Attenzione: i dati modificati non sono ancora stati salvati. Per confermare inserimenti o cancellazioni di voci è necessario confermare con il tasto SALVA/INSERISCI in fondo alla pagina
In this paper we present the results of the first low frequency all-sky
search of continuous gravitational wave signals conducted on Virgo VSR2
and VSR4 data. The search covered the full sky, a frequency range
between 20 and 128 Hz with a range of spin-down between -1.0 x 10(-10)
and +1.5 x 10(-11) Hz/s, and was based on a hierarchical approach. The
starting point was a set of short fast Fourier transforms, of length
8192 s, built from the calibrated strain data. Aggressive data cleaning,
in both the time and frequency domains, has been done in order to
remove, as much as possible, the effect of disturbances of instrumental
origin. On each data set a number of candidates has been selected, using
the Frequency Hough transform in an incoherent step. Only coincident
candidates among VSR2 and VSR4 have been examined in order to strongly
reduce the false alarm probability, and the most significant candidates
have been selected. The criteria we have used for candidate selection
and for the coincidence step greatly reduce the harmful effect of large
instrumental artifacts. Selected candidates have been subject to a
follow-up by constructing a new set of longer fast Fourier transforms
followed by a further incoherent analysis, still based on the Frequency
Hough transform. No evidence for continuous gravitational wave signals
was found, and therefore we have set a population-based joint VSR2-VSR4
90\% confidence level upper limit on the dimensionless gravitational
wave strain in the frequency range between 20 and 128 Hz. This is the
first all-sky search for continuous gravitational waves conducted, on
data of ground-based interferometric detectors, at frequencies below 50
Hz. We set upper limits in the range between about 10(-24) and 2 x
10(-23) at most frequencies. Our upper limits on signal strain show an
improvement of up to a factor of similar to 2 with respect to the
results of previous all-sky searches at frequencies below 80 Hz.
First low frequency all-sky search for continuous gravitational wave signals
Aasi, J.;Abbott, B. P.;Abbott, R.;Abbott, T. D.;Abernathy, M. R.;Acernese, F.;Ackley, K.;Adams, C.;Adams, T.;Addesso, P.;Adhikari, R. X.;Adya, V. B.;Affeldt, C.;Agathos, M.;Agatsuma, K.;Aggarwal, N.;Aguiar, O. D.;Ain, A.;Ajith, P.;Allen, B.;Allocca, A.;Amariutei, D. V.;Andersen, M.;Anderson, S. B.;Anderson, W. G.;Arai, K.;Araya, M. C.;Arceneaux, C. C.;Areeda, J. S.;Arnaud, N.;Ashton, G.;Aston, S. M.;Astone, P.;Aufmuth, P.;Aulbert, C.;Babak, S.;Baker, P. T.;Baldaccini, F.;Ballardin, G.;Ballmer, S. W.;Barayoga, J. C.;Barclay, S. E.;Barish, B. C.;Barker, D.;Barone, F.;Barr, B.;Barsotti, L.;Barsuglia, M.;Bartlett, J.;Barton, M. A.;Bartos, I.;Bassiri, R.;Basti, A.;Batch, J. C.;Baune, C.;Bavigadda, V.;Behnke, B.;Bejger, M.;Belczynski, C.;Bell, A. S.;Berger, B. K.;Bergman, J.;Bergmann, G.;Berry, C. P. L.;Bersanetti, D.;Bertolini, A.;Betzwieser, J.;Bhagwat, S.;Bhandare, R.;Bilenko, I. A.;Billingsley, G.;Birch, J.;Birney, R.;Biscans, S.;Bitossi, M.;Biwer, C.;Bizouard, M. A.;Blackburn, J. K.;Blair, C. D.;Blair, D.;Bloemen, S.;Bock, O.;Bodiya, T. P.;Boer, M.;Bogaert, G.;Bojtos, P.;Bond, C.;Bondu, F.;Bonnand, R.;Bork, R.;Born, M.;Boschi, V.;Bose, Sukanta;Brady, P. R.;Braginsky, V. B.;Branchesi, M.;Branco, V.;Brau, J. E.;Briant, T.;Brillet, A.;Brinkmann, M.;Brisson, V.;Brockill, P.;Brooks, A. F.;Brown, D. A.;Brown, D.;Brown, D. D.;Brown, N. M.;Buchanan, C. C.;Buikema, A.;Bulik, T.;Bulten, H. J.;Buonanno, A.;Buskulic, D.;Buy, C.;Byer, R. L.;Cadonati, L.;Cagnoli, G.;Bustillo, J. Calderon;Camp, J. B.;Cannon, K. C.;Cao, J.;Capano, C. D.;Capocasa, E.;Carbognani, F.;Caride, S.;Diaz, J. Casanueva;Caudill, S.;Cavaglia, M.;Cavalier, F.;Cavalieri, R.;Celerier, C.;Cella, G.;Cepeda, C.;Baiardi, L. Cerboni;Cesarini, E.;Chakraborty, R.;Chalermsongsak, T.;Chamberlin, S. J.;Chao, S.;Charlton, P.;Chassande Mottin, E.;Chen, X.;Chen, Y.;Cheng, C.;Chincarini, A.;Chiummo, A.;Cho, H. S.;Cho, M.;Chow, J. H.;Christensen, N.;Chu, Q.;Chua, S.;Chung, S.;Ciani, G.;Clara, F.;Clark, J. A.;Cleva, F.;Coccia, E.;Cohadon, P. F.;Colla, A.;Collette, C. G.;Colombini, M.;Constancio, M.;Conte, A.;Conti, L.;Cook, D.;Corbitt, T. R.;Cornish, N.;Corsi, A.;Costa, C. A.;Coughlin, M. W.;Coughlin, Scott;Countryman, S. T.;Couvares, P.;Coward, D. M.;Cowart, M. J.;Coyne, D. C.;Coyne, R.;Craig, K.;Creighton, J. D. E.;Cripe, J.;Crowder, S. G.;Cumming, A.;Cunningham, L.;Cuoco, E.;Dal Canton, T.;Damjanic, M. D.;Danilishin, S. L.;D'Antonio, S.;Danzmann, K.;Darman, N. S.;Dattilo, V.;Dave, I.;Daveloza, H. P.;Davier, M.;Davies, G. S.;Daw, E. J.;Day, R.;Debra, D.;Debreczeni, G.;Degallaix, J.;De laurentis, M.;Deleglise, S.;Del Pozzo, W.;Denker, T.;Dent, T.;Dereli, H.;Dergachev, V.;De Rosa, R.;Derosa, R. T.;Desalvo, R.;Dhurandhar, S.;Dia, M. C.;Di Fiore, L.;Di Giovanni, M.;Di Lieto, A.;Di Palma, I.;Di Virgilio, A.;Dojcinoski, G.;Dolique, V.;Dominguez, E.;Donovan, F.;Dooley, K. L.;Doravari, S.;Douglas, R.;Downes, T. P.;Drago, M.;Drever, R. W. P.;Driggers, J. C.;Du, Z.;Ducrot, M.;Dwyer, S. E.;Edo, T. B.;Edwards, M. C.;Edwards, M.;Effler, A.;Eggenstein, H. B.;Ehrens, P.;Eichholz, J. M.;Eikenberry, S. S.;Essick, R. C.;Etzel, T.;Evans, M.;Evans, T. M.;Everett, R.;Factourovich, M.;Fafone, V.;Fairhurst, Stephen;Farinon, S.;Farr, B.;Farr, W. M.;Favata, M.;Fays, Maxime;Fejer, M