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##  Appendix A The ALICE Collaboration

S. Acharya 126, D. Adamová 86, G. Aglieri Rinella 33, M. Agnello 30, N.
Agrawal 51, Z. Ahammed 134, S. Ahmad16, S.U. Ahn 71, I. Ahuja 38, A. Akindinov
142, M. Al-Turany 97, D. Aleksandrov 142, B. Alessandro 56, H.M. Alfanda 6, R.
Alfaro Molina 67, B. Ali 16, A. Alici 26, N. Alizadehvandchali 115, A. Alkin
33, J. Alme 21, G. Alocco 52, T. Alt 64, A.R. Altamura 50, I. Altsybeev 95,
M.N. Anaam 6, C. Andrei 46, N. Andreou 114, A. Andronic 137, V. Anguelov 94,
F. Antinori 54, P. Antonioli 51, N. Apadula 74, L. Aphecetche 103, H.
Appelshäuser 64, C. Arata 73, S. Arcelli 26, M. Aresti 23, R. Arnaldi 56,
J.G.M.C.A. Arneiro 110, I.C. Arsene 20, M. Arslandok 139, A. Augustinus 33, R.
Averbeck 97, M.D. Azmi16, H. Baba123, A. Badalà 53, J. Bae 104, Y.W. Baek 41,
X. Bai 119, R. Bailhache 64, Y. Bailung 48, A. Balbino 30, A. Baldisseri 129,
B. Balis 2, D. Banerjee 4, Z. Banoo 91, R. Barbera 27, F. Barile 32, L.
Barioglio 95, M. Barlou78, B. Barman42, G.G. Barnaföldi 138, L.S. Barnby 85,
V. Barret 126, L. Barreto 110, C. Bartels 118, K. Barth 33, E. Bartsch 64, N.
Bastid 126, S. Basu 75, G. Batigne 103, D. Battistini 95, B. Batyunya 143, D.
Bauri47, J.L. Bazo Alba 101, I.G. Bearden 83, C. Beattie 139, P. Becht 97, D.
Behera 48, I. Belikov 128, A.D.C. Bell Hechavarria 137, F. Bellini 26, R.
Bellwied 115, S. Belokurova 142, Y.A.V. Beltran 45, G. Bencedi 138, S. Beole
25, Y. Berdnikov 142, A. Berdnikova 94, L. Bergmann 94, M.G. Besoiu 63, L.
Betev 33, P.P. Bhaduri 134, A. Bhasin 91, M.A. Bhat 4, B. Bhattacharjee 42, L.
Bianchi 25, N. Bianchi 49, J. Bielčík 36, J. Bielčíková 86, J. Biernat 107,
A.P. Bigot 128, A. Bilandzic 95, G. Biro 138, S. Biswas 4, N. Bize 103, J.T.
Blair 108, D. Blau 142, M.B. Blidaru 97, N. Bluhme39, C. Blume 64, G. Boca
22,55, F. Bock 87, T. Bodova 21, A. Bogdanov142, S. Boi 23, J. Bok 58, L.
Boldizsár 138, M. Bombara 38, P.M. Bond 33, G. Bonomi 133,55, H. Borel 129, A.
Borissov 142, A.G. Borquez Carcamo 94, H. Bossi 139, E. Botta 25, Y.E.M.
Bouziani 64, L. Bratrud 64, P. Braun-Munzinger 97, M. Bregant 110, M. Broz 36,
G.E. Bruno 96,32, M.D. Buckland 24, D. Budnikov 142, H. Buesching 64, S.
Bufalino 30, P. Buhler 102, N. Burmasov 142, Z. Buthelezi 68,122, A. Bylinkin
21, S.A. Bysiak107, M. Cai 6, H. Caines 139, A. Caliva 29, E. Calvo Villar
101, J.M.M. Camacho 109, P. Camerini 24, F.D.M. Canedo 110, M. Carabas 125,
A.A. Carballo 33, F. Carnesecchi 33, R. Caron 127, L.A.D. Carvalho 110, J.
Castillo Castellanos 129, F. Catalano 33,25, C. Ceballos Sanchez 143, I.
Chakaberia 74, P. Chakraborty 47, S. Chandra 134, S. Chapeland 33, M. Chartier
118, S. Chattopadhyay 134, S. Chattopadhyay 99, T.G. Chavez 45, T. Cheng 97,6,
C. Cheshkov 127, B. Cheynis 127, V. Chibante Barroso 33, D.D. Chinellato 111,
E.S. Chizzali I,95, J. Cho 58, S. Cho 58, P. Chochula 33, D. Choudhury42, P.
Christakoglou 84, C.H. Christensen 83, P. Christiansen 75, T. Chujo 124, M.
Ciacco 30, C. Cicalo 52, F. Cindolo 51, M.R. Ciupek97, G. ClaiII,51, F.
Colamaria 50, J.S. Colburn100, D. Colella 96,32, M. Colocci 26, M. Concas
III,33, G. Conesa Balbastre 73, Z. Conesa del Valle 130, G. Contin 24, J.G.
Contreras 36, M.L. Coquet 129, P. Cortese 132,56, M.R. Cosentino 112, F. Costa
33, S. Costanza 22,55, C. Cot 130, J. Crkovská 94, P. Crochet 126, R. Cruz-
Torres 74, P. Cui 6, A. Dainese 54, M.C. Danisch 94, A. Danu 63, P. Das 80, P.
Das 4, S. Das 4, A.R. Dash 137, S. Dash 47, R.M.H. David45, A. De Caro 29, G.
de Cataldo 50, J. de Cuveland39, A. De Falco 23, D. De Gruttola 29, N. De
Marco 56, C. De Martin 24, S. De Pasquale 29, R. Deb 133, R. Del Grande 95, L.
Dello Stritto 29, W. Deng 6, P. Dhankher 19, D. Di Bari 32, A. Di Mauro 33, B.
Diab 129, R.A. Diaz 143,7, T. Dietel 113, Y. Ding 6, J. Ditzel 64, R. Divià
33, D.U. Dixit 19, Ø. Djuvsland21, U. Dmitrieva 142, A. Dobrin 63, B. Dönigus
64, J.M. Dubinski 135, A. Dubla 97, S. Dudi 90, P. Dupieux 126, M. Durkac106,
N. Dzalaiova13, T.M. Eder 137, R.J. Ehlers 74, F. Eisenhut 64, R. Ejima92, D.
Elia 50, B. Erazmus 103, F. Ercolessi 26, B. Espagnon 130, G. Eulisse 33, D.
Evans 100, S. Evdokimov 142, L. Fabbietti 95, M. Faggin 28, J. Faivre 73, F.
Fan 6, W. Fan 74, A. Fantoni 49, M. Fasel 87, A. Feliciello 56, G. Feofilov
142, A. Fernández Téllez 45, L. Ferrandi 110, M.B. Ferrer 33, A. Ferrero 129,
C. Ferrero 56, A. Ferretti 25, V.J.G. Feuillard 94, V. Filova 36, D. Finogeev
142, F.M. Fionda 52, E. Flatland33, F. Flor 115, A.N. Flores 108, S. Foertsch
68, I. Fokin 94, S. Fokin 142, E. Fragiacomo 57, E. Frajna 138, U. Fuchs 33,
N. Funicello 29, C. Furget 73, A. Furs 142, T. Fusayasu 98, J.J. Gaardhøje 83,
M. Gagliardi 25, A.M. Gago 101, T. Gahlaut47, C.D. Galvan 109, D.R.
Gangadharan 115, P. Ganoti 78, C. Garabatos 97, A.T. Garcia 130, J.R.A. Garcia
45, E. Garcia-Solis 9, C. Gargiulo 33, P. Gasik 97, A. Gautam 117, M.B. Gay
Ducati 66, M. Germain 103, A. Ghimouz124, C. Ghosh134, M. Giacalone 51, G.
Gioachin 30, P. Giubellino 97,56, P. Giubilato 28, A.M.C. Glaenzer 129, P.
Glässel 94, E. Glimos 121, D.J.Q. Goh76, V. Gonzalez 136, P. Gordeev142, M.
Gorgon 2, K. Goswami 48, S. Gotovac34, V. Grabski 67, L.K. Graczykowski 135,
E. Grecka 86, A. Grelli 59, C. Grigoras 33, V. Grigoriev 142, S. Grigoryan
143,1, F. Grosa 33, J.F. Grosse-Oetringhaus 33, R. Grosso 97, D. Grund 36,
N.A. Grunwald94, G.G. Guardiano 111, R. Guernane 73, M. Guilbaud 103, K.
Gulbrandsen 83, T. Gündem 64, T. Gunji 123, W. Guo 6, A. Gupta 91, R. Gupta
91, R. Gupta 48, S.P. Guzman 45, K. Gwizdziel 135, L. Gyulai 138, C.
