Skip to main content
Log in

Herschel celestial calibration sources

Four large main-belt asteroids as prime flux calibrators for the far-IR/sub-mm range

  • Original Article
  • Published:
Experimental Astronomy Aims and scope Submit manuscript

Abstract

Celestial standards play a major role in observational astrophysics. They are needed to characterise the performance of instruments and are paramount for photometric calibration. During the Herschel Calibration Asteroid Preparatory Programme approximately 50 asteroids have been established as far-IR/sub-mm/mm calibrators for Herschel. The selected asteroids fill the flux gap between the sub-mm/mm calibrators Mars, Uranus and Neptune, and the mid-IR bright calibration stars. All three Herschel instruments observed asteroids for various calibration purposes, including pointing tests, absolute flux calibration, relative spectral response function, observing mode validation, and cross-calibration aspects. Here we present newly established models for the four large and well characterized main-belt asteroids (1) Ceres, (2) Pallas, (4) Vesta, and (21) Lutetia which can be considered as new prime flux calibrators. The relevant object-specific properties (size, shape, spin-properties, albedo, thermal properties) are well established. The seasonal (distance to Sun, distance to observer, phase angle, aspect angle) and daily variations (rotation) are included in a new thermophysical model setup for these targets. The thermophysical model predictions agree within 5 % with the available (and independently calibrated) Herschel measurements. The four objects cover the flux regime from just below 1,000 Jy (Ceres at mid-IR N-/Q-band) down to fluxes below 0.1 Jy (Lutetia at the longest wavelengths). Based on the comparison with PACS, SPIRE and HIFI measurements and pre-Herschel experience, the validity of these new prime calibrators ranges from mid-infrared to about 700 μm, connecting nicely the absolute stellar reference system in the mid-IR with the planet-based calibration at sub-mm/mm wavelengths.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10

Similar content being viewed by others

Notes

  1. Near-IR filter leaks are photometrically problematic when near-IR bright objects -like stars- are observed in far-IR bands.

  2. http://en.wikipedia.org/wiki/Atacama_Large_Millimeter_Array

  3. Hubble Space Telescope

  4. Herschel unique observation identifier

  5. The Vesta SED requires a colour correction value of 1.03 in the green band

  6. The PACS photometric calibration is based on the assumption of a constant energy spectrum of the observed source ν × F ν = λ × F λ . Asteroid SEDs deviate from this assumption and colour-corrections are required

  7. HIPE is a joint development by the Herschel Science Ground Segment Consortium, consisting of ESA, the NASA Herschel Science Center, and the HIFI, PACS and SPIRE consortia.

  8. http://www.lesia.obspm.fr/perso/emmanuel-lellouch/mars/

  9. The thermal inertia Γ is defined as \(\sqrt {\kappa \rho c}\), where κ is the thermal conductivity, ρ the density, and c the heat capacity.

  10. Database of Asteroid Models from Inversion Techniques, http://astro.troja.mff.cuni.cz/projects/asteroids3D/

  11. http://dawn.jpl.nasa.gov/

  12. http://www.esa.int/Our_Activities/Space_Science/Rosetta

  13. Database of Asteroid Models from Inversion Techniques, http://astro.troja.mff.cuni.cz/projects/asteroids3D/

  14. http://www.ir.isas.jaxa.jp/SPICA/SPICA_HP/index_English.html

References

  1. Altieri, B.: PACS Observer’s Manual, HERSCHEL-HSC-DOC-0832, vol. 2.4 (2011) , http://herschel.esac.esa.int/Docs/PACS/html/pacs_om.html

  2. Balog, Z., Müller, T.G., Nielbock, M., et al.: The Herschel-PACS photometer calibration: point-source flux calibration. (2013), Exp. Astron. this issue

  3. Beichman, C.A., Neugebauer, G., Habing, H.J., Clegg, P.E., Chester, T.J.: Infrared Astronomical Satellite (IRAS) Catalogs and Atlases, vol. 1: Explanatory Supplement. (1988)

  4. Belskaya, I.N., Fornasier, S., Krugly, Yu.N., et al.: Puzzling asteroid 21 Lutetia: our knowledge prior to the Rosetta fly-by. A&A 515, 29 (2010)