. M.;Feldbaum, D.;Ferrante, I.;Ferreira, E. C.;Ferrini, F.;Fidecaro, F.;Fiori, I.;Fisher, R. P.;Flaminio, R.;Fournier, J. D.;Franco, S.;Frasca, S.;Frasconi, F.;Frede, M.;Frei, Z.;Freise, A.;Frey, R.;Fricke, T. T.;Fritschel, P.;Frolov, V. V.;Fulda, P.;Fyffe, M.;Gabbard, H. A. G.;Gair, J. R.;Gammaitoni, L.;Gaonkar, S. G.;Garufi, F.;Gatto, A.;Gehrels, N.;Gemme, G.;Gendre, B.;Genin, E.;Gennai, A.;Gergely, L. A.;Germain, V.;Ghosh, A.;Ghosh, S.;Giaime, J. A.;Giardina, K. D.;Giazotto, A.;Gleason, J. R.;Goetz, E.;Goetz, R.;Gondan, L.;Gonzalez, G.;Gonzalez, J.;Gopakumar, A.;Gordon, N. A.;Gorodetsky, M. L.;Gossan, S. E.;Gosselin, M.;Gossler, S.;Gouaty, R.;Graef, C.;Graff, P. B.;Granata, M.;Grant, A.;Gras, S.;Gray, C.;Greco, G.;Groot, P.;Grote, H.;Grover, K.;Grunewald, S.;Guidi, G. M.;Guido, C. J.;Guo, X.;Gupta, A.;Gupta, M. K.;Gushwa, K. E.;Gustafson, E. K.;Gustafson, R.;Hacker, J. J.;Hall, B. R.;Hall, E. D.;Hammer, D.;Hammond, G.;Haney, M.;Hanke, M. M.;Hanks, J.;Hanna, C.;Hannam, Mark;Hardwick, T.;Harms, J.;Harry, G. M.;Harry, I. W.;Hart, M. J.;Hartman, M. T.;Haster, C. J.;Haughian, K.;Heidmann, A.;Heintze, M. C.;Heitmann, H.;Hello, P.;Hemming, G.;Hendry, M.;Heng, I. S.;Hennig, J.;Heptonstall, A. W.;Heurs, M.;Hild, S.;Hoak, D.;Hodge, K. A.;Hoelscher Obermaier, J.;Hofman, D.;Hollitt, S. E.;Holt, K.;Hopkins, Paul;Hough, J.;Houston, E. A.;Howell, E. J.;Hu, Y. M.;Huang, S.;Huerta, E. A.;Huet, D.;Hughey, B.;Husa, S.;Huttner, S. H.;Huynh, M.;Huynh Dinh, T.;Idrisy, A.;Indik, N.;Ingram, D. R.;Inta, R.;Islas, G.;Isler, J. C.;Isogai, T.;Iyer, B. R.;Izumi, K.;Jacobson, M. B.;Jang, H.;Jaranowski, P.;Jawahar, S.;Ji, Y.;Jimenez Forteza, F.;Johnson, W. W.;Jones, D. I.;Jones, R.;Jonker, R. J. G.;Ju, L.;Haris, K.;Kalogera, V.;Kandhasamy, S.;Kang, G.;Kanner, J. B.;Karki, S.;Karlen, J. L.;Kasprzack, M.;Katsavounidis, E.;Katzman, W.;Kaufer, S.;Kaur, T.;Kawabe, K.;Kawazoe, F.;Kefelian, F.;Kehl, M. S.;Keitel, D.;Kelley, D. B.;Kells, W.;Kerrigan, J.;Key, J. S.;Khalili, F. Y.;Khan, Z.;Khazanov, E. A.;Kijbunchoo, N.;Kim, C.;Kim, K.;Kim, N. G.;Kim, N.;Kim, Y. M.;King, E. J.;King, P. J.;Kinzel, D. L.;Kissel, J. S.;Klimenko, S.;Kline, J. T.;Koehlenbeck, S. M.;Kokeyama, K.;Koley, S.;Kondrashov, V.;Korobko, M.;Korth, W. Z.;Kowalska, I.;Kozak, D. B.;Kringel, V.;Krishnan, B.;Krolak, A.;Krueger, C.;Kuehn, G.;Kumar, A.;Kumar, P.;Kuo, L.;Kutynia, A.;Lackey, B. D.;Landry, M.;Lantz, B.;Lasky, P. D.;Lazzarini, A.;Lazzaro, C.;Leaci, P.;Leavey, S.;Lebigot, E. O.;Lee, C. H.;Lee, H. K.;Lee, H. M.;Lee, J.;Lee, J. P.;Leonardi, M.;Leong, J. R.;Leroy, N.;Letendre, N.;Levin, Y.;Levine, B. M.;Lewis, J. B.;Li, T. G. F.;Libson, A.;Lin, A. C.;Littenberg, T. B.;Lockerbie, N. A.;Lockett, V.;Lodhia, D.;Logue, J.;Lombardi, A. L.;Lorenzini, M.;Loriette, V.;Lormand, M.;Losurdo, G.;Lough, J. D.;Lubinski, M. J.;Luck, H.;Lundgren, A. P.;Luo, J.;Lynch, R.;Ma, Y.;Macarthur, J.;Macdonald, E. P.;Macdonald, T.;Machenschalk, B.;Macinnis, M.;Macleod, D. M.;Madden