Hadjidakis 130, F.U. Haider 91, S. Haidlova 36, H. Hamagaki 76, A. Hamdi 74,
Y. Han 140, B.G. Hanley 136, R. Hannigan 108, J. Hansen 75, M.R. Haque 135,
J.W. Harris 139, A. Harton 9, H. Hassan 116, D. Hatzifotiadou 51, P. Hauer 43,
L.B. Havener 139, S.T. Heckel 95, E. Hellbär 97, H. Helstrup 35, M. Hemmer 64,
T. Herman 36, G. Herrera Corral 8, F. Herrmann137, S. Herrmann 127, K.F.
Hetland 35, B. Heybeck 64, H. Hillemanns 33, B. Hippolyte 128, F.W. Hoffmann
70, B. Hofman 59, G.H. Hong 140, M. Horst 95, A. Horzyk2, Y. Hou 6, P. Hristov
33, C. Hughes 121, P. Huhn64, L.M. Huhta 116, T.J. Humanic 88, A. Hutson 115,
D. Hutter 39, R. Ilkaev142, H. Ilyas 14, M. Inaba 124, G.M. Innocenti 33, M.
Ippolitov 142, A. Isakov 84,86, T. Isidori 117, M.S. Islam 99, M. Ivanov 97,
M. Ivanov13, V. Ivanov 142, K.E. Iversen 75, M. Jablonski 2, B. Jacak 74, N.
Jacazio 26, P.M. Jacobs 74, S. Jadlovska106, J. Jadlovsky106, S. Jaelani 82,
C. Jahnke 110, M.J. Jakubowska 135, M.A. Janik 135, T. Janson70, S. Ji 17, S.
Jia 10, A.A.P. Jimenez 65, F. Jonas 87, D.M. Jones 118, J.M. Jowett 33,97, J.
Jung 64, M. Jung 64, A. Junique 33, A. Jusko 100, M.J. Kabus 33,135, J.
Kaewjai105, P. Kalinak 60, A.S. Kalteyer 97, A. Kalweit 33, V. Kaplin 142, A.
Karasu Uysal 72, D. Karatovic 89, O. Karavichev 142, T. Karavicheva 142, P.
Karczmarczyk 135, E. Karpechev 142, U. Kebschull 70, R. Keidel 141, D.L.D.
Keijdener59, M. Keil 33, B. Ketzer 43, S.S. Khade 48, A.M. Khan 119, S. Khan
16, A. Khanzadeev 142, Y. Kharlov 142, A. Khatun 117, A. Khuntia 36, B. Kileng
35, B. Kim 104, C. Kim 17, D.J. Kim 116, E.J. Kim 69, J. Kim 140, J.S. Kim 41,
J. Kim 58, J. Kim 69, M. Kim 19, S. Kim 18, T. Kim 140, K. Kimura 92, S.
Kirsch 64, I. Kisel 39, S. Kiselev 142, A. Kisiel 135, J.P. Kitowski 2, J.L.
Klay 5, J. Klein 33, S. Klein 74, C. Klein-Bösing 137, M. Kleiner 64, T.
Klemenz 95, A. Kluge 33, A.G. Knospe 115, C. Kobdaj 105, T. Kollegger97, A.
Kondratyev 143, N. Kondratyeva 142, E. Kondratyuk 142, J. Konig 64, S.A.
Konigstorfer 95, P.J. Konopka 33, G. Kornakov 135, S.D. Koryciak 2, A.
Kotliarov 86, V. Kovalenko 142, M. Kowalski 107, V. Kozhuharov 37, I. Králik
60, A. Kravčáková 38, L. Krcal 33,39, M. Krivda 100,60, F. Krizek 86, K.
Krizkova Gajdosova 33, M. Kroesen 94, M. Krüger 64, D.M. Krupova 36, E.
Kryshen 142, V. Kučera 58, C. Kuhn 128, P.G. Kuijer 84, T. Kumaoka124, D.
Kumar134, L. Kumar 90, N. Kumar90, S. Kumar 32, S. Kundu 33, P. Kurashvili 79,
A. Kurepin 142, A.B. Kurepin 142, A. Kuryakin 142, S. Kushpil 86, V.
Kuskov142, M.J. Kweon 58, Y. Kwon 140, S.L. La Pointe 39, P. La Rocca 27, A.
Lakrathok105, M. Lamanna 33, R. Langoy 120, P. Larionov 33, E. Laudi 33, L.
Lautner 33,95, R. Lavicka 102, R. Lea 133,55, H. Lee 104, I. Legrand 46, G.
Legras 137, J. Lehrbach 39, T.M. Lelek2, R.C. Lemmon 85, I. León Monzón 109,
M.M. Lesch 95, E.D. Lesser 19, P. Lévai 138, X. Li10, J. Lien 120, R. Lietava
100, I. Likmeta 115, B. Lim 25, S.H. Lim 17, V. Lindenstruth 39, A. Lindner46,
C. Lippmann 97, D.H. Liu 6, J. Liu 118, G.S.S. Liveraro 111, I.M. Lofnes 21,
C. Loizides 87, S. Lokos 107, J. Lomker 59, P. Loncar 34, X. Lopez 126, E.
López Torres 7, P. Lu 97,119, F.V. Lugo 67, J.R. Luhder 137, M. Lunardon 28,
G. Luparello 57, Y.G. Ma 40, M. Mager 33, A. Maire 128, M.V. Makariev 37, M.
Malaev 142, G. Malfattore 26, N.M. Malik 91, Q.W. Malik20, S.K. Malik 91, L.
Malinina VI,143, D. Mallick 130,80, N. Mallick 48, G. Mandaglio 31,53, S.K.
Mandal 79, V. Manko 142, F. Manso 126, V. Manzari 50, Y. Mao 6, R.W. Marcjan
2, G.V. Margagliotti 24, A. Margotti 51, A. Marín 97, C. Markert 108, P.
Martinengo 33, M.I. Martínez 45, G. Martínez García 103, M.P.P. Martins 110,
S. Masciocchi 97, M. Masera 25, A. Masoni 52, L. Massacrier 130, O. Massen 59,
A. Mastroserio 131,50, O. Matonoha 75, S. Mattiazzo 28, A. Matyja 107, C.
Mayer 107, A.L. Mazuecos 33, F. Mazzaschi 25, M. Mazzilli 33, J.E. Mdhluli
122, Y. Melikyan 44, A. Menchaca-Rocha 67, J.E.M. Mendez 65, E. Meninno
102,29, A.S. Menon 115, M. Meres 13, S. Mhlanga113,68, Y. Miake124, L.
Micheletti 33, D.L. Mihaylov 95, K. Mikhaylov 143,142, A.N. Mishra 138, D.
Miśkowiec 97, A. Modak 4, B. Mohanty80, M. Mohisin KhanIV,16, M.A. Molander
44, S. Monira 135, C. Mordasini 116, D.A. Moreira De Godoy 137, I. Morozov
142, A. Morsch 33, T. Mrnjavac 33, V. Muccifora 49, S. Muhuri 134, J.D.
Mulligan 74, A. Mulliri23, M.G. Munhoz 110, R.H. Munzer 64, H. Murakami 123,
S. Murray 113, L. Musa 33, J. Musinsky 60, J.W. Myrcha 135, B. Naik 122, A.I.
Nambrath 19, B.K. Nandi 47, R. Nania 51, E. Nappi 50, A.F. Nassirpour 18, A.
Nath 94, C. Nattrass 121, M.N. Naydenov 37, A. Neagu20, A. Negru125, E.
Nekrasova142, L. Nellen 65, R. Nepeivoda 75, S. Nese 20, G. Neskovic 39, N.
Nicassio 50, B.S. Nielsen 83, E.G. Nielsen 83, S. Nikolaev 142, S. Nikulin
142, V. Nikulin 142, F. Noferini 51, S. Noh 12, P. Nomokonov 143, J. Norman
118, N. Novitzky 87, P. Nowakowski 135, A. Nyanin 142, J. Nystrand 21, M.
Ogino 76, S. Oh 18, A. Ohlson 75, V.A. Okorokov 142, J. Oleniacz 135, A.C.
Oliveira Da Silva 121, A. Onnerstad 116, C. Oppedisano 56, A. Ortiz Velasquez
65, J. Otwinowski 107, M. Oya92, K. Oyama 76, Y. Pachmayer 94, S. Padhan 47,
D. Pagano 133,55, G. Paić 65, A. Palasciano 50, S. Panebianco 129, H. Park
124, H. Park 104, J. Park 58, J.E. Parkkila 33, Y. Patley 47, R.N. Patra91, B.
Paul 23, H. Pei 6, T. Peitzmann 59, X. Peng 11, M. Pennisi 25, S. Perciballi
25, D. Peresunko 142, G.M. Perez 7, Y. Pestov142, V. Petrov 142, M. Petrovici
46, R.P. Pezzi 103,66, S. Piano 57, M. Pikna 13, P. Pillot 103, O. Pinazza