    Article  ADS  Google Scholar 

  5. Bendo, G.J., Griffin, M.J., Bock, J.J., et al.: Flux calibration of the Herschel-SPIRE photometer. MNRAS 433, 3062 (2013)

    Article  ADS  Google Scholar 

  6. Bowell, E., Hapke, B., Domingue, D., et al.: Application of photometric models to asteroids. In: Binzel, R.P., et al. (eds.) Asteroids II, pp. 524–556. University of Arizona Press (1989)

  7. Carry, B., Dumas, C., Fulchignoni, M., et al.: Near-infrared mapping and physical properties of the dwarf-planet Ceres. A&A 478, 235 (2008)

    Article  ADS  Google Scholar 

  8. Carry, B., Dumas, C., Kaasalainen, M., et al.: Physical properties of (2) Pallas. Icarus 205, 460 (2010)

    Article  ADS  Google Scholar 

  9. Carry, B., Kaasalainen, M., Merline, W.J., et al.: Shape modeling technique KOALA validated by ESA Rosetta at (21) Lutetia. P&SS 66, 200 (2012)

    Article  ADS  Google Scholar 

  10. Chamberlain, M.A., Boynton, W.V.: Ceres lightcurve analysis Period determination. Icarus 188, 451 (2007)

    Article  ADS  Google Scholar 

  11. Cohen, M.: Stellar calibration in the infrared: extending the legacy of the KAO, ISO, and MSX to SIRTF and beyond. In: Metcalfe, L., Salama, A., Peschke, S.B., Kessler, M.F. (eds.) Proceedings of the Conference “The Calibration Legacy of the ISO Mission”, p. 135. ESA SP-481 (2003)

  12. Decin, L.: Stellar models in IR calibration. In: Metcalfe, L., Salama, A., Peschke, S.B., Kessler, M.F. (eds.) Proceedings of the Conference “The Calibration Legacy of the ISO Mission”, p. 141. ESA SP-481 (2003)

  13. de Graauw, Th., Helmich, F.P., Phillips, T.G., et al.: The Herschel-Heterodyne instrument for the far-infrared (HIFI). A&A 518, 6D (2010)

    Article  Google Scholar 

  14. Dehaes, S., Bauwens, E., Decin, L., et al.: Structure of the outer layers of cool standard stars. A&A 533, A107 (2011)

    Article  ADS  Google Scholar 

  15. Drummond, J.D., Weidenschilling, S.J., Chapman, C.R., Davis, D.R.: Photometric geodesy of main-belt asteroids. II - Analysis of lightcurves for poles, periods, and shapes. Icarus 76, 19 (1988)

    Article  ADS  Google Scholar 

  16. Drummond, J.D., Christou, J., Nelson, J.: Triaxial ellipsoid dimensions and poles of asteroids from AO observations at the Keck-II telescope. Icarus 202, 147 (2009)

    Article  ADS  Google Scholar 

  17. Drummond, J.D., Carry, B., Merline, W.J., et al.: The size and pole of Ceres from nine years of adaptive optics observations at Keck and the VLT, AAS, DPS meeting #45, #208.06 (2013)

  18. Dunham, D.W., Dunham, J.B., Binzel, R.P., et al.: The size and shape of (2) Pallas from the 1983 occultation of 1 Vulpeculae. AJ 99, 1636 (1990)

    Article  ADS  Google Scholar 

  19. Engelbracht, C.W., Blaylock, M., Su, K.Y.L., et al.: Absolute calibration and characterization of the multiband imaging photometer for spitzer. I. The stellar calibrator sample and the 24 micron calibration. PASP 119, 994–1018 (2007)

    Article  ADS  Google Scholar 

  20. Gordon, K.D., Engelbracht, C.W., Fadda, D., et al.: Absolute calibration and characterization of the multiband imaging photometer for Spitzer. II. 70 micron Imaging. PASP 119, 1019–1037 (2007)

    Article  ADS  Google Scholar 

  21. Griffin, M.J., North, C.E., Schulz, B., et al.: Flux calibration of broadband far infrared and submillimetre photometric instruments: Theory and application to Herschel-SPIRE. MNRAS 434, 992–1004 (2013)

