A search for transiting planets around hot subdwarfs

Astronomy and Astrophysics, vol. 650, pp. A205

Abstract

Context. Hot subdwarfs experienced strong mass loss on the red giant branch (RGB) and are now hot and small He-burning objects. These stars constitute excellent opportunities for addressing the question of the evolution of exoplanetary systems directly after the RGB phase of evolution. Aims. In this project we aim to perform a transit survey in all available light curves of hot subdwarfs from space-based telescopes (Kepler, K2, TESS, and CHEOPS) with our custom-made pipeline SHERLOCK in order to determine the occurrence rate of planets around these stars as a function of orbital period and planetary radius. We also aim to determine whether planets that were previously engulfed in the envelope of their red giant host star can survive, even partially, as a planetary remnant. Methods. For this first paper, we performed injection-and-recovery tests of synthetic transits for a selection of representative Kepler, K2, and TESS light curves to determine which transiting bodies in terms of object radius and orbital period we will be able to detect with our tools. We also provide estimates for CHEOPS data, which we analyzed with the pycheops package. Results. Transiting objects with a radius ≲1.0 R⊕ can be detected in most of the Kepler, K2, and CHEOPS targets for the shortest orbital periods (1 d and shorter), reaching values as low as ~0.3 R⊕ in the best cases. Sub-Earth-sized bodies are only reached for the brightest TESS targets and for those that were observed in a significant number of sectors. We also give a series of representative results for larger planets at greater distances, which strongly depend on the target magnitude and on the length and quality of the data. Conclusions. The TESS sample will provide the most important statistics for the global aim of measuring the planet occurrence rate around hot subdwarfs. The Kepler, K2, and CHEOPS data will allow us to search for planetary remnants, that is, very close and small (possibly disintegrating) objects.

Authors 82

1. V. Van Grootel

   University of Liège

   Affiliation as printed

   Space sciences, Technologies and Astrophysics Research (STAR) Institute, Université de Liège, 19C Allée du 6 Août, 4000 Liège, Belgium

2. F. J. Pozuelos

   University of Liège

   Affiliation as printed

   Astrobiology Research Unit, Université de Liège, Allée du 6 Août 19C, 4000 Liège, Belgium

   Space sciences, Technologies and Astrophysics Research (STAR) Institute, Université de Liège, 19C Allée du 6 Août, 4000 Liège, Belgium

3. Antoine Thuillier

   University of Liège

   Affiliation as printed

   Space sciences, Technologies and Astrophysics Research (STAR) Institute, Université de Liège, 19C Allée du 6 Août, 4000 Liège, Belgium

4. S. Charpinet

   Centre National de la Recherche Scientifique · Institut de Recherche en Astrophysique et Planétologie · Université de Toulouse

   Affiliation as printed

   Institut de Recherche en Astrophysique et Planétologie, CNRS, Université de Toulouse, CNES, 14 avenue Edouard Belin, 31400 Toulouse, France

5. L. Delrez

   University of Geneva · University of Liège

   Affiliation as printed

   Astrobiology Research Unit, Université de Liège, Allée du 6 Août 19C, 4000 Liège, Belgium

   Observatoire Astronomique de l’Université de Genève, Chemin Pegasi 51, Versoix, Switzerland

   Space sciences, Technologies and Astrophysics Research (STAR) Institute, Université de Liège, 19C Allée du 6 Août, 4000 Liège, Belgium

   Observatoire Astronomique de l'Université de Genève, Chemin Pegasi 51, Versoix, Switzerland

6. M. Beck

   University of Geneva

   Affiliation as printed

   Observatoire Astronomique de l’Université de Genève, Chemin Pegasi 51, Versoix, Switzerland

   Observatoire Astronomique de l'Université de Genève, Chemin Pegasi 51, Versoix, Switzerland

7. A. Fortier

   University of Bern · Center for Space and Habitability

   Affiliation as printed

   Center for Space and Habitability, Gesellsschaftstrasse 6, 3012 Bern, Switzerland

   Physikalisches Institut, University of Bern, Gesellsschaftstrasse 6, 3012 Bern, Switzerland

8. S. Hoyer

   Centre National de la Recherche Scientifique · Aix-Marseille Université

   Affiliation as printed

   Aix Marseille Univ, CNRS, CNES, LAM, Marseille, France

9. S. G. Sousa

   Universidade do Porto · Institute of Astrophysics and Space Sciences · Centro de Astrofísica da Universidade do Porto