51,33, L. Pinsky115, C. Pinto 95, S. Pisano 49, M. Płoskoń 74, M. Planinic89,
F. Pliquett64, M.G. Poghosyan 87, B. Polichtchouk 142, S. Politano 30, N.
Poljak 89, A. Pop 46, S. Porteboeuf-Houssais 126, V. Pozdniakov 143, I.Y.
Pozos 45, K.K. Pradhan 48, S.K. Prasad 4, S. Prasad 48, R. Preghenella 51, F.
Prino 56, C.A. Pruneau 136, I. Pshenichnov 142, M. Puccio 33, S. Pucillo 25,
Z. Pugelova106, S. Qiu 84, L. Quaglia 25, S. Ragoni 15, A. Rai 139, A.
Rakotozafindrabe 129, L. Ramello 132,56, F. Rami 128, S.A.R. Ramirez 45, T.A.
Rancien73, M. Rasa 27, S.S. Räsänen 44, R. Rath 51, M.P. Rauch 21, I.
Ravasenga 84, K.F. Read 87,121, C. Reckziegel 112, A.R. Redelbach 39, K.
Redlich V,79, C.A. Reetz 97, A. Rehman21, F. Reidt 33, H.A. Reme-Ness 35, Z.
Rescakova38, K. Reygers 94, A. Riabov 142, V. Riabov 142, R. Ricci 29, M.
Richter 20, A.A. Riedel 95, W. Riegler 33, A.G. Riffero 25, C. Ristea 63, M.V.
Rodriguez 33, M. Rodríguez Cahuantzi 45, K. Røed 20, R. Rogalev 142, E.
Rogochaya 143, T.S. Rogoschinski 64, D. Rohr 33, D. Röhrich 21, P.F. Rojas45,
S. Rojas Torres 36, P.S. Rokita 135, G. Romanenko 26, F. Ronchetti 49, A.
Rosano 31,53, E.D. Rosas65, K. Roslon 135, A. Rossi 54, A. Roy 48, S. Roy 47,
N. Rubini 26, D. Ruggiano 135, R. Rui 24, P.G. Russek 2, R. Russo 84, A.
Rustamov 81, E. Ryabinkin 142, Y. Ryabov 142, A. Rybicki 107, H. Rytkonen 116,
J. Ryu 17, W. Rzesa 135, O.A.M. Saarimaki 44, S. Sadhu 32, S. Sadovsky 142, J.
Saetre 21, K. Šafařík 36, P. Saha42, S.K. Saha 4, S. Saha 80, B. Sahoo 47, B.
Sahoo 48, R. Sahoo 48, S. Sahoo61, D. Sahu 48, P.K. Sahu 61, J. Saini 134, K.
Sajdakova38, S. Sakai 124, M.P. Salvan 97, S. Sambyal 91, D. Samitz 102, I.
Sanna 33,95, T.B. Saramela110, P. Sarma 42, V. Sarritzu 23, V.M. Sarti 95,
M.H.P. Sas 33, S. Sawan80, J. Schambach 87, H.S. Scheid 64, C. Schiaua 46, R.
Schicker 94, F. Schlepper 94, A. Schmah97, C. Schmidt 97, H.R. Schmidt93, M.O.
Schmidt 33, M. Schmidt93, N.V. Schmidt 87, A.R. Schmier 121, R. Schotter 128,
A. Schröter 39, J. Schukraft 33, K. Schweda 97, G. Scioli 26, E. Scomparin 56,
J.E. Seger 15, Y. Sekiguchi123, D. Sekihata 123, M. Selina 84, I. Selyuzhenkov
97, S. Senyukov 128, J.J. Seo 94,58, D. Serebryakov 142, L. Šerkšnytė 95, A.
Sevcenco 63, T.J. Shaba 68, A. Shabetai 103, R. Shahoyan33, A. Shangaraev 142,
A. Sharma90, B. Sharma 91, D. Sharma 47, H. Sharma 54, M. Sharma 91, S. Sharma
76, S. Sharma 91, U. Sharma 91, A. Shatat 130, O. Sheibani115, K. Shigaki 92,
M. Shimomura77, J. Shin12, S. Shirinkin 142, Q. Shou 40, Y. Sibiriak 142, S.
Siddhanta 52, T. Siemiarczuk 79, T.F. Silva 110, D. Silvermyr 75, T.
Simantathammakul105, R. Simeonov 37, B. Singh91, B. Singh 95, K. Singh 48, R.
Singh 80, R. Singh 91, R. Singh 48, S. Singh 16, V.K. Singh 134, V. Singhal
134, T. Sinha 99, B. Sitar 13, M. Sitta 132,56, T.B. Skaali20, G. Skorodumovs
94, M. Slupecki 44, N. Smirnov 139, R.J.M. Snellings 59, E.H. Solheim 20, J.
Song 17, C. Sonnabend 33,97, F. Soramel 28, A.B. Soto-hernandez 88, R.
Spijkers 84, I. Sputowska 107, J. Staa 75, J. Stachel 94, I. Stan 63, P.J.
Steffanic 121, S.F. Stiefelmaier 94, D. Stocco 103, I. Storehaug 20, P.
Stratmann 137, S. Strazzi 26, A. Sturniolo 31,53, C.P. Stylianidis84, A.A.P.
Suaide 110, C. Suire 130, M. Sukhanov 142, M. Suljic 33, R. Sultanov 142, V.
Sumberia 91, S. Sumowidagdo 82, S. Swain61, I. Szarka 13, M. Szymkowski 135,
S.F. Taghavi 95, G. Taillepied 97, J. Takahashi 111, G.J. Tambave 80, S. Tang
6, Z. Tang 119, J.D. Tapia Takaki 117, N. Tapus125, L.A. Tarasovicova 137,
M.G. Tarzila 46, G.F. Tassielli 32, A. Tauro 33, G. Tejeda Muñoz 45, A.
Telesca 33, L. Terlizzi 25, C. Terrevoli 115, S. Thakur 4, D. Thomas 108, A.
Tikhonov 142, N. Tiltmann 33,137, A.R. Timmins 115, M. Tkacik106, T. Tkacik
106, A. Toia 64, R. Tokumoto92, K. Tomohiro92, N. Topilskaya 142, M. Toppi 49,
T. Tork 130, P.V. Torres65, V.V. Torres 103, A.G. Torres Ramos 32, A. Trifiró
31,53, A.S. Triolo 33,31,53, S. Tripathy 51, T. Tripathy 47, S. Trogolo 33, V.
Trubnikov 3, W.H. Trzaska 116, T.P. Trzcinski 135, A. Tumkin 142, R. Turrisi
54, T.S. Tveter 20, K. Ullaland 21, B. Ulukutlu 95, A. Uras 127, G.L. Usai 23,
M. Vala38, N. Valle 22, L.V.R. van Doremalen59, M. van Leeuwen 84, C.A. van
Veen 94, R.J.G. van Weelden 84, P. Vande Vyvre 33, D. Varga 138, Z. Varga 138,
M. Vasileiou 78, A. Vasiliev 142, O. Vázquez Doce 49, O. Vazquez Rueda 115, V.
Vechernin 142, E. Vercellin 25, S. Vergara Limón45, R. Verma47, L. Vermunt 97,
R. Vértesi 138, M. Verweij 59, L. Vickovic34, Z. Vilakazi122, O. Villalobos
Baillie 100, A. Villani 24, A. Vinogradov 142, T. Virgili 29, M.M.O. Virta
116, V. Vislavicius75, A. Vodopyanov 143, B. Volkel 33, M.A. Völkl 94, K.
Voloshin142, S.A. Voloshin 136, G. Volpe 32, B. von Haller 33, I. Vorobyev 95,
N. Vozniuk 142, J. Vrláková 38, J. Wan40, C. Wang 40, D. Wang40, Y. Wang 40,
Y. Wang 6, A. Wegrzynek 33, F.T. Weiglhofer39, S.C. Wenzel 33, J.P. Wessels
137, S.L. Weyhmiller 139, J. Wiechula 64, J. Wikne 20, G. Wilk 79, J.
Wilkinson 97, G.A. Willems 137, B. Windelband 94, M. Winn 129, J.R. Wright
108, W. Wu40, Y. Wu 119, R. Xu 6, A. Yadav 43, A.K. Yadav 134, S. Yalcin 72,
Y. Yamaguchi 92, S. Yang21, S. Yano 92, Z. Yin 6, I.-K. Yoo 17, J.H. Yoon 58,
H. Yu12, S. Yuan21, A. Yuncu 94, V. Zaccolo 24, C. Zampolli 33, F. Zanone 94,
N. Zardoshti 33, A. Zarochentsev 142, P. Závada 62, N. Zaviyalov142, M. Zhalov
142, B. Zhang 6, C. Zhang 129, L. Zhang 40, S. Zhang 40, X. Zhang 6, Y.
Zhang119, Z. Zhang 6, M. Zhao 10, V. Zherebchevskii 142, Y. Zhi10, D. Zhou 6,
Y. Zhou 83, J. Zhu 97,6, Y. Zhu6, S.C. Zugravel 56, N. Zurlo 133,55