    Article  ADS  Google Scholar 

  22. Griffin, M.J., Orton, G.S.: The near-millimeter brightness temperature spectra of Uranus and Neptune. Icarus 105, 537 (1993)

    Article  ADS  Google Scholar 

  23. Griffin, M.J., Abergel, A., Abreu, A., et al.: The Herschel-SPIRE instrument and its in-flight performance. A&A 518, 3G (2010)

    Article  ADS  Google Scholar 

  24. Hammersley, P.L., Jourdain de Muizon, M.: The development of stellar photometric standards for ISO. In: Metcalfe, L., Salama, A., Peschke, S.B., Kessler, M.F. (eds.) Proceedings of the Conference “The Calibration Legacy of the ISO Mission”, p. 129. ESA SP-481 (2003)

  25. Horner, J., Müller, T.G., Lykawka, P.S.: (1173) Anchises—thermophysical and dynamical studies of a dynamically unstable Jovian Trojan. MNRAS 423, 2587–2596 (2012)

    Article  ADS  Google Scholar 

  26. Kawada, M., Baba, H., Barthel, P.D., et al.: The Far-Infrared Surveyor (FIS) for AKARI. PASJ 59, 389 (2007)

    ADS  Google Scholar 

  27. Keihm, S.J.: Interpretation of the lunar microwave brightness temperature spectrum - feasibility of orbital heat flow mapping. Icarus 60, 568 (2013)

    Article  ADS  Google Scholar 

  28. Kessler, M.F., Steinz, J.A., Anderegg, M.E., et al.: The Infrared Space Observatory (ISO) mission. A&A 315, 27 (1996)

    ADS  Google Scholar 

  29. Klaas, U., Abraham, P., Acosta-Pulido, J.A., et al.: ISOPHOT in-flight calibration strategies. In: Metcalfe, L., Salama, A., Peschke, S.B., Kessler, M.F. (eds.) Proceedings of the Conference “The Calibration Legacy of the ISO Missions”, p. 19. ESA SP-481 (2003)

  30. Lagerkvist, C.-I., Magnusson, P., Williams, I.P., et al.: Physical studies of asteroids. XXIV - phase relations for 48 asteroids obtained with the Carlsberg Meridian circle. A&AS 94, 43 (1992)

    ADS  Google Scholar 

  31. Lagerkvist, C.-I., Piironen, J., Erikson, A. (eds.): Asteroid Photometric Catalogue, p. 201, 5th edn. Astronomical Observatory, Uppsala University, Sweden (2001)

    Google Scholar 

  32. Lagerros, J.S.V.: Thermal physics of asteroids. I. Effects of shape, heat conduction and beaming. A&A 310, 1011 (1996)

    ADS  Google Scholar 

  33. Lagerros, J.S.V.: Thermal physics of asteroids. III. Irregular shapes and albedo variegations. A&A 325, 1226 (1997)

    ADS  Google Scholar 

  34. Lagerros, J.S.V.: Thermal physics of asteroids. IV. Thermal infrared beaming. A&A 332, 1123 (1998)

    ADS  Google Scholar 

  35. Lamy, P.L., Faury, G., Jorda, L., et al.: Multi-color, rotationally resolved photometry of asteroid 21 Lutetia from OSIRIS/Rosetta observations. A&A 521, 19 (2010)

    Article  ADS  Google Scholar 

  36. Lebofsky, L.A., Sykes, M.V., Tedesco, E.T., et al.: A refined “Standard” thermal model for asteroids based on observations of 1 ceres and 2 pallas. Icarus 68, 239–251 (1986)

    Article  ADS  Google Scholar 

  37. Li, J.-Y., McFadden, L.A., Parker, J.W., et al.: Photometric analysis of 1 ceres and surface mapping from HST observations. Icarus 182, 143 (2006)

    Article  ADS  Google Scholar 

  38. Lim, T.L., Swinyard, B.M., Burgdorf, M.J., et al.: The LWS calibration strategy. In: Metcalfe, L., Salama, A., Peschke, S.B., Kessler, M.F. (eds.) Proceedings of the Conference “The Calibration Legacy of the ISO Missions”, p. 13. ESA SP-481 (2003)

  39. Lim, T.L., Stansberry, J., Müller, T.G., et al.: “TNOs are Cool”: a survey of the trans-Neptunian region. III. Thermophysical properties of 90482 Orcus and 136472 makemake. A&A 518, 148–152 (2010)