Fong, D. X.;Magana Sandoval, F.;Magee, R. M.;Mageswaran, M.;Majorana, E.;Maksimovic, I.;Malvezzi, V.;Man, N.;Mandel, I.;Mandic, V.;Mangano, V.;Mangini, N. M.;Mansell, G. L.;Manske, M.;Mantovani, M.;MARCHESONI, Fabio;Marion, F.;Marka, S.;Ma, Z.;Markosyan, A. S.;Maros, E.;Martelli, F.;Martellini, L.;Martin, I. W.;Martin, R. M.;Martynov, D. V.;Marx, J. N.;Mason, K.;Masserot, A.;Massinger, T. J.;Mastrogiovanni, S.;Matichard, F.;Matone, L.;Mavalvala, N.;Mazumder, N.;Mazzolo, G.;Mccarthy, R.;Mcclelland, D. E.;Mccormick, S.;Mcguire, S. C.;Mcintyre, G.;Mciver, J.;Mcwilliams, S. T.;Meacher, D.;Meadors, G. D.;Mehmet, M.;Meidam, J.;Meinders, M.;Melatos, A.;Mendell, G.;Mercer, R. A.;Merzougui, M.;Meshkov, S.;Messenger, C.;Messick, C.;Meyers, P. M.;Mezzani, F.;Miao, H.;Michel, C.;Middleton, H.;Mikhailov, E. E.;Milano, L.;Miller, J.;Millhouse, M.;Minenkov, Y.;Ming, J.;Mirshekari, S.;Mishra, C.;Mitra, S.;Mitrofanov, V. P.;Mitselmakher, G.;Mittleman, R.;Moe, B.;Moggi, A.;Mohan, M.;Mohapatra, S. R. P.;Montani, M.;Moore, B. C.;Moraru, D.;Moreno, G.;Morriss, S. R.;Mossavi, K.;Mours, B.;Mow Lowry, C. M.;Mueller, C. L.;Mueller, G.;Mukherjee, A.;Mukherjee, S.;Mullavey, A.;Munch, J.;Murphy, D. J.;Murray, P. G.;Mytidis, A.;Nagy, M. F.;Nardecchia, I.;Naticchioni, L.;Nayak, R. K.;Necula, V.;Nedkova, K.;Nelemans, G.;Neri, M.;Newton, G.;Nguyen, T. T.;Nielsen, A. B.;Nitz, A.;Nocera, F.;Nolting, D.;Normandin, M. E. N.;Nuttall, L. K.;Ochsner, E.;O'Dell, J.;Oelker, E.;Ogin, G. H.;Oh, J. J.;Oh, S. H.;Ohme, Frank;Oppermann, P.;Oram, R.;O'Reilly, B.;Ortega, W. E.;O'Shaughnessy, R.;Ott, C. D.;Ottaway, D. J.;Ottens, R. S.;Overmier, H.;Owen, B. J.;Padilla, C. T.;Pai, A.;Pai, S. A.;Palamos, J. R.;Palashov, O.;Palomba, C.;Pal Singh, A.;Pan, H.;Pan, Y.;Pankow, C.;Pannarale Greco, Francesco;Paoletti, F.;Papa, M. A.;Paris, H. R.;Pasqualetti, A.;Passaquieti, R.;Passuello, D.;Patrick, Z.;Pedraza, M.;Pekowsky, L.;Pele, A.;Penn, S.;Perreca, A.;Phelps, M.;Piccinni, O.;Pichot, M.;Pickenpack, M.;Piergiovanni, F.;Pierro, V.;Pillant, G.;Pinard, L.;Pinto, I. M.;Pitkin, M.;Poeld, J. H.;Poggiani, R.;Post, A.;Powell, J.;Prasad, J.;Predoi, Valeriu;Prestegard, T.;Price, L. R.;Prijatelj, M.;Principe, M.;Privitera, S.;Prix, R.;Prodi, G. A.;Prokhorov, L.;Puncken, O.;Punturo, M.;Puppo, P.;Purrer, M.;Qin, J.;Quetschke, V.;Quintero, E. A.;Quitzow