulations. Methods: We employ the Pencil Code, a 3D high-order finite-difference MHD code using Cartesian coordinates. We solve the equations of ideal MHD with a local isothermal equation of state. Planets and stars are treated as particles evolved with an N-body scheme. Solid boulders are treated as individual superparticles that couple to the gas through a drag force that is linear in the local relative velocity between gas and particle. Results: We find that Cartesian grids are well-suited for accretion disk problems. The disk-in-a-box models based on Cartesian grids presented here develop and sustain MHD turbulence, in good agreement with published results achieved with cylindrical codes. Models without an inner boundary do not show the spurious build-up of magnetic pressure and Reynolds stress seen in the models with boundaries, but the global stresses and alpha viscosities are similar in the two cases. We investigate the dependence of the magnetorotational instability on disk scale height, finding evidence that the turbulence generated by the magnetorotational instability grows with thermal pressure. The turbulent stresses depend on the thermal pressure obeying a power law of 0.24 ± 0.03, compatible with the value of 0.25 found in shearing box calculations. The ratio of Maxwell to Reynolds stresses decreases with increasing temperature, dropping from 5 to 1 when the sound speed was raised by a factor 4, maintaing the same field strength. We also study the dynamics of solid boulders in the hydromagnetic turbulence, by making use of 106 Lagrangian particles embedded in the Eulerian grid. The effective diffusion provided by the turbulence prevents settling of the solids in a infinitesimally thin layer, forming instead a layer of solids of finite vertical thickness. The measured scale height of this diffusion-supported layer of solids implies turbulent vertical diffusion coefficients with globally averaged Schmidt numbers of 1.0 ± 0.2 for a model with α≈10-3 and 0.78 ± 0.06 for a model with α≈10-1. That is, the vertical turbulent diffusion acting on the solids phase is comparable to the turbulent viscosity acting on the gas phase. The average bulk density of solids in the turbulent flow is quite low (ρp = 6.0×10-11 kg m-3), but in the high pressure regions, significant overdensities are observed, where the solid-to-gas ratio reached values as great as 85, corresponding to 4 orders of magnitude higher than the initial interstellar value of 0.01
|
|
| 16. |
- Lyra, Wladimir, et al.
(author)
-
Planet formation bursts at the borders of the dead zone in 2D numerical simulations of circumstellar disks
- 2009
-
In: Astronomy and Astrophysics. - : EDP Sciences. - 0004-6361 .- 1432-0746. ; 497:3, s. 869-888
-
Journal article (peer-reviewed)abstract
- Context: As accretion in protoplanetary disks is enabled by turbulent viscosity, the border between active and inactive (dead) zones constitutes a location where there is an abrupt change in the accretion flow. The gas accumulation that ensues triggers the Rossby wave instability, which in turn saturates into anticyclonic vortices. It has been suggested that the trapping of solids within them leads to a burst of planet formation on very short timescales. Aims: We study in the formation and evolution of the vortices in greater detail, focusing on the implications for the dynamics of embedded solid particles and planet formation. Methods: We performed two-dimensional global simulations of the dynamics of gas and solids in a non-magnetized thin protoplanetary disk with the Pencil code. We used multiple particle species of radius 1, 10, 30, and 100 cm. We computed the particles' gravitational interaction by a particle-mesh method, translating the particles' number density into surface density and computing the corresponding self-gravitational potential via fast Fourier transforms. The dead zone is modeled as a region of low viscosity. Adiabatic and locally isothermal equations of state are used. Results: The Rossby wave instability is triggered under a variety of conditions, thus making vortex formation a robust process. Inside the vortices, fast accumulation of solids occurs and the particles collapse into objects of planetary mass on timescales as short as five orbits. Because the drag force is size-dependent, aerodynamical sorting ensues within the vortical motion, and the first bound structures formed are composed primarily of similarly-sized particles. In addition to erosion due to ram pressure, we identify gas tides from the massive vortices as a disrupting agent of formed protoplanetary embryos. We find evidence that the backreaction of the drag force from the particles onto the gas modifies the evolution of the Rossby wave instability, with vortices being launched only at later times if this term is excluded from the momentum equation. Even though the gas is not initially gravitationally unstable, the vortices can grow to Q ≈ 1 in locally isothermal runs, which halts the inverse cascade of energy towards smaller wavenumbers. As a result, vortices in models without self-gravity tend to rapidly merge towards a m = 2 or m =1 mode, while models with self-gravity retain dominant higher order modes (m = 4 or m = 3) for longer times. Non-selfgravitating disks thus show fewer and stronger vortices. We also estimate the collisional velocity history of the particles that compose the most massive embryo by the end of the simulation, finding that the vast majority of them never experienced a collision with another particle at speeds faster than 1 m s-1. This result lends further support to previous studies showing that vortices provide a favorable environment for planet formation.