    Article  ADS  Google Scholar 

  40. Lutz, D.: PACS Photometer Point Spread Function. PICC-ME-TN-033, vol. 2.0., http://herschel.esac.esa.int/twiki/pub/Public/PacsCalibrationWeb/bolopsf_20.pdf (2012)

  41. Marston, A.P., Teyssier, D.: HIFI Observer’s Manual, HERSCHEL-HSC-DOC-0784, vol. 2.4., http://herschel.esac.esa.int/Docs/HIFI/html/hifi_om.html (2011)

  42. Millis, R.L., Wassermann, L.H., Franz, O.G., et al.: The size, shape, density, and Abledo of ceres from its occultation of BD+8deg471. Icarus 72, 507–518 (1987)

    Article  ADS  Google Scholar 

  43. Moór, A., Müller, T.G., Kiss, C., et al.: PACS photometer calibration block analysis. Exp. Astron. this issue (2013)

  44. Moreno, R.: PhD thesis. Université de Paris (1998)

  45. Moreno, R.: Neptune and Uranus planetary brightness temperature tabulation. Tech. rep., ESA Herschel Science Centre. available from ftp://ftp.sciops.esa.int/pub/hsc-calibration/PlanetaryModels/ESA4/ (2012)

  46. Müller, T.G.: Asteroids as IR standards for ISOPHOT. A&A 338, 340–352 (1998)

    Google Scholar 

  47. Müller, T.G., Lagerros, J.SV., Burgdorf, M., et al.: Fundamental thermal emission parameters of main-belt asteroids derived from ISO. In: Cox, P., Kessler, M.F. (eds.) The Universe as Seen by ISO, p. 141. ESA SP-427 (1999)

  48. Müller, T.G., Lagerros, J.SV.: Asteroids as calibration standards in the thermal infrared for space observatories. A&A 381, 324–339 (2002)

    Article  ADS  Google Scholar 

  49. Müller, T.G.: Thermophysical analysis of infrared observations of asteroids. M&PS 37, 1919 (2002)

    ADS  Google Scholar 

  50. Müller, T.G., Lagerros, J.S.V.: Asteroids as calibration standards in the thermal infrared—applications and results from ISO. In: Metcalfe, L., Salama, A., Peschke, S.B., Kessler, M.F. (eds.) Proceedings of the Conference “The Calibration Legacy of the ISO Missions”, p. 157. ESA SP-481 (2003)

  51. Müller, T.G., Blommaert, J.A.D.L.: 65 Cybele in the thermal infrared: multiple observations and thermophysical analysis. A&A 418, 347–356 (2004)

    Article  ADS  Google Scholar 

  52. Müller, T.G., Sterzik, M.F., Schütz, O., et al.: Thermal infrared observations of near-Earth asteroid 2002 NY40. A&A 424, 1075–1080 (2004)

    Article  ADS  Google Scholar 

  53. Müller, T.G.: The Asteroid Preparatory Programme for HERSCHEL, ASTRO-F & ALMA. In: Wilson, A. (ed.) Proceedings of the Dusty and Molecular Universe: A Prelude to Herschel and ALMA, p. 471. ESA SP-577 (2005)

  54. Müller, T.G., Sekiguchi, T., Kaasalainen, M., et al.: Thermal infrared observations of the Hayabusa spacecraft target asteroid 25143 Itokawa. A&A 443, 347–355 (2005)

    Article  ADS  Google Scholar 

  55. Müller, T.G., Ďurech, J., Hasegawa, S., et al.: Thermo-physical properties of 162173 (1999 JU3), a potential flyby and rendezvous target for interplanetary missions—Based on the experience from the Hayabusa flyby target 25143 Itokawa. A&A 525, 145 (2011)

    Article  Google Scholar 

  56. Müller, T.G., Okumura, K., Klaas, U.: PACS Photometer Passbands and Colour Correction Factors for Various Source SEDs, PICC-ME-TN-038, vol. 1.0., http://herschel.esac.esa.int/twiki/pub/Public/PacsCalibrationWeb/cc_report_v1.pdf (2011)