   Affiliation as printed

   Instituto de Astrofísica e Ciências do Espaço, Universidade do Porto, CAUP, Rua das Estrelas, 4150-762 Porto, Portugal

10. B. N. Barlow

    High Point University

    Affiliation as printed

    Department of Physics, High Point University, One University Parkway, High Point, NC 27268, USA

11. N. Billot

    University of Geneva

    Affiliation as printed

    Observatoire Astronomique de l’Université de Genève, Chemin Pegasi 51, Versoix, Switzerland

    Observatoire Astronomique de l'Université de Genève, Chemin Pegasi 51, Versoix, Switzerland

12. M. Dévora-Pajares

    Universidad de Granada · Valencian International University

    Affiliation as printed

    Dpto. Física Teórica y del Cosmos, Universidad de Granada, 18071 Granada, Spain

    Universidad Internacional de Valencia (VIU), Carrer del Pintor Sorolla 21, 46002 Valencia, Spain

13. R. H. Østensen

    Missouri State University

    Affiliation as printed

    Department of Physics, Astronomy and Materials Science, Missouri State University, 901 S. National, Springfield, MO 65897, USA

14. Yann Alibert

    University of Bern

    Affiliation as printed

    Physikalisches Institut, University of Bern, Gesellsschaftstrasse 6, 3012 Bern, Switzerland

15. R. Alonso

    Universidad de La Laguna · Instituto de Astrofísica de Canarias

    Affiliation as printed

    Departamento de Astrofísica, Universidad de La Laguna, 38206 La Laguna, Tenerife, Spain

    Instituto de Astrofísica de Canarias, 38200 La Laguna, Tenerife, Spain

16. Guillem Anglada-Escudé

    Institute of Space Sciences · Institut d'Estudis Espacials de Catalunya

    Affiliation as printed

    Institut de Ciències de l’Espai (ICE, CSIC), Campus UAB, Can Magrans s/n, 08193 Bellaterra, Spain

    Institut d’Estudis Espacials de Catalunya (IEEC), 08034 Barcelona, Spain

    Institut d'Estudis Espacials de Catalunya (IEEC), 08034 Barcelona, Spain

    Institut de Ciències de l'Espai (ICE, CSIC), Campus UAB, Can Magrans s/n, 08193 Bellaterra, Spain

17. Joel Asquier

    European Space Agency · European Space Research and Technology Centre

    Affiliation as printed

    ESTEC, European Space Agency, 2201AZ, Noordwijk, NL, The Netherlands

18. D. Barrado

    Centro de Astrobiología

    Affiliation as printed

    Depto. de Astrofísica, Centro de Astrobiologia (CSIC-INTA), ESAC campus, 28692 Villanueva de la Cãda (Madrid), Spain

19. S. C. C. Barros

    Universidade do Porto · Institute of Astrophysics and Space Sciences · Centro de Astrofísica da Universidade do Porto

    Affiliation as printed

    Departamento de Física e Astronomia, Faculdade de Ciências, Universidade do Porto, Rua do Campo Alegre, 4169-007 Porto, Portugal

    Instituto de Astrofísica e Ciências do Espaço, Universidade do Porto, CAUP, Rua das Estrelas, 4150-762 Porto, Portugal

20. W. Baumjohann

    Austrian Academy of Sciences · Space Research Institute

    Affiliation as printed

    Space Research Institute, Austrian Academy of Sciences, Schmiedlstrasse 6, 8042 Graz, Austria

21. T. Beck

    University of Bern

    Affiliation as printed

    Physikalisches Institut, University of Bern, Gesellsschaftstrasse 6, 3012 Bern, Switzerland

22. A. Bekkelien

    University of Geneva

    Affiliation as printed

    Observatoire Astronomique de l’Université de Genève, Chemin Pegasi 51, Versoix, Switzerland

    Observatoire Astronomique de l'Université de Genève, Chemin Pegasi 51, Versoix, Switzerland

23. W. Benz

    University of Bern · Center for Space and Habitability

    Affiliation as printed

    Center for Space and Habitability, Gesellsschaftstrasse 6, 3012 Bern, Switzerland

    Physikalisches Institut, University of Bern, Gesellsschaftstrasse 6, 3012 Bern, Switzerland