James, R.;Raab, F. J.;Rabeling, D. S.;Racz, I.;Radkins, H.;Raffai, P.;Raja, S.;Rakhmanov, M.;Rapagnani, P.;Raymond, V.;Razzano, M.;Re, V.;Reed, C. M.;Regimbau, T.;Rei, L.;Reid, S.;Reitze, D. H.;Ricci, F.;Riles, K.;Robertson, N. A.;Robie, R.;Robinet, F.;Rocchi, A.;Rodger, A. S.;Rolland, L.;Rollins, J. G.;Roma, V. J.;Romano, J. D.;Romano, R.;Romanov, G.;Romie, J. H.;Rosins, D.;Rowan, S.;Rud, A.;Ruggi, P.;Ryan, K.;Sachdev, S.;Sadecki, T.;Sadeghian, L.;Saleem, M.;Salemi, F.;Sammut, L.;Sanchez, E.;Sandberg, V.;Sanders, J. R.;Santiago Prieto, I.;Sassolas, B.;Sathyaprakash, Bangalore S.;Savage, R.;Sawadsky, A.;Schale, P.;Schilling, R.;Schmidt, P.;Schnabel, R.;Schofield, R. M. S.;Schonbeck, A.;Schreiber, E.;Schuette, D.;Schutz, Bernard Frederick;Scott, S. M.;Sellers, D.;Sentenac, D.;Sequino, V.;Sergeev, A.;Serna, G.;Sevigny, A.;Shaddock, D. A.;Shaffery, P.;Shah, S.;Shahriar, M. S.;Shaltev, M.;Shao, Z.;Shapiro, B.;Shawhan, P.;Shoemaker, D. H.;Sidery, T. L.;Siellez, K.;Siemens, X.;Sigg, D.;Silva, A. D.;Simakov, D.;Singer, A.;Singer, L. P.;Singh, R.;Sintes, A. M.;Slagmolen, B. J. J.;Smith, J. R.;Smith, N. D.;Smith, R. J. E.;Son, E. J.;Sorazu, B.;Souradeep, T.;Srivastava, A. K.;Staley, A.;Steinke, M.;Steinlechner, J.;Steinlechner, S.;Steinmeyer, D.;Stephens, B. C.;Steplewski, S.;Stevenson, S. P.;Stone, R.;Strain, K. A.;Straniero, N.;Strauss, N. A.;Strigin, S.;Sturani, R.;Stuver, A. L.;Summerscales, T. Z.;Sun, L.;Sutton, Patrick J.;Szczepanczyk, M. J.;Tacca, M.;Talukder, D.;Tanner, D. B.;Tap, M.;Tarabrin, S. P.;Taracchini, A.;Taylor, R.;Theeg, T.;Thirugnanasambandam, M. P.;Thomas, M.;Thomas, P.;Thorne, K. A.;Thorne, K. S.;Thrane, E.;Tiwari, S.;Tiwari, Vaibhav;Tomlinson, C.;Tonelli, M.;Torres, C. V.;Torrie, C. I.;Travasso, F.;Traylor, G.;Trifiro, D.;Tringali, M. C.;Tse, M.;Turconi, M.;Ugolini, D.;Unnikrishnan, C. S.;Urban, A. L.;Usman, S. A.;Vahlbruch, H.;Vajente, G.;Valdes, G.;Vallisneri, M.;Van Bakel, N.;Van Beuzekom, M.;Van den Brand, J. F. J.;Van den Broeck, C.;Van der Schaaf, L.;Van der Sluys, M. V.;Eijningen, J. V.;Eggel, A. A. V.;Vardaro, M.;Vass, S.;Vasuth, M.;Vaulin, R.;Vecchio, A.;Vedovato, G.;Veitch, J.;Veitch, P. J.;Venkateswara, K.;Verkindt, D.;Vetrano, F.;Vicere, A.;Vinet, J. Y;Vo, T.;Vocca, H.;Vorvick, C.;Vousden, W. D.;Vyatchanin, S. P.;Wade, A. R.;Wade, M.;Wade, L. E.;Walker, M.;Wallace, L.;Walsh, S.;Wang, G.;Wang, H.;Wang, M.;Wang, X.;Ward, R. L.;Warner, J.;Was, M.;Weaver, B.;Wei, L. W.;Weinert, M.;Weinstein, A. J.;Weiss, R.;Welborn, T.;Wen, L.;Wessels, P.;Westphal, T.;Wette, K.;Whelan, J. T.;Whitcomb, S. E.;White, D. J.;Whiting, B. F.;Williams, K. J.;Williams, L.;Williams, R. D.;Williamson, Andrew;Willke, B.;Wimmer, M. H.;Winkler, W.;Wipf, C. C.;Wittel, H.;Woan, G.;Worden, J.;Yablon, J.;Yakushin, I.;Yam, W.;Yamamoto, H.;Yancey, C. C.;Yvert, M.;Zadrozny, A.;Zangrando, L.;Zanolin, M.;Zendri, J. P.;Zhang, Fan;Zhang, M.;Zhang, Y.;Zhao, C.;Zhou, M.;Zhu, X. J.;Zucker, M. E.;Zuraw, S. E.;Zweizig, J.