  57. Müller, T.G., Nielbock, M., Balog, Z., et al.: PACS Photometer—Point-Source Flux Calibration, PICC-ME-TN-037, vol. 1.0. (2011)

  58. Müller, T.G., O’Rourke, L., Barucci, A.M., et al.: Physical properties of OSIRIS-REx target asteroid (101955) 1999 RQ36. Derived from Herschel, VLT/ VISIR, and Spitzer observations. A&A 548, 36–45 (2012)

    Article  Google Scholar 

  59. Murakami, H., Baba, H., Barthel, P., et al.: The infrared astronomical mission AKARI. PASJ 59, 369 (2007)

    ADS  Google Scholar 

  60. Nielbock, M., Müller, T.G., Balog, Z., et al. The Herschel-PACS photometer calibration: a time dependent flux calibration for the PACS chopped photometry AOT mode. Exp. Astron. this issue (2013)

  61. O’Rourke, L., Müller, T.G., Valtchanov, I., et al.: Thermal & shape properties of asteroid (21) Lutetia from Herschel observations around the Rosetta flyby. P&SS 55, 192 (2012)

    Article  Google Scholar 

  62. Orton, G.S., Burgdorf, M.J.: Planetary Spectral Models as References for Calibration. In: Metcalfe, L., Salama, A., Peschke, S.B., Kessler, M.F. (eds.) Proceedings of the Conference “The Calibration Legacy of the ISO Mission”, p. 147. ESA SP-481 (2003)

  63. Pearson, C., Lim, T., North, C.E., et al.: SPIRE point source photometry within the Herschel interactive processing environment (HIPE). Exp. Astron. (2013) this issue

  64. Pilbratt, G.L., Riedinger, J.R., Passvogel, T., et al.: Herschel Space Observatory. An ESA facility for far-infrared and submillimetre astronomy. A&A 518, L1 (2010)

    Article  ADS  Google Scholar 

  65. Poglitsch, A., Waelkens, C., Geis, N., et al.: The Photodetector Array Camera and Spectrometer (PACS) on the Herschel Space Observatory. A&A 518, L2 (2010)

    Article  ADS  Google Scholar 

  66. Reach, W.T., Megeath, S.T., Cohen, M., et al.: Absolute calibration of the infrared array camera on the Spitzer space telescope. PASP 117, 978–990 (2005)

    Article  ADS  Google Scholar 

  67. Rieke, G.H., Lebofsky, M.J., Low, F.J.: An absolute photometric system at 10 and 20 microns. AJ 90, 900 (1985)

    Article  ADS  Google Scholar 

  68. Rieke, G.H., Young, E.T., Engelbracht, C.W., et al.: The multiband imaging photometer for Spitzer (MIPS). ApJS 154, 25 (2004)

    Article  ADS  Google Scholar 

  69. Roelfsema, R.R., Helmich, F.P., Teyssier, D., et al.: In-orbit performance of Herschel-HIFI. A&A 537, A17 (2012)

    Article  ADS  Google Scholar 

  70. Russell, C.T., Raymond, C.A., Coradini, A., et al.: Dawn at vesta: testing the protoplanetary paradigm. Science 336, 684 (2012)

    Article  ADS  Google Scholar 

  71. Sandell, G.: Secondary calibrators at submillimetre wavelengths. MNRAS 271, 75 (1994)

    ADS  Google Scholar 

  72. Sandell, G.: Submillimetre calibration—experience from ground-based observations. In: Metcalfe, L., Salama, A., Peschke, S.B., Kessler, M.F. (eds.) Proceedings of the Conference “The Calibration Legacy of the ISO Mission”, p. 439. ESA SP-481 (2003)

  73. Schmidt, B.E., Thomas, P.C., Bauer, J.M., et al.: The shape and surface variation of 2 Pallas from the hubble space telescope. Sci. 326, 275 (2009)

    Article  ADS  Google Scholar 

  74. Schulz, B.: ISOPHOT point source calibration—wrapping it up. In: Metcalfe, L., Salama, A., Peschke, S.B., Kessler, M.F. (eds.) Proceedings of the Conference “The Calibration Legacy of the ISO Mission”, p. 83. ESA SP-481 (2003)