24. Xavier Bonfils

    Centre National de la Recherche Scientifique · Institut de Planétologie et d'Astrophysique de Grenoble · Université Grenoble Alpes

    Affiliation as printed

    Université Grenoble Alpes, CNRS, IPAG, 38000 Grenoble, France

25. A. Brandeker

    Stockholm University · AlbaNova

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    Department of Astronomy, Stockholm University, AlbaNova University Center, 10691 Stockholm, Sweden

26. Christopher H. Broeg

    University of Bern

    Affiliation as printed

    Physikalisches Institut, University of Bern, Gesellsschaftstrasse 6, 3012 Bern, Switzerland

27. G. Bruno

    Osservatorio Astrofisico di Catania

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    INAF, Osservatorio Astrofisico di Catania, Via S. Sofia 78, 95123 Catania, Italy

28. T. Bárczy
    Affiliation as printed

    ADMATIS, 3534 Miskolc, Kandó Kálmán u. 5., Hungary

29. J. Cabrera

    Deutsches Zentrum für Luft- und Raumfahrt e. V. (DLR)

    Affiliation as printed

    Institute of Planetary Research, German Aerospace Center (DLR), Rutherfordstrasse 2, 12489 Berlin, Germany

30. Andrew Collier Cameron

    University of St Andrews

    Affiliation as printed

    Centre for Exoplanet Science, SUPA School of Physics and Astronomy, University of St Andrews, North Haugh, St Andrews KY16 9SS, UK

31. S. Charnoz

    Centre National de la Recherche Scientifique · Institut de physique du globe de Paris · Université Paris Cité

    Affiliation as printed

    Université de Paris, Institut de physique du globe de Paris, CNRS, 75005 Paris, France

32. M. B. Davies

    Lund University

    Affiliation as printed

    Lund Observatory, Dept. of Astronomy and Theoretical Physics, Lund University, Box 43, 22100 Lund, Sweden

33. M. Deleuil

    Centre National de la Recherche Scientifique · Aix-Marseille Université

    Affiliation as printed

    Aix Marseille Univ, CNRS, CNES, LAM, Marseille, France

34. Olivier D S Demangeon

    Universidade do Porto · Institute of Astrophysics and Space Sciences · Centro de Astrofísica da Universidade do Porto

    Affiliation as printed

    Departamento de Física e Astronomia, Faculdade de Ciências, Universidade do Porto, Rua do Campo Alegre, 4169-007 Porto, Portugal

    Instituto de Astrofísica e Ciências do Espaço, Universidade do Porto, CAUP, Rua das Estrelas, 4150-762 Porto, Portugal

35. Brice-Olivier Demory

    University of Bern

    Affiliation as printed

    Physikalisches Institut, University of Bern, Gesellsschaftstrasse 6, 3012 Bern, Switzerland

36. David Ehrenreich

    University of Geneva

    Affiliation as printed

    Observatoire Astronomique de l’Université de Genève, Chemin Pegasi 51, Versoix, Switzerland

    Observatoire Astronomique de l'Université de Genève, Chemin Pegasi 51, Versoix, Switzerland

37. Anders Erikson

    Deutsches Zentrum für Luft- und Raumfahrt e. V. (DLR)

    Affiliation as printed

    Institute of Planetary Research, German Aerospace Center (DLR), Rutherfordstrasse 2, 12489 Berlin, Germany

38. L. Fossati

    Austrian Academy of Sciences · Space Research Institute

    Affiliation as printed

    Space Research Institute, Austrian Academy of Sciences, Schmiedlstrasse 6, 8042 Graz, Austria

39. Malcolm C. Fridlund Aachen Leiden Observatory

    Leiden University · Leiden Observatory · Chalmers University of Technology

    Affiliation as printed

    Department of Space, Earth and Environment, Chalmers University of Technology, Onsala Space Observatory, 43992 Onsala, Sweden

    Leiden Observatory, University of Leiden, PO Box 9513, 2300 RA Leiden, The Netherlands

40. D. Futyan

    University of Geneva

    Affiliation as printed

    Observatoire Astronomique de l’Université de Genève, Chemin Pegasi 51, Versoix, Switzerland

    Observatoire Astronomique de l'Université de Genève, Chemin Pegasi 51, Versoix, Switzerland