|
|
| 17. |
- Lyra, Wladimir, et al.
(author)
-
Standing on the shoulders of giants : Trojan Earths and vortex trapping in low mass self-gravitating protoplanetary disks of gas and solids
- 2009
-
In: Astronomy and Astrophysics. - : EDP Sciences. - 0004-6361 .- 1432-0746. ; 493:3, s. 1125-1139
-
Journal article (peer-reviewed)abstract
- Context: Centimeter and meter-sized solid particles in protoplanetary disks are trapped within long-lived, high-pressure regions, creating opportunities for collapse into planetesimals and planetary embryos. Aims: We aim to study the effect of the high-pressure regions generated in the gaseous disks by a giant planet perturber. These regions consist of gas retained in tadpole orbits around the stable Lagrangian points as a gap is carved, and the Rossby vortices launched at the edges of the gap. Methods: We performed global simulations of the dynamics of gas and solids in a low mass non-magnetized self-gravitating thin protoplanetary disk. We employed the Pencil code to solve the Eulerian hydro equations, tracing the solids with a large number of Lagrangian particles, usually 100 000. To compute the gravitational potential of the swarm of solids, we solved the Poisson equation using particle-mesh methods with multiple fast Fourier transforms. Results: Huge particle concentrations are seen in the Lagrangian points of the giant planet, as well as in the vortices they induce at the edges of the carved gaps. For 1 cm to 10 cm radii, gravitational collapse occurs in the Lagrangian points in less than 200 orbits. For 5 cm particles, a 2M⊕ planet is formed. For 10 cm, the final maximum collapsed mass is around 3M⊕. The collapse of the 1 cm particles is indirect, following the timescale of gas depletion from the tadpole orbits. Vortices are excited at the edges of the gap, primarily trapping particles of 30 cm radii. The rocky planet that is formed is as massive as 17M⊕, constituting a Super-Earth. Collapse does not occur for 40 cm onwards. By using multiple particle species, we find that gas drag modifies the streamlines in the tadpole region around the classical L4 and L5 points. As a result, particles of different radii have their stable points shifted to different locations. Collapse therefore takes longer and produces planets of lower mass. Three super-Earths are formed in the vortices, the most massive having 4.5M⊕. Conclusions: A Jupiter-mass planet can induce the formation of other planetary embryos at the outer edge of its gas gap. Trojan Earth-mass planets are readily formed; although not existing in the solar system, might be common in the exoplanetary zoo.
|
|
| 18. |
- Lyra, Wladimir, 1981-
(author)
-
Turbulence-Assisted Planetary Growth : Hydrodynamical Simulations of Accretion Disks and Planet Formation
- 2009
-
Doctoral thesis (other academic/artistic)abstract
- The current paradigm in planet formation theory is developed around a hierarquical growth of solid bodies, from interstellar dust grains to rocky planetary cores. A particularly difficult phase in the process is the growth from meter-size boulders to planetary embryos of the size of our Moon or Mars. Objects of this size are expected to drift extremely rapid in a protoplanetary disk, so that they would generally fall into the central star well before larger bodies can form.In this thesis, we used numerical simulations to find a physical mechanism that may retain solids in some parts of protoplanetary disks long enough to allow for the formation of planetary embryos. We found that such accumulation can happen at the borders of so-called dead zones. These dead zones would be regions where the coupling to the ambient magnetic field is weaker and the turbulence is less strong, or maybe even absent in some cases. We show by hydrodynamical simulations that material accumulating between the turbulent active and dead regions would be trapped into vortices to effectively form planetary embryos of Moon to Mars mass.We also show that in disks that already formed a giant planet, solid matter accumulates on the edges of the gap the planet carves, as well as at the stable Lagrangian points. The concentration is strong enough for the solids to clump together and form smaller, rocky planets like Earth. Outside our solar system, some gas giant planets have been detected in the habitable zone of their stars. Their wakes may harbour rocky, Earth-size worlds.
