2020MNRAS.492.3143T

Query : 2020MNRAS.492.3143T

2020MNRAS.492.3143T - Mon. Not. R. Astron. Soc., 492, 3143-3155 (2020/March-1)

Rotation and spots in normal A and Am/Fm stars.

TRUST O., JURUA E., DE CAT P. and JOSHI S.

Abstract (from CDS):

Frequency analysis of long-term ultraprecise photometry can lead to precise values of rotation frequencies of rotating stars with `hump and spike' features in their periodograms. Using these features, we computed the rotation frequencies and amplitudes. The corresponding equatorial rotational velocity (V_{{rot}}) and spot size were estimated. On fitting the autocorrelation functions of the light curves with the appropriate model, we determined the star-spot decay time-scale. The V_{rot} agrees well with the projected rotational velocity (ν ∼sin i) in the literature. Considering a single circular and black spot, we estimate its radius from the amplitude of the `spike'. No evidence for a significant difference in the average `spike' amplitude and spot radius was found for Am/Fm and normal A stars. Indeed, we derived an average value of ∼ 21± 2 and ∼ 19± 2 ppm for the photometric amplitude and of 1.01 ± 0.13 and 1.16 ± 0.12 R_ E for the spot radius (where R_E is the Earth radius), respectively. We do find a significant difference for the average spot decay time-scale, which amounts to 3.6 ± 0.2 and 1.5 ± 0.2 d for Am/Fm and normal A stars, respectively. In general, spots on normal A stars are similar in size to those on Am/Fm stars, and both are weaker than previously estimated. The existence of the `spikes' in the frequency spectra may not be strongly dependent on the appearance of star-spots on the stellar surface. In comparison with G, K, and M stars, spots in normal A and Am/Fm stars are weak, which may indicate the presence of a weak magnetic field.

Abstract Copyright: © 2020 The Author(s) Published by Oxford University Press on behalf of the Royal Astronomical Society

Journal keyword(s): stars: chemically peculiar - stars: general - stars: rotation

VizieR on-line data: <Available at CDS (J/MNRAS/492/3143): tablea1.dat tablea2.dat>

CDS comments: KIC 4557097 is a probable misprint for KIC 4567097.