41. D. Gandolfi

    University of Turin

    Affiliation as printed

    Dipartimento di Fisica, Università degli Studi di Torino, via Pietro Giuria 1, 10125 Torino, Italy

42. M. Gillon

    University of Liège

    Affiliation as printed

    Astrobiology Research Unit, Université de Liège, Allée du 6 Août 19C, 4000 Liège, Belgium

43. M. Güdel

    University of Vienna

    Affiliation as printed

    University of Vienna, Department of Astrophysics, Türkenschanzstrasse 17, 1180 Vienna, Austria

44. Kevin Heng

    University of Warwick · Center for Space and Habitability

    Affiliation as printed

    Center for Space and Habitability, Gesellsschaftstrasse 6, 3012 Bern, Switzerland

    Department of Physics, University of Warwick, Gibbet Hill Road, Coventry CV4 7AL, UK

45. K. G. Isaak

    European Space Agency · European Space Research and Technology Centre

    Affiliation as printed

    ESTEC, European Space Agency, 2201AZ, Noordwijk, NL, The Netherlands

46. L. L. Kiss

    Eötvös Loránd University · The University of Sydney · Konkoly Observatory · HUN-REN Research Centre for Astronomy and Earth Sciences

    Affiliation as printed

    ELTE Eötvös Loránd University, Institute of Physics, Pázmány Péter sétány 1/A, 1117 Budapest, Hungary

    Konkoly Observatory, Research Centre for Astronomy and Earth Sciences, 1121 Budapest, Konkoly Thege Miklós út 15-17, Hungary

    Sydney Institute for Astronomy, School of Physics A29, University of Sydney, NSW 2006, Australia

47. J. Laskar

    Centre National de la Recherche Scientifique · Observatoire de Paris · Institut de Mécanique Céleste et de Calcul des Éphémérides

    Affiliation as printed

    IMCCE, UMR8028 CNRS, Observatoire de Paris, PSL Univ., Sorbonne Univ., 77 av. Denfert-Rochereau, 75014 Paris, France

48. Alain Lecavelier des Etangs

    Centre National de la Recherche Scientifique · Institut d'Astrophysique de Paris · Université Pierre-et-Marie-Curie

    Affiliation as printed

    Institut d’astrophysique de Paris, UMR7095 CNRS, Université Pierre & Marie Curie, 98bis blvd. Arago, 75014 Paris, France

    Institut d'astrophysique de Paris, UMR7095 CNRS, Université Pierre & Marie Curie, 98bis blvd. Arago, 75014 Paris, France

49. M. Lendl

    University of Geneva

    Affiliation as printed

    Observatoire Astronomique de l’Université de Genève, Chemin Pegasi 51, Versoix, Switzerland

    Observatoire Astronomique de l'Université de Genève, Chemin Pegasi 51, Versoix, Switzerland

50. Christophe Lovis

    University of Geneva

    Affiliation as printed

    Observatoire Astronomique de l’Université de Genève, Chemin Pegasi 51, Versoix, Switzerland

    Observatoire Astronomique de l'Université de Genève, Chemin Pegasi 51, Versoix, Switzerland

51. Demetrio Magrin

    Osservatorio Astronomico di Padova

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    INAF, Osservatorio Astronomico di Padova, Vicolo dell’Osservatorio 5, 35122 Padova, Italy

    INAF, Osservatorio Astronomico di Padova, Vicolo dell'Osservatorio 5, 35122 Padova, Italy

52. Pierre F L Maxted

    Keele University

    Affiliation as printed

    Astrophysics Group, Keele University, Staffordshire, ST5 5BG, UK

53. M. Mečina

    University of Vienna · Austrian Academy of Sciences · Space Research Institute

    Affiliation as printed

    Space Research Institute, Austrian Academy of Sciences, Schmiedlstrasse 6, 8042 Graz, Austria

    University of Vienna, Department of Astrophysics, Türkenschanzstrasse 17, 1180 Vienna, Austria

54. Alexander James Mustill

    Lund University

    Affiliation as printed

    Lund Observatory, Dept. of Astronomy and Theoretical Physics, Lund University, Box 43, 22100 Lund, Sweden

55. V. Nascimbeni

    Osservatorio Astronomico di Padova

    Affiliation as printed

    INAF, Osservatorio Astronomico di Padova, Vicolo dell’Osservatorio 5, 35122 Padova, Italy