## Affiliation Notes

I Also at: Max-Planck-Institut für Physik, Munich, Germany
II Also at: Italian National Agency for New Technologies, Energy and
Sustainable Economic Development (ENEA), Bologna, Italy
III Also at: Dipartimento DET del Politecnico di Torino, Turin, Italy
IV Also at: Department of Applied Physics, Aligarh Muslim University, Aligarh,
India
V Also at: Institute of Theoretical Physics, University of Wroclaw, Poland
VI Also at: An institution covered by a cooperation agreement with CERN

## Collaboration Institutes

1 A.I. Alikhanyan National Science Laboratory (Yerevan Physics Institute)
Foundation, Yerevan, Armenia
2 AGH University of Krakow, Cracow, Poland
3 Bogolyubov Institute for Theoretical Physics, National Academy of Sciences
of Ukraine, Kiev, Ukraine
4 Bose Institute, Department of Physics and Centre for Astroparticle Physics
and Space Science (CAPSS), Kolkata, India
5 California Polytechnic State University, San Luis Obispo, California, United
States
6 Central China Normal University, Wuhan, China
7 Centro de Aplicaciones Tecnológicas y Desarrollo Nuclear (CEADEN), Havana,
Cuba
8 Centro de Investigación y de Estudios Avanzados (CINVESTAV), Mexico City and
Mérida, Mexico
9 Chicago State University, Chicago, Illinois, United States
10 China Institute of Atomic Energy, Beijing, China
11 China University of Geosciences, Wuhan, China
12 Chungbuk National University, Cheongju, Republic of Korea
13 Comenius University Bratislava, Faculty of Mathematics, Physics and
Informatics, Bratislava, Slovak Republic
14 COMSATS University Islamabad, Islamabad, Pakistan
15 Creighton University, Omaha, Nebraska, United States
16 Department of Physics, Aligarh Muslim University, Aligarh, India
17 Department of Physics, Pusan National University, Pusan, Republic of Korea
18 Department of Physics, Sejong University, Seoul, Republic of Korea
19 Department of Physics, University of California, Berkeley, California,
United States
20 Department of Physics, University of Oslo, Oslo, Norway
21 Department of Physics and Technology, University of Bergen, Bergen, Norway
22 Dipartimento di Fisica, Università di Pavia, Pavia, Italy
23 Dipartimento di Fisica dell’Università and Sezione INFN, Cagliari, Italy
24 Dipartimento di Fisica dell’Università and Sezione INFN, Trieste, Italy
25 Dipartimento di Fisica dell’Università and Sezione INFN, Turin, Italy
26 Dipartimento di Fisica e Astronomia dell’Università and Sezione INFN,
Bologna, Italy
27 Dipartimento di Fisica e Astronomia dell’Università and Sezione INFN,
Catania, Italy
28 Dipartimento di Fisica e Astronomia dell’Università and Sezione INFN,
Padova, Italy
29 Dipartimento di Fisica ‘E.R. Caianiello’ dell’Università and Gruppo
Collegato INFN, Salerno, Italy
30 Dipartimento DISAT del Politecnico and Sezione INFN, Turin, Italy
31 Dipartimento di Scienze MIFT, Università di Messina, Messina, Italy
32 Dipartimento Interateneo di Fisica ‘M. Merlin’ and Sezione INFN, Bari,
Italy
33 European Organization for Nuclear Research (CERN), Geneva, Switzerland
34 Faculty of Electrical Engineering, Mechanical Engineering and Naval
Architecture, University of Split, Split, Croatia
35 Faculty of Engineering and Science, Western Norway University of Applied
Sciences, Bergen, Norway
36 Faculty of Nuclear Sciences and Physical Engineering, Czech Technical
University in Prague, Prague, Czech Republic
37 Faculty of Physics, Sofia University, Sofia, Bulgaria
38 Faculty of Science, P.J. Šafárik University, Košice, Slovak Republic
39 Frankfurt Institute for Advanced Studies, Johann Wolfgang Goethe-
Universität Frankfurt, Frankfurt, Germany
40 Fudan University, Shanghai, China
41 Gangneung-Wonju National University, Gangneung, Republic of Korea
42 Gauhati University, Department of Physics, Guwahati, India
43 Helmholtz-Institut für Strahlen- und Kernphysik, Rheinische Friedrich-
Wilhelms-Universität Bonn, Bonn, Germany
44 Helsinki Institute of Physics (HIP), Helsinki, Finland
45 High Energy Physics Group, Universidad Autónoma de Puebla, Puebla, Mexico
46 Horia Hulubei National Institute of Physics and Nuclear Engineering,
Bucharest, Romania
47 Indian Institute of Technology Bombay (IIT), Mumbai, India
48 Indian Institute of Technology Indore, Indore, India
49 INFN, Laboratori Nazionali di Frascati, Frascati, Italy
50 INFN, Sezione di Bari, Bari, Italy
51 INFN, Sezione di Bologna, Bologna, Italy
52 INFN, Sezione di Cagliari, Cagliari, Italy
53 INFN, Sezione di Catania, Catania, Italy
54 INFN, Sezione di Padova, Padova, Italy
55 INFN, Sezione di Pavia, Pavia, Italy
56 INFN, Sezione di Torino, Turin, Italy
57 INFN, Sezione di Trieste, Trieste, Italy
58 Inha University, Incheon, Republic of Korea
59 Institute for Gravitational and Subatomic Physics (GRASP), Utrecht
University/Nikhef, Utrecht, Netherlands
60 Institute of Experimental Physics, Slovak Academy of Sciences, Košice,
Slovak Republic
61 Institute of Physics, Homi Bhabha National Institute, Bhubaneswar, India
62 Institute of Physics of the Czech Academy of Sciences, Prague, Czech
Republic
63 Institute of Space Science (ISS), Bucharest, Romania
64 Institut für Kernphysik, Johann Wolfgang Goethe-Universität Frankfurt,
Frankfurt, Germany
65 Instituto de Ciencias Nucleares, Universidad Nacional Autónoma de México,
Mexico City, Mexico
66 Instituto de Física, Universidade Federal do Rio Grande do Sul (UFRGS),
Porto Alegre, Brazil
67 Instituto de Física, Universidad Nacional Autónoma de México, Mexico City,