2016-01-01
Abstract
In this paper we present the results of the first low frequency all-sky
search of continuous gravitational wave signals conducted on Virgo VSR2
and VSR4 data. The search covered the full sky, a frequency range
between 20 and 128 Hz with a range of spin-down between -1.0 x 10(-10)
and +1.5 x 10(-11) Hz/s, and was based on a hierarchical approach. The
starting point was a set of short fast Fourier transforms, of length
8192 s, built from the calibrated strain data. Aggressive data cleaning,
in both the time and frequency domains, has been done in order to
remove, as much as possible, the effect of disturbances of instrumental
origin. On each data set a number of candidates has been selected, using
the Frequency Hough transform in an incoherent step. Only coincident
candidates among VSR2 and VSR4 have been examined in order to strongly
reduce the false alarm probability, and the most significant candidates
have been selected. The criteria we have used for candidate selection
and for the coincidence step greatly reduce the harmful effect of large
instrumental artifacts. Selected candidates have been subject to a
follow-up by constructing a new set of longer fast Fourier transforms
followed by a further incoherent analysis, still based on the Frequency
Hough transform. No evidence for continuous gravitational wave signals
was found, and therefore we have set a population-based joint VSR2-VSR4
90\% confidence level upper limit on the dimensionless gravitational
wave strain in the frequency range between 20 and 128 Hz. This is the
first all-sky search for continuous gravitational waves conducted, on
data of ground-based interferometric detectors, at frequencies below 50
Hz. We set upper limits in the range between about 10(-24) and 2 x
10(-23) at most frequencies. Our upper limits on signal strain show an
improvement of up to a factor of similar to 2 with respect to the
results of previous all-sky searches at frequencies below 80 Hz.
I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.
Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11581/400496
Citazioni
ND
33
44
social impact
Conferma cancellazione
Sei sicuro che questo prodotto debba essere cancellato?
simulazione ASN
Il report seguente simula gli indicatori relativi alla propria produzione scientifica in relazione alle soglie ASN 2023-2025 del proprio SC/SSD. Si ricorda che il superamento dei valori soglia (almeno 2 su 3) è requisito necessario ma non sufficiente al conseguimento dell'abilitazione. La simulazione si basa sui dati IRIS e sugli indicatori bibliometrici alla data indicata e non tiene conto di eventuali periodi di congedo obbligatorio, che in sede di domanda ASN danno diritto a incrementi percentuali dei valori. La simulazione può differire dall'esito di un’eventuale domanda ASN sia per errori di catalogazione e/o dati mancanti in IRIS, sia per la variabilità dei dati bibliometrici nel tempo. Si consideri che Anvur calcola i valori degli indicatori all'ultima data utile per la presentazione delle domande.
La presente simulazione è stata realizzata sulla base delle specifiche raccolte sul tavolo ER del Focus Group IRIS coordinato dall’Università di Modena e Reggio Emilia e delle regole riportate nel DM 589/2018 e allegata Tabella A. Cineca, l’Università di Modena e Reggio Emilia e il Focus Group IRIS non si assumono alcuna responsabilità in merito all’uso che il diretto interessato o terzi faranno della simulazione. Si specifica inoltre che la simulazione contiene calcoli effettuati con dati e algoritmi di pubblico dominio e deve quindi essere considerata come un mero ausilio al calcolo svolgibile manualmente o con strumenti equivalenti.