    INAF, Osservatorio Astronomico di Padova, Vicolo dell'Osservatorio 5, 35122 Padova, Italy

56. Göran Olofsson

    Stockholm University · AlbaNova

    Affiliation as printed

    Department of Astronomy, Stockholm University, AlbaNova University Center, 10691 Stockholm, Sweden

57. R. Ottensamer

    University of Vienna

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    University of Vienna, Department of Astrophysics, Türkenschanzstrasse 17, 1180 Vienna, Austria

58. I. Pagano

    Osservatorio Astrofisico di Catania

    Affiliation as printed

    INAF, Osservatorio Astrofisico di Catania, Via S. Sofia 78, 95123 Catania, Italy

59. Ε. Πάλλη

    Universidad de La Laguna · Instituto de Astrofísica de Canarias

    Affiliation as printed

    Departamento de Astrofísica, Universidad de La Laguna, 38206 La Laguna, Tenerife, Spain

    Instituto de Astrofísica de Canarias, 38200 La Laguna, Tenerife, Spain

60. Gisbert Peter

    Deutsches Zentrum für Luft- und Raumfahrt e. V. (DLR)

    Affiliation as printed

    Institute of Optical Sensor Systems, German Aerospace Center (DLR), Rutherfordstrasse 2, 12489 Berlin, Germany

61. G. Piotto

    University of Padua · Osservatorio Astronomico di Padova

    Affiliation as printed

    Dipartimento di Fisica e Astronomia, Università degli Studi di Padova, Vicolo dell’Osservatorio 3, 35122 Padova, Italy

    INAF, Osservatorio Astronomico di Padova, Vicolo dell’Osservatorio 5, 35122 Padova, Italy

    Dipartimento di Fisica e Astronomia, Università degli Studi di Padova, Vicolo dell'Osservatorio 3, 35122 Padova, Italy

    INAF, Osservatorio Astronomico di Padova, Vicolo dell'Osservatorio 5, 35122 Padova, Italy

62. J.-Y. Plesseria

    University of Liège

    Affiliation as printed

    Centre Spatial de Liège, STAR institute, Université de Liège, avenue du Pré Aily, 4031 Angleur (Liège), Belgium

63. D. Pollacco

    University of Warwick

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    Department of Physics, University of Warwick, Gibbet Hill Road, Coventry CV4 7AL, UK

64. Didier P. Queloz

    University of Geneva · University of Cambridge

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    Cavendish Laboratory, JJ Thomson Avenue, Cambridge CB3 0HE, UK

    Observatoire Astronomique de l’Université de Genève, Chemin Pegasi 51, Versoix, Switzerland

    Observatoire Astronomique de l'Université de Genève, Chemin Pegasi 51, Versoix, Switzerland

65. Roberto Ragazzoni

    University of Padua · Osservatorio Astronomico di Padova

    Affiliation as printed

    Dipartimento di Fisica e Astronomia, Università degli Studi di Padova, Vicolo dell’Osservatorio 3, 35122 Padova, Italy

    INAF, Osservatorio Astronomico di Padova, Vicolo dell’Osservatorio 5, 35122 Padova, Italy

    Dipartimento di Fisica e Astronomia, Università degli Studi di Padova, Vicolo dell'Osservatorio 3, 35122 Padova, Italy

    INAF, Osservatorio Astronomico di Padova, Vicolo dell'Osservatorio 5, 35122 Padova, Italy

66. N. Rando

    European Space Agency · European Space Research and Technology Centre

    Affiliation as printed

    ESTEC, European Space Agency, 2201AZ, Noordwijk, NL, The Netherlands

67. Heike Rauer

    Deutsches Zentrum für Luft- und Raumfahrt e. V. (DLR) · Technische Universität Berlin · Freie Universität Berlin

    Affiliation as printed

    Center for Astronomy and Astrophysics, Technical University Berlin, Hardenberstrasse 36, 10623 Berlin, Germany

    Institut für Geologische Wissenschaften, Freie Universität Berlin, 12249 Berlin, Germany

    Institute of Planetary Research, German Aerospace Center (DLR), Rutherfordstrasse 2, 12489 Berlin, Germany