Simbad objects: 174

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Number of rows : 174
N Identifier Otype ICRS (J2000)
RA 		ICRS (J2000)
DEC 		Mag U Mag B Mag V Mag R Mag I Sp type #ref
1850 - 2026 		#notes
1 * gam Gem SB* 06 37 42.71050 +16 23 57.4095 1.98 1.92 1.92 1.86 1.87 A1.5IV+ 525 1
2 * alf CMa SB* 06 45 08.91728 -16 42 58.0171 -1.51 -1.46 -1.46 -1.46 -1.43 A0mA1Va 1610 0
3 * rho Pup dS* 08 07 32.6488235 -24 18 15.567946 3.43 3.24 2.81 2.46 2.25 F5IIkF2IImF5II 322 0
4 * bet UMa PM* 11 01 50.4797515135 +56 22 56.761138187 2.35 2.35 2.37 2.31 2.35 A1IVps 462 0
5 * tet Leo PM* 11 14 14.4056705348 +15 25 46.455911128 3.40 3.33 3.35 3.32 3.34 A2IV 353 0
6 * alf Lyr dS* 18 36 56.33635 +38 47 01.2802 0.03 0.03 0.03 0.07 0.10 A0V 2787 1
7 TYC 3131-538-1 * 18 50 50.4991933968 +43 42 15.351942816   10.61 10.29     A7III 10 0
8 TYC 3548-231-1 * 18 55 55.3560840168 +49 23 06.379530180   10.93 10.74     ~ 8 0
9 TYC 3127-2016-1 Pe* 18 56 58.4055303480 +41 36 45.328123872   10.20 9.87     A2IVSrSiEu 11 0
10 HD 176708 * 18 59 00.9221766144 +49 19 09.893749800   7.54 7.46     A2V 21 0
11 TYC 3545-131-1 V* 19 00 23.4061857624 +48 40 42.164096520   12.10 11.77     F6V 3 0
12 HD 177458 * 19 02 49.8010744032 +43 21 21.177046548   9.77 9.63     A 14 0
13 HD 177592 * 19 03 39.2580729312 +39 54 39.240699372   8.12 8.07     A0 25 0
14 TYC 3541-3001-1 dS* 19 04 36.0623324904 +46 03 37.850502924   10.73 10.63     A4Vn 12 0
15 TYC 3549-2630-1 * 19 07 04.8915842280 +49 05 28.035076740   11.81 11.68     A5IV 7 0
16 2MASS J19072867+4708240 Pe* 19 07 28.6621192368 +47 08 24.071679384           A7III 5 0
17 TYC 3542-1092-1 * 19 07 35.0049509352 +46 32 46.899956976   12.22 12.56     A8V 6 0
18 UCAC4 688-066818 Er* 19 07 44.7637576584 +47 32 42.421026516   13.232 13.274 13.399   ~ 6 0
19 HD 179337 * 19 10 19.3054971408 +41 30 58.417912836   9.54 9.31     A6IV-V 14 0
20 HD 179458 * 19 10 33.7958850816 +45 45 03.996048816   9.15 8.95     A3V 30 0
21 TYC 3121-1076-1 * 19 10 38.1562555896 +38 23 57.112831896   12.40 11.58     A1V 9 0
22 2MASS J19111437+4931178 V* 19 11 14.3789169816 +49 31 17.912388144           F1V 4 0
23 TYC 3542-2292-1 * 19 12 18.0903410544 +45 55 37.672096548   12.01 11.78     A2V 8 0
24 BD+38 3478 * 19 12 28.5269143992 +38 44 51.692254728   10.20 10.06     A4V 16 0
25 BD+45 2858 V* 19 14 10.5209542752 +45 43 16.549285296   10.03 9.66     A9III 13 0
26 HD 180637 * 19 15 08.0740075704 +46 10 25.781011212   9.28 9.29     B9.5V 18 0
27 BD+40 3639 * 19 15 14.7707708928 +40 43 45.300778752   9.42 9.36     A0.5Va+ 22 0
28 TYC 3129-2256-1 * 19 15 39.8713975680 +41 29 51.184207608   11.76 11.84     ~ 9 0
29 HD 180888 * 19 16 01.6625689872 +47 00 11.225939616   9.16 9.05     A2IV 18 0
30 TYC 3554-525-1 dS* 19 16 49.7011515768 +50 46 41.539777968   11.91 11.73     ~ 5 0
31 TYC 3546-1602-1 * 19 17 27.7375639320 +47 44 36.512867328   11.36 11.29     A9V 9 0
32 TYC 3129-337-1 Pe* 19 18 29.5989270048 +41 29 36.003045120   12.20 12.37     F2IV 8 0
33 TYC 3134-1831-1 * 19 20 12.3872871696 +38 29 27.202373124   12.39 12.13     A7V 8 0
34 TYC 3555-966-1 * 19 21 47.6956612968 +51 40 30.378348444   11.30 11.13     A4V 9 0
35 TYC 3142-1504-1 Pe* 19 22 46.3885144680 +41 19 01.539377112   10.60 10.36     F0V 9 0
36 HD 182613 * 19 23 00.5621181216 +51 02 02.883087492 8.880 8.760 8.590     A2 20 0
37 KOI-3191 Ro* 19 23 15.8495938992 +47 40 58.423162548   11.39 11.32     A4V 13 0
38 2MASS J19231896+3730270 Pe* 19 23 18.9711299256 +37 30 26.960480820           kA2hF0mF1SiEu 9 0
39 TYC 3547-1116-1 dS* 19 23 47.5143774216 +48 31 27.993438804   11.70 11.25     B9V 10 0
40 TYC 3134-1835-1 * 19 23 59.3225909592 +37 31 19.228558836   11.40 11.20     ~ 6 0
41 KOI-5016 Ro* 19 24 15.4194634248 +38 58 14.309152716   10.23 9.99     ~ 19 0
42 TYC 3547-1796-1 * 19 24 18.6040657536 +47 44 02.810380488   12.44 12.39     B9III 7 0