Mexico
68 iThemba LABS, National Research Foundation, Somerset West, South Africa
69 Jeonbuk National University, Jeonju, Republic of Korea
70 Johann-Wolfgang-Goethe Universität Frankfurt Institut für Informatik,
Fachbereich Informatik und Mathematik, Frankfurt, Germany
71 Korea Institute of Science and Technology Information, Daejeon, Republic of
Korea
72 KTO Karatay University, Konya, Turkey
73 Laboratoire de Physique Subatomique et de Cosmologie, Université Grenoble-
Alpes, CNRS-IN2P3, Grenoble, France
74 Lawrence Berkeley National Laboratory, Berkeley, California, United States
75 Lund University Department of Physics, Division of Particle Physics, Lund,
Sweden
76 Nagasaki Institute of Applied Science, Nagasaki, Japan
77 Nara Women’s University (NWU), Nara, Japan
78 National and Kapodistrian University of Athens, School of Science,
Department of Physics , Athens, Greece
79 National Centre for Nuclear Research, Warsaw, Poland
80 National Institute of Science Education and Research, Homi Bhabha National
Institute, Jatni, India
81 National Nuclear Research Center, Baku, Azerbaijan
82 National Research and Innovation Agency - BRIN, Jakarta, Indonesia
83 Niels Bohr Institute, University of Copenhagen, Copenhagen, Denmark
84 Nikhef, National institute for subatomic physics, Amsterdam, Netherlands
85 Nuclear Physics Group, STFC Daresbury Laboratory, Daresbury, United Kingdom
86 Nuclear Physics Institute of the Czech Academy of Sciences, Husinec-Řež,
Czech Republic
87 Oak Ridge National Laboratory, Oak Ridge, Tennessee, United States
88 Ohio State University, Columbus, Ohio, United States
89 Physics department, Faculty of science, University of Zagreb, Zagreb,
Croatia
90 Physics Department, Panjab University, Chandigarh, India
91 Physics Department, University of Jammu, Jammu, India
92 Physics Program and International Institute for Sustainability with Knotted
Chiral Meta Matter (SKCM2), Hiroshima University, Hiroshima, Japan
93 Physikalisches Institut, Eberhard-Karls-Universität Tübingen, Tübingen,
Germany
94 Physikalisches Institut, Ruprecht-Karls-Universität Heidelberg, Heidelberg,
Germany
95 Physik Department, Technische Universität München, Munich, Germany
96 Politecnico di Bari and Sezione INFN, Bari, Italy
97 Research Division and ExtreMe Matter Institute EMMI, GSI Helmholtzzentrum
für Schwerionenforschung GmbH, Darmstadt, Germany
98 Saga University, Saga, Japan
99 Saha Institute of Nuclear Physics, Homi Bhabha National Institute, Kolkata,
India
100 School of Physics and Astronomy, University of Birmingham, Birmingham,
United Kingdom
101 Sección Física, Departamento de Ciencias, Pontificia Universidad Católica
del Perú, Lima, Peru
102 Stefan Meyer Institut für Subatomare Physik (SMI), Vienna, Austria
103 SUBATECH, IMT Atlantique, Nantes Université, CNRS-IN2P3, Nantes, France
104 Sungkyunkwan University, Suwon City, Republic of Korea
105 Suranaree University of Technology, Nakhon Ratchasima, Thailand
106 Technical University of Košice, Košice, Slovak Republic
107 The Henryk Niewodniczanski Institute of Nuclear Physics, Polish Academy of
Sciences, Cracow, Poland
108 The University of Texas at Austin, Austin, Texas, United States
109 Universidad Autónoma de Sinaloa, Culiacán, Mexico
110 Universidade de São Paulo (USP), São Paulo, Brazil
111 Universidade Estadual de Campinas (UNICAMP), Campinas, Brazil
112 Universidade Federal do ABC, Santo Andre, Brazil
113 University of Cape Town, Cape Town, South Africa
114 University of Derby, Derby, United Kingdom
115 University of Houston, Houston, Texas, United States
116 University of Jyväskylä, Jyväskylä, Finland
117 University of Kansas, Lawrence, Kansas, United States
118 University of Liverpool, Liverpool, United Kingdom
119 University of Science and Technology of China, Hefei, China
120 University of South-Eastern Norway, Kongsberg, Norway
121 University of Tennessee, Knoxville, Tennessee, United States
122 University of the Witwatersrand, Johannesburg, South Africa
123 University of Tokyo, Tokyo, Japan
124 University of Tsukuba, Tsukuba, Japan
125 University Politehnica of Bucharest, Bucharest, Romania
126 Université Clermont Auvergne, CNRS/IN2P3, LPC, Clermont-Ferrand, France
127 Université de Lyon, CNRS/IN2P3, Institut de Physique des 2 Infinis de
Lyon, Lyon, France
128 Université de Strasbourg, CNRS, IPHC UMR 7178, F-67000 Strasbourg, France,
Strasbourg, France
129 Université Paris-Saclay, Centre d’Etudes de Saclay (CEA), IRFU, Départment
de Physique Nucléaire (DPhN), Saclay, France
130 Université Paris-Saclay, CNRS/IN2P3, IJCLab, Orsay, France
131 Università degli Studi di Foggia, Foggia, Italy
132 Università del Piemonte Orientale, Vercelli, Italy
133 Università di Brescia, Brescia, Italy
134 Variable Energy Cyclotron Centre, Homi Bhabha National Institute, Kolkata,
India
135 Warsaw University of Technology, Warsaw, Poland
136 Wayne State University, Detroit, Michigan, United States
137 Westfälische Wilhelms-Universität Münster, Institut für Kernphysik,
Münster, Germany
138 Wigner Research Centre for Physics, Budapest, Hungary
139 Yale University, New Haven, Connecticut, United States
140 Yonsei University, Seoul, Republic of Korea
141 Zentrum für Technologie und Transfer (ZTT), Worms, Germany
142 Affiliated with an institute covered by a cooperation agreement with CERN
143 Affiliated with an international laboratory covered by a cooperation
agreement with CERN.