  75. Stansberry, J.A., Gordon, K.D., Bhattacharya, B., et al.: Absolute calibration and characterization of the multiband imaging photometer for Spitzer. III. An asteroid-based calibration of MIPS at 160 micron. PASP 119, 1038 (2007)

    Article  ADS  Google Scholar 

  76. Swinyard, B.M., Ade, P., Baluteau, J.-P., et al.: In-flight calibration of the Herschel-SPIRE instrument. A&A 518, 4S (2010)

    Article  ADS  Google Scholar 

  77. Thomas, P.C., Binzel, R.P., Gaffey, M.J., et al.: Vesta: spin pole, size, and shape from HST images. Icarus 128, 88 (1997)

    Article  ADS  Google Scholar 

  78. Thomas, P.C., Parker, J.W., McFadden, L.A., et al.: Differentiation of the asteroid Ceres as revealed by its shape. Nature 437, 224–226 (2005)

    Article  ADS  Google Scholar 

  79. Valtchanov, I.: SPIRE Observer’s Manual, HERSCHEL-DOC-0798, vol. 2.4., http://herschel.esac.esa.int/Docs/SPIRE/html/spire_om.html (2011)

  80. Wasserman, L.H., Millis, R.L., Franz, O.G., et al.: The diameter of Pallas from its occultation of SAO 85009. AJ 84, 259 (1979)

    Article  ADS  Google Scholar 

  81. Werner, M.W., Roellig, T.L., Low, F.J., et al.: The Spitzer space telescope mission. ApJS 154, 1 (2004)

    Article  ADS  Google Scholar 

  82. Wright, E.L., Eisenhardt, P.RM., Mainzer, A.K., et al.: The wide-field infrared survey explorer (WISE): mission description and initial on-orbit performance. AJ 140, 1868–1881 (2010)

    Article  ADS  Google Scholar 

Download references

Acknowledgments

We would like to thank the PIs of the various scientific projects for permission to use their Herschel science data in the context of our calibration work.

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Thomas Müller.

Appendix

Appendix

1.1 Overview of available Herschel photometric measurements

In the following tables we list the available photometric observations (calibration and science observations) with one of the four asteroids in the field of view. Some of the early measurements were used with very different instrument settings and non-standard observing modes. The corresponding fluxes are not well calibrated and we excluded them from our analysis.

Table 2 Overview of all relevant Herschel-PACS photometer scan-map observations of (1) Ceres
Table 3 Overview of all relevant Herschel-PACS photometer chop-nod observations of (1) Ceres
Table 4 Overview of all relevant Herschel-SPIRE photometer observations of (1) Ceres
Table 5 Overview of all relevant Herschel-HIFI point observations of (1) Ceres
Table 6 Overview of all relevant Herschel-PACS photometer scan-map observations of (2) Pallas
Table 7 Overview of all relevant Herschel-PACS photometer chop-nod observations of (2) Pallas
Table 8 Overview of all relevant Herschel-SPIRE photometer observations of (2) Pallas, like in Table 4
Table 9 Overview of all relevant Herschel-PACS photometer scan-map observations of (4) Vesta
Table 10 Overview of all relevant Herschel-PACS photometer chop-nod observations of (4) Vesta
Table 11 Overview of all relevant Herschel-Spire photometer observations of (4) Vesta like in Table 4
Table 12 Overview of all relevant Herschel-PACS photometer scan-map observations of (21) Lutetia
Table 13 Overview of all relevant Herschel-PACS photometer chop-nod observations of (21) Lutetia
Table 14 Overview of all relevant Herschel-SPIRE photometer observations of (21) Lutetia like in Table 4
Table 15 Additional Herschel fixed position photometer observations (no tracking)

1.2 Observational results of the Herschel photometric measurements

Table 16 Photometric Herschel data of (1) Ceres together with the observing geometry and the TPM predictions
Table 17 Photometric Herschel data of (2) Pallas
Table 18 Photometric Herschel data of (4) Vesta
Table 19 Photometric Herschel data of (21) Lutetia

Rights and permissions

Reprints and permissions

About this article

Cite this article

Müller, T., Balog, Z., Nielbock, M. et al. Herschel celestial calibration sources. Exp Astron 37, 253–330 (2014). https://doi.org/10.1007/s10686-013-9357-y

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10686-013-9357-y

Keywords

Navigation