|
|
| 19. |
- Piskunov, Nikolai
(author)
-
Doppler imaging
- 2008
-
In: Physica Scripta. - 0031-8949 .- 1402-4896. ; T133, s. 014017-
-
Journal article (peer-reviewed)abstract
- In this paper, I present a short review of the history and modern status of Doppler imaging techniques, highlighting their dependence on the knowledge of the fundamental stellar parameters, the quality of stellar atmospheric models and the accuracy of spectral synthesis.
|
|
| 20. |
|
|
| 21. |
|
|
| 22. |
- Piskunov, Nikolai
(author)
-
Optimal Extraction of Echelle Spectra
- 2007
-
In: Precision Spectroscopy in Astrophysics. - Berlin, Heidelberg : Springer Berlin Heidelberg. ; , s. 129-
-
Conference paper (other academic/artistic)
|
|
| 23. |
- Ryabchikova, T., et al.
(author)
-
A comparative analysis of the laboratory and theoretical transition probabilities of the Fe-peak elements for a new release of VALD
- 2008
-
In: Journal of Physics, Conference Series. - : IOP Publishing. - 1742-6588 .- 1742-6596. ; 130, s. 012017-
-
Journal article (peer-reviewed)abstract
- We carried out a comparative analysis of the recent atomic data for iron-peak elements, mainly Ti, Cr and Fe, for a new release of the Vienna Atomic Line Database (VALD3). New data were compared with those available in VALD2 and were checked using high-resolution, high signal-to-noise spectra of sharp-lined chemically normal stars including the Sun, and the zero-rotation extremely Cr- and Fe-rich chemically peculiar star HD 133792. The observed spectrum of the latter star allowed for comparison with transition probability calculations based on the orthogonal operator technique with the Cowan code for Cr II and Fe II lines for lower level energies between 2 eV and 11 eV in the wavelength region 3100 to 9000 Å. In general, the agreement between the new experimental transition probabilities and those currently available in VALD2 is fairly good, which helps to validate the stellar abundance data derived with the VALD2 atomic parameters. We also found that, for a few important Ti II and Fe II lines in the visible spectral region, new transition probabilities are not consistent within their quoted accuracy. In a series of recent works on experimental f-values for Fe II it was shown that calculations based on the orthogonal operator technique agree better with the experimental data than the Cowan code calculations and, hence, should have preference for stellar spectroscopy. Our analysis of the Ap star HD 133792 spectrum clearly demonstrates that there are quite a number of high-excitation Cr II and Fe II lines which are fitted reasonably well when using the transition probabilities calculated with the Cowan code. As a rule these lines have their upper energy levels classified differently in both methods of calculations.
|
|
| 24. |
- Sahu, Kailash C., et al.
(author)
-
Transiting extrasolar planetary candidates in the Galactic bulge
- 2006
-
In: Nature. - : Springer Science and Business Media LLC. - 0028-0836 .- 1476-4687. ; 443:7111, s. 534-540
-
Journal article (peer-reviewed)abstract
- More than 200 extrasolar planets have been discovered around relatively nearby stars, primarily through the Doppler line shifts owing to reflex motions of their host stars, and more recently through transits of some planets across the faces of the host stars. The detection of planets with the shortest known periods, 1.2 - 2.5 days, has mainly resulted from transit surveys which have generally targeted stars more massive than 0.75 M-., where M-. is the mass of the Sun. Here we report the results from a planetary transit search performed in a rich stellar field towards the Galactic bulge. We discovered 16 candidates with orbital periods between 0.4 and 4.2 days, five of which orbit stars of masses in the range 0.44 - 0.75 M-.. In two cases, radial-velocity measurements support the planetary nature of the companions. Five candidates have orbital periods below 1.0 day, constituting a new class of ultra-short-period planets, which occur only around stars of less than 0.88 M-.. This indicates that those orbiting very close to more-luminous stars might be evaporatively destroyed or that jovian planets around stars of lower mass might migrate to smaller radii.
|
|
| 25. |
|
|