# The Stability of Core-Envelope Models for the Crab and the Vela pulsars

P. S. Negi  Department of Physics, Kumaun University, Nainital 263 002, India
<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

The core-envelope analytic models of Negi et al. (1989) have been found to be
consistent with both the mechanisms of glitch generation in pulsars, namely -
(i) the starquake model and (ii) the vortex unpinning model. In Negi (2019),
the author has been able to reproduce the observed values of glitch healing
parameter, $G_{h}(=I_{\rm core}/I_{\rm total};G_{h}$ represents the fractional
moment of inertia of the core component in the starquake mechanism of glitch
generation) for the Crab as well as for the Vela pulsar. In another study, by
using the models discussed in Negi et al. (1989), the author has obtained the
minimum value of fractional moment of inertia of the crust about 7.4% and
larger for all the values of masses in the range - $1M_{\odot}-1.96M_{\odot}$
considered for the Vela pulsar (Negi 2020a). The latter study of the author is
found to be consistent with the recent requirement (on the basis of vortex
unpinning model of the glitch generation) which refers: The minimum fractional
crustal moment of inertia of the Vela pulsar should be about 7% for a mass
higher than about 1$M_{\odot}$. However, an important study which requires
investigation of pulsational stability and gravitational binding of the models
of Negi et al. (1989) has not been carried out so far. The present paper deals
with such a study of the models (Negi et al. 1989) for all permissible values
of the compactness parameter $u(\equiv M/a$, total mass to size ratio in
geometrized units) and compressibility factor $Q$ (defined in Tolman’s VII
solution as: $x=r^{2}/K^{2}=r^{2}/a^{2}Q)$. It is seen that the configurations
remain pulsationally stable and gravitationally bound for all permissible
values of $u(\leq 0.25)$ and $Q$ ($0<Q\leq 1.2$).

Static Spherical Structures: Analytic Solutions – Neutron Stars: Pulsars -
Crab - Vela: Gravitational Binding, Pulsational Stability

##  1 Introduction

Ever since the pioneering work of Oppenheimer and Volkoff (1939), the study
regarding internal structure of Neutron Stars (NSs) has been a topic of vast
astrophysical interest. Various models based on Equations of State (EOSs),
derived from the knowledge of minimum physical constraints, are available in
the literature (Rhoades and Ruffini 1974 , Brecher and Caporaso 1976, Hartle
1978). Since the details of the nuclear interactions are still not well known
beyond the density of $\sim 10^{14}\rm{gcm}^{-3}$, the NS models based on
extrapolated EOSs around nuclear saturation density $E_{n}\sim
2.7\times\rm{gcm}^{-3}$ and higher are widely discussed in the literature
(Wiringa et al. 1988, Kalogera and Baym 1996, Haensel et al. 2006; and
references therein). However, the importance and elegance of NS models based
on analytic exact solutions of Einstein’s field equations (Stephani et al.
2003) can’t be denied. Among a variety of solutions for a static and
spherically symmetric mass distribution, Tolman’s VII solution is found to be
the most realistic one (Durgapal et al. 1984, Lattimer and Prakash 2005). Negi
and Durgapal (1999, 2001) have shown that the Tolman’s VII solution with
vanishing surface density remains stable and gravitationally bound for
$u>(1/3)$ and may be used to model NSs and pulsars. Later on, Moustakidis
(2017) has verified the claim of Negi and Durgapal (1999, 2001) regarding the
stability of Tolman’s VII solution with vanishing surface density.

Study of the phenomena of sudden spin-up rates in many pulsars (known as a
glitch) is found to be a powerful tool for understanding the internal
structure of NSs. The most studied pulsars in this regard are the Crab and the
Vela pulsars. At present, it is believed that the starquake mechanism of
glitch generation (Baym and Pines 1971, Ruderman 1976, Alpar et al. 1996) is
responsible for the Crab glitches (small size glitches, $\Delta\nu/\nu\sim
10^{-9}$) and another mechanism, known as vortex unpinning model (Anderson and
Itoh 1975, Alpar et al. 1993) is appropriate for the glitch generation in the
Vela pulsar (large size glitches, $\Delta\nu/\nu\sim 10^{-6}$).

The study of Negi et al. (1989) deals with the construction of core-envelope
models based on the exact solutions of Einstein’s field equations for static
and spherical mass distribution. The core of the models is described by
Tolman’s VII solution, matched smoothly at the core-envelope boundary. The
region of envelope is described by Tolman’s VI solution which is finally
matched to vacuum Schwarzschild solution at the surface. The core-envelope
boundary of the models is assured by matching of all the four variables -
pressure ($P$), energy density ($E$) and both of the metric parameters $\nu$
and $\lambda$ without recourse to any computational method. The complete
solutions for both the regions (the core and the envelope) are available in
Negi et al. (1989). Recently, the author has shown that the models of Negi et
al. (1989) can reproduce the observed values of glitch healing parameter for
both the pulsars (that is, the Crab and the Vela pulsars) on the basis of the
starquake model of glitch generation (Negi 2019). This study of the author
(Negi 2019) predicts the mass of the Crab pulsar in the range :
1.79$M_{\odot}-1.88M_{\odot}$ whereas the mass of the Vela pulsar corresponds
to a value $\sim$ 1.96$M_{\odot}$. In another study based on the models of
Negi et al. (1989), the author has been able to construct the models of the
Vela pulsar which can reproduce the minimum fractional crustal moment of
inertia about 7.4% and larger for the mass range - 1$M_{\odot}-1.96M_{\odot}$
considered for the Vela pulsar (Negi 2020a). It is to be pointed out here that
the requirement of minimum fractional crustal moment of inertia about 7% for a
mass significantly higher than about 1$M_{\odot}$ for the Vela pulsar has
recently been imposed due to crustal ‘entrainment’ effect, discussed in the
literature on the basis of the vortex unpinning explanation for the large Vela
glitches (Andersson et al. 2012, Chamel 2013).