68. I. Ribas

    Institute of Space Sciences · Institut d'Estudis Espacials de Catalunya

    Affiliation as printed

    Institut de Ciències de l’Espai (ICE, CSIC), Campus UAB, Can Magrans s/n, 08193 Bellaterra, Spain

    Institut d’Estudis Espacials de Catalunya (IEEC), 08034 Barcelona, Spain

    Institut d'Estudis Espacials de Catalunya (IEEC), 08034 Barcelona, Spain

    Institut de Ciències de l'Espai (ICE, CSIC), Campus UAB, Can Magrans s/n, 08193 Bellaterra, Spain

69. Nuno C. Santos

    Universidade do Porto · Institute of Astrophysics and Space Sciences · Centro de Astrofísica da Universidade do Porto

    Affiliation as printed

    Departamento de Física e Astronomia, Faculdade de Ciências, Universidade do Porto, Rua do Campo Alegre, 4169-007 Porto, Portugal

    Instituto de Astrofísica e Ciências do Espaço, Universidade do Porto, CAUP, Rua das Estrelas, 4150-762 Porto, Portugal

70. G. Scandariato

    Osservatorio Astrofisico di Catania

    Affiliation as printed

    INAF, Osservatorio Astrofisico di Catania, Via S. Sofia 78, 95123 Catania, Italy

71. Damien Ségransan

    University of Geneva

    Affiliation as printed

    Observatoire Astronomique de l’Université de Genève, Chemin Pegasi 51, Versoix, Switzerland

    Observatoire Astronomique de l'Université de Genève, Chemin Pegasi 51, Versoix, Switzerland

72. Roberto Silvotti

    Osservatorio Astrofisico di Torino

    Affiliation as printed

    INAF – Osservatorio Astrofisico di Torino, Strada dell’Osservatorio 20, 10025 Pino Torinese, Italy

    INAF – Osservatorio Astrofisico di Torino, Strada dell'Osservatorio 20, 10025 Pino Torinese, Italy

73. Attila E. Simon

    University of Bern

    Affiliation as printed

    Physikalisches Institut, University of Bern, Gesellsschaftstrasse 6, 3012 Bern, Switzerland

74. Alexis M. S. Smith

    Deutsches Zentrum für Luft- und Raumfahrt e. V. (DLR)

    Affiliation as printed

    Institute of Planetary Research, German Aerospace Center (DLR), Rutherfordstrasse 2, 12489 Berlin, Germany

75. M. Steller

    Austrian Academy of Sciences · Space Research Institute

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    Space Research Institute, Austrian Academy of Sciences, Schmiedlstrasse 6, 8042 Graz, Austria

76. Gyula M. Szabó

    Eötvös Loránd University

    Affiliation as printed

    ELTE Eötvös Loránd University, Gothard Astrophysical Observatory, 9700 Szombathely, Szent Imre h. u. 112, Hungary

    MTA-ELTE Exoplanet Research Group, 9700 Szombathely, Szent Imre h. u. 112, Hungary

77. N. Thomas

    University of Bern

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    Physikalisches Institut, University of Bern, Gesellsschaftstrasse 6, 3012 Bern, Switzerland

78. Stéphane Udry

    University of Geneva

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    Observatoire Astronomique de l’Université de Genève, Chemin Pegasi 51, Versoix, Switzerland

    Observatoire Astronomique de l'Université de Genève, Chemin Pegasi 51, Versoix, Switzerland

79. Valentina Viotto

    Osservatorio Astronomico di Padova

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    INAF, Osservatorio Astronomico di Padova, Vicolo dell’Osservatorio 5, 35122 Padova, Italy

    INAF, Osservatorio Astronomico di Padova, Vicolo dell'Osservatorio 5, 35122 Padova, Italy

80. Nicholas A. Walton

    University of Cambridge

    Affiliation as printed

    Institute of Astronomy, University of Cambridge, Madingley Road, Cambridge, CB3 0HA, UK

81. Karsten Westerdorff

    Deutsches Zentrum für Luft- und Raumfahrt e. V. (DLR)

    Affiliation as printed

    Institute of Optical Sensor Systems, German Aerospace Center (DLR), Rutherfordstrasse 2, 12489 Berlin, Germany

82. Thomas G Wilson

    University of St Andrews

    Affiliation as printed

    Centre for Exoplanet Science, SUPA School of Physics and Astronomy, University of St Andrews, North Haugh, St Andrews KY16 9SS, UK