Therefore, it follows that the models of Negi et al. (1989) have been able to
explain the observational data on the basis of the starquake as well as those
on the basis of the vortex unpinning models for the Crab and the Vela pulsars
(Negi 2019, Negi 2020a). However, the gravitational binding and pulsational
stability of the models of Negi et al. (1989) have not been discussed so far.
Therefore, it would be interesting and also important to check whether the
models discussed in Negi et al. (1989) posses gravitationally bound structures
and remain stable against small radial pulsations in the state of hydrostatic
equilibrium ? Sec. 3 of the present paper is devoted to such a study. Results
of this study are presented in Sec. 4. Sec. 5 summarizes the main findings of
this study.

##  2 Gravitational Binding and Pulsational Stability of the Structures

We consider the metric corresponding to a static and spherically symmetric
mass distribution in the following form

$ds^{2}=e^{\nu}dt^{2}-e^{\lambda}dr^{2}-r^{2}(d\theta^{2}+{\rm sin}^{2}\theta
d\phi^{2})$ (1)

where $G=c=1$ (that is, we are using geometrized units) and $\nu$ and
$\lambda$ are functions of $r$ alone. The relevant equations governing the
core ($0\leq r\leq b$) and the crust ($b\leq r\leq a$) are described
respectively by Tolman’s VII and Tolman’s VI solutions of the Einstein’s field
equations for the metric (Eq.1) and are available in Negi et al. (1989).
However, some relations relevant to the present study are defined as:

$u\equiv M/a$ is called the compactness parameter which is defined earlier;
where the total mass is given by $M=\int_{0}^{a}4\pi Er^{2}dr$; and $y=r/a$ is
called the radial coordinate measured in units of configuration size.

$Q(\equiv K^{2}/a^{2})$ is known as the compressibility factor which is also
defined earlier; $K$ is a constant appearing in the Tolman’s VII solution. The
matching of various parameters at the core-envelope boundary yields (Negi et
al. 1989)

$b^{2}/a^{2}Q=5/6$, thus $(b/a)$ represents the boundary, $y_{b}$, of the
core-envelope models and $b$ represents the core radius. The analytic relation
between central to boundary (energy) density, $E_{0}/E_{b}$, and central to
surface (energy) density are given by: $(E_{0}/E_{b})=6$;
$(E_{0}/E_{a})=7.2/Q$.

The coefficient of gravitational binding $\alpha$ (that is, the gravitational
binding energy per unit rest-mass) and the ratio of gravitational packing
$\alpha_{p}$ ( that is, the ratio of gravitational energy to the total energy)
can be obtained by using the equations (Zeldovich and Novikov 1978, Negi
2020b)

$\alpha=(M_{r}-M)/M_{r}=[(M_{r}/a)-(M/a)]/(M_{r}/a)$ (2)

$\alpha_{p}=(M_{p}-M)/M=[(M_{p}/a)-(M/a)]/(M/a)$ (3)

where $M/a,M_{r}/a$ and $M_{p}/a$ are given by the relations

$M/a=4\pi\int_{0}^{1}Ea^{2}y^{2}dy$ (4) $M_{r}/a=4\pi\int_{0}^{1}\rho
a^{2}y^{2}e^{\lambda/2}dy$ (5)
$M_{p}/a=4\pi\int_{0}^{1}Ea^{2}y^{2}e^{\lambda/2}dy$ (6)

where $\rho=(P+E)e^{(\nu-\nu_{a})/2}$ is called the rest-mass density
(Durgapal and Pande 1980).The various quantities appearing in eqs. (2) - (6)
may be calculated for various permissible values of $Q$ and $u$ by numerical
integration from the centre of the configuration ($y=0$) to the core-envelope
boundary ($y_{b}=b/a$) by using Tolman’s VII solution and from this boundary
to the surface ($y=1$) by employing Tolman’s VI solution.

A core-envelope model may undergo a gravitational collapse for higher
densities. To check whether the configuration is stable under small radial
pulsations, a radial purturbation is introduced in the equilibrium
configuration which leads to radial oscillations in the structure. The
pulsational stability of the structures under such small radial perturbations
can be judged by using the variational method (Chandrasekhar 1964, Negi 2020b;
and references therein). For a stable configuration the pulsational frequency
is given by

$f=(1/2\pi)(A/B)^{1/2}$ (7)

where the functions $A$ and $B$ represent respectively, the potential energy
and the kinetic energy with velocities replaced by displacements and are given
by111For simplification these expressions are obtained by using the ‘trial
function’ $\xi=re^{\nu/2}$, because this trial function is sufficient to judge
the pulsational stability as obtained by using the trial function of the form
of a power series ( Negi and Durgapal 1999, Negi 2007; and references therein)
$\xi=b_{1}r(1+a_{1}r^{2}+a_{2}r^{4}+a_{3}r^{6})e^{\nu/2}$, where
$a_{1},a_{2},$ and $a_{3}$ are arbitrary constants. Furthermore, the study of
Knutsen (1989) also shows that the use of the trial function of the form of
power series mentioned above (with suitable values of the arbitrary constants
$a_{1},a_{2},$ and $a_{3}$ such that the appropriate boundary conditions may
be satisfied) provide the results similar to those obtained by using the trial
function $\xi=re^{\nu/2}$.

$8\pi B/a^{3}=\int_{0}^{1}(8\pi Pa^{2}+8\pi
Ea^{2})y^{4}e^{(3\lambda+\nu)/2}dy$ (8)

and

$\displaystyle 8\pi A/a$ $\displaystyle=$
$\displaystyle\int_{0}^{1}y^{2}e^{3(\lambda+\nu)/2}\\{e^{-\lambda}[9(8\pi
Pa^{2}+8\pi Ea^{2})(dP/dE)$ (9) $\displaystyle+4(8\pi a^{2}dP/dy)y-\frac{(8\pi
a^{2}dP/dy)^{2}y^{2}}{(8\pi Pa^{2}+8\pi Ea^{2})}]$ $\displaystyle+8\pi
Pa^{2}(8\pi Pa^{2}+8\pi Ea^{2})y^{2}\\}dy$

Eqs. (8) and (9) may be computed by employing a fourth order Runge-Kutta
method from the centre ($y=0$) to the boundary ($y=b/a$) by using Tolman’s VII
solution and from the boundary ($y=b/a$) to the surface ($y=1$) by using
Tolman’s VI solution which yield the values of function $(8\pi B/a^{3})$ and
$(8\pi a/A)$. On dividing values obtained by using eq.(9) by eq.(8), one gets
the value of $a\omega$, where $\omega$ being the angular frequency of
pulsation which follows from eq.(7). On computation, the positive values of
pulsation frequencies would show that the average (constant) value of
adiabatic index, $\gamma_{\rm ave}$, is larger than the minimum (critical)
value of (constant) adiabatic index, $\gamma_{\rm crit}$, required for the
stability of the structures (that is, $\gamma_{\rm ave}\geq\gamma_{\rm
crit}$). Thus, we can safely conclude that the structures are stable under
small radial perturbations. This is to be pointed out here that the use of the
trial function $\xi=re^{\nu/2}$ in the above eqs. (8) and (9) safely assures
the pulsational stability of the models considered in this study, because the
present models correspond to the value of $u<1/3$. For $u\geq 1/3$, the
optimal trial function $\xi=re^{\nu/4}$ may be used for ascertaining the
pulsational stability (see, e.g. Negi and Durgapal 2001, Moustakidis 2017,
Negi 2020b) which is not required in the present study. The various variables
appear in eqs. (2) - (9) may be obtained from Negi et al. (1989) in the
following form

###  2.1 The Core: $0\leq y\leq(b/a)$

$8\pi Ea^{2}=8\pi E_{0}a^{2}(1-x)$ (10)

where $x=y^{2}/Q$; and $8\pi E_{0}a^{2}=72u/5Q$

$e^{-\lambda}=1-(24u/25)(5-3x)x$ (11) $e^{\nu/2}=C_{1}{\rm cos}(w/2)+C_{2}{\rm
sin}(w/2)$ (12) $8\pi Pa^{2}=Ma^{2}\frac{C_{2}{\rm cos}(w/2)-C_{1}{\rm
sin}(w/2)}{C_{1}{\rm cos}(w/2)+C_{2}{\rm sin}(w/2)}-Na^{2}$ (13)
$Ma^{2}=12(2u)^{1/2}e^{-\lambda/2}/5Q$ (14) $Na^{2}=(24/25Q)u(5-3x)$ (15)
$\displaystyle w$ $\displaystyle=$ $\displaystyle\rm{ln}\\{x-(5/6)$ (16)
$\displaystyle+\left[(25/72u)-(1/3)(5-3x)x\right]^{1/2}\\}$ $-8\pi
a^{2}\frac{dE}{dy}=8\pi E_{0}a^{2}(2y/Q)$ (17)

$-8\pi a^{2}\frac{dP}{dy}=(1/2)(8\pi Pa^{2}+8\pi Ea^{2})(8\pi
Pa^{2}+Na^{2})e^{\lambda}y$ (18) $8\pi\rho a^{2}=(8\pi Pa^{2}+8\pi
Ea^{2})e^{\nu/2}(1-2u)^{-1/2}$ (19)

###  2.2 The Envelope: $(b/a)\leq y\leq 1$

$8\pi Ea^{2}=2u/y^{2}$ (20) $e^{-\lambda}=(1-2u)$ (21)
$e^{\nu/2}=\frac{y^{(1-n)}[(1+n)^{2}-(1-n)^{2}y^{2n}]}{4n(2-n^{2})^{1/2}}$
(22) $8\pi
Pa^{2}=\frac{2u(1-n^{2})(1-y^{2n})}{y^{2}[(1+n)^{2}-(1-n)^{2}y^{2n}]}$ (23)

$-8\pi a^{2}\frac{dE}{dy}=4u/y^{3}$ (24)

$-8\pi a^{2}\frac{dP}{dy}=(1/2)(8\pi Pa^{2}+8\pi Ea^{2})(8\pi
Pa^{2}y^{2}+1-e^{-\lambda})e^{\lambda}y^{-1}$ (25) $8\pi\rho a^{2}=(8\pi
Pa^{2}+8\pi Ea^{2})e^{\nu/2}(1-2u)^{-1/2}$ (26)

where $n^{2}=(1-4u)/(1-2u)$; and the constants $C_{1}$ and $C_{2}$ are given
[1]

$C_{1}=A_{1}{\rm cos}(w_{b}/2)-B_{1}{\rm sin}(w_{b}/2)$ (27) $C_{2}=A_{1}{\rm
sin}(w_{b}/2)+B_{1}{\rm cos}(w_{b}/2)$ (28)

where

$w_{b}=\rm{ln}(5/6)[(1/2u)-1]^{1/2}$ (29)
$A_{1}=\frac{y_{b}^{(1-n)}[(1+n)^{2}-(1-n)^{2}y_{b}^{2n}]}{4n(2-n^{2})^{1/2}}$
(30)
$B_{1}=(A_{1}/M_{b}b^{2})\left(\frac{2u(1-n^{2})(1-y_{b}^{2n})}{(1+n)^{2}-(1-n)^{2}y_{b}^{2n}}+N_{b}b^{2}\right)$
(31)

and

$M_{b}b^{2}=2(2u)^{1/2}(1-2u)^{1/2}$ (32) $N_{b}b^{2}=2u$ (33)

##  3 Results and Conclusions

Figures 1 - 4 give the plots between $a\omega$ and $Q$ for $u$ values 0.10,
0.15, 0.20 and 0.24999 respectively. The positive values of $a\omega$ indicate
that the structures are pulsationally stable for all values of $u$ permissible
for the models considered in the study of Negi et al.(1989). The binding
energy coefficients, $\alpha$ and $\alpha_{p}$, of the models considered in
the present study are shown is Figures 5 - 8 for $u$ values in the range 0.10
- 0.24999. The positive values of $\alpha$ (since $\alpha_{p}$ always remains
positive) indicate that the structures are gravitationally bound for all
possible values of $Q$ and $u$. As $Q\rightarrow 1.2$ the values of $\alpha$
and $\alpha_{p}$ become closer to each other. However, this value of $Q$
corresponds to a slower variation of density inside the structure (which
corresponds to a structure with a negligible envelope, i.e. the entire
configuration is represented by Tolman’s VII solution) so much so that the
rest- mass density and the energy density become almost equal. Furthermore, it
may be noted that $\alpha$ is continuously increasing with $Q$ for all the
values of $u$ considered in the range 0.10 - 0.24999 which means that the
structures are also pulsationally stable together with the fact that they are
also gravitationally bound which is the outcome of binding energy criterion of
fluid stars which states that the configurations remain pulsationally stable
upto the first maxima in the binding energy curve (Zeldovich and Novikov 1978,
Shapiro and Teukolsky 1983).

##  4 Summary

A core-envelope massive configuration corresponding to a core described by
Tolman’s VII solution and the envelope is given by Tolman’s VI solution has
been investigated under small radial pulsations and the binding-energy
coefficients ($\alpha$ and $\alpha_{p}$) have been calculated for various
allowed values of compactness parameters, $u$ and the compressibility factor
$Q$. The study shows that the models of Negi et al.(1989) for which all the
four variables $P,E,\nu$ and $\lambda$ along with ($\rm d\nu/\rm dr$) and
($\rm d\lambda/\rm dr$) are continuous at the core-envelope boundary $r=b$
represent causally consistent, gravitationally bound and pulsationally stable
structures.

Thus, these gravitationally bound and pulsationally stable models are capable
of explaining the glitch healing parameters on the basis of starquake model of
glitch generation for both the pulsars, namely the Crab pulsar and the Vela
pulsar and predict the mass for the Crab pulsar about $1.8M_{\odot}$ and for
the Vela pulsar about $1.96M_{\odot}$ for the surface density $2\times
10^{14}\rm{gcm}^{-3}$ (like, Brecher and Caporaso 1978) considered in the
study of Negi (2019). Furthermore, on the basis of vortex unpinning model of
glitch generation, the models considered in this study are capable of
providing the recent required minimum value of fractional crustal moment of
inertia about 7% and higher for all values of the masses in the range
$1M_{\odot}-1.96M_{\odot}$ assigned for the Vela pulsar(Negi 2020a).

##  5 Conflict of Interest Statement:

The Author declares that there is no conflict of interest.

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Figure 1: Variation of $a\omega$ with $Q$ for $u$ value 0.10. Figure 2:
Variation of $a\omega$ with $Q$ for $u$ value 0.15. Figure 3: Variation of
$a\omega$ with $Q$ for $u$ value 0.20. Figure 4: Variation of $a\omega$ with
$Q$ for $u$ value 0.24999. Figure 5: Variation of $\alpha,\alpha_{p}$ with $Q$
for the value of $u=0.10$. Figure 6: Variation of $\alpha,\alpha_{p}$ with $Q$
for the value of $u=0.15$. Figure 7: Variation of $\alpha,\alpha_{p}$ with $Q$
for the value of $u=0.20$. Figure 8: Variation of $\alpha,\alpha_{p}$ with $Q$
for the value of $u=0.24999$.