43 TYC 3138-106-1 * 19 24 58.9048760544 +40 27 46.510004160   11.53 11.48     A3V 10 0
44 2MASS J19245958+4724483 * 19 24 59.5824228096 +47 24 48.264672924           ~ 4 0
45 BD+42 3334 Pe* 19 25 04.7657391384 +42 41 31.267296348   10.38 10.06     kA3hA7mA7Sr 28 0
46 2MASS J19252407+4506283 * 19 25 24.0708891552 +45 06 28.353132828           ~ 6 0
47 TYC 3547-1172-1 dS* 19 25 39.8956963224 +48 26 39.318631404   11.91 11.68     A6V 7 0
48 TYC 3146-1441-1 dS* 19 25 43.1379233400 +44 26 26.217339000   10.87 10.61     G1V 13 0
49 TYC 3551-561-1 * 19 26 20.2917366480 +49 00 25.692730920   11.44 11.38     A5IV 9 0
50 TYC 3146-398-1 Pe* 19 26 33.5537992056 +43 45 49.453048476   10.13 9.85     A7III 14 0
51 BD+38 3604 * 19 28 17.0362900896 +39 09 14.531776380   10.60 10.53     B9IV 15 0
52 TYC 3134-726-1 gD* 19 28 18.4567930872 +39 03 37.340158860   12.87 12.60     Am 5 0
53 TYC 3543-2208-1 Pe* 19 28 37.5075000936 +46 29 17.938594392   12.50 12.16     kA1hA5mA8Si 9 0
54 TYC 3547-1850-1 * 19 29 10.1175946752 +47 31 32.176601832   11.24 10.86     A2V 11 0
55 2MASS J19293196+3823578 V* 19 29 31.9670736576 +38 23 57.816778128           F4II-III 3 0
56 TYC 2667-800-1 Pe* 19 30 20.9892528888 +37 19 02.088009948   11.89 11.91     A4III 11 0
57 HD 184024 * 19 30 29.7571552584 +43 10 00.746357520   8.33 8.31     A0.5IVnn 24 0
58 KOI-3185 Pe* 19 31 06.3194961960 +38 23 38.248501524   11.25 10.98     kA5hA8mA9 15 0
59 TYC 3139-1899-1 V* 19 31 56.5149487056 +39 59 15.365072076   11.13 10.72     F2V 6 0
60 HD 184469 * 19 32 55.9357568688 +39 39 45.727299792   7.67 7.75     B9 27 0
61 2MASS J19331553+4133195 V* 19 33 15.5330715456 +41 33 19.542410388           F5IV 3 0
62 TYC 3560-2311-1 Pe* 19 33 55.8943590984 +46 58 22.468573236   11.33 11.37     A7III 14 0
63 KOI-1622 Ro* 19 34 11.7374802384 +42 53 19.853397456   12.06 12.35     F0IV 33 0
64 2MASS J19341879+4248114 Pe* 19 34 18.7957124304 +42 48 11.334124332     13.679     A9V 7 0
65 2MASS J19344449+4439030 * 19 34 44.4939763296 +44 39 03.070199736           A7III 7 0
66 BD+45 2925 * 19 34 46.6854222096 +45 37 11.807030640   10.34 10.17     A1V 19 0
67 BD+39 3825 Pe* 19 35 01.8877324176 +39 45 52.014539556   10.01 9.75     kA2hA5mA8 12 0
68 TYC 3139-2292-1 dS* 19 35 09.0558089760 +40 17 10.051473120   11.60 11.24     ~ 5 0
69 TYC 3139-2394-1 * 19 36 08.9636438832 +40 41 05.479507644   10.34 10.09     ~ 10 0
70 HD 185265 * 19 36 26.9352692016 +41 53 33.896196492   7.53 7.53     A1IV-s 25 0
71 TYC 3556-1380-1 dS* 19 36 47.7621015596 +46 17 33.944342061   12.81 12.64     A9V 19 0
72 Cl* NGC 6811 SAN 77 Ro* 19 36 48.1625887392 +46 19 37.976623622   12.24 11.883   11.57 A5IV 27 0
73 TYC 3143-80-1 * 19 36 53.4030563304 +41 57 57.678977376   12.35 12.56     ~ 6 0
74 TYC 3143-1384-1 * 19 37 04.4824042680 +41 29 57.858413136   11.56 11.13     A2III 7 0
75 Cl* NGC 6811 SAN 118 * 19 37 12.1279149499 +46 10 12.874692186   12.83 12.519     ~ 15 0
76 Cl* NGC 6811 SAN 147 gD* 19 37 25.2380114270 +46 19 35.587893758   12.62 12.255   11.79 ~ 21 0
77 2MASS J19372738+4222284 dS* 19 37 27.3840645792 +42 22 28.395363900           ~ 8 0
78 HD 185658 Em* 19 38 05.1897089136 +45 53 03.825938724   8.33 8.11     F2IIIs 31 0
79 TYC 3139-1375-1 * 19 38 13.0425962592 +39 40 06.569615136   9.96 9.84     A3Van(wkmetA2) 10 0
80 TYC 3560-2452-1 * 19 38 29.6853436584 +47 14 00.757434192   11.74 11.54     A8III 8 0
81 TYC 3564-2119-1 * 19 38 58.1515860240 +49 01 11.218258668   11.5 11.4     Am 11 0
82 UCAC4 651-075039 * 19 39 33.5740606848 +40 03 27.619967844   12.853 12.764 12.895   ~ 4 0
83 2MASS J19394700+5036388 * 19 39 46.9979481096 +50 36 38.763126936           F0IV 7 0
84 2MASS J19395437+4717107 * 19 39 54.3863199984 +47 17 10.775584320           ~ 5 0
85 TYC 3560-2857-1 * 19 40 17.2604531736 +48 36 41.061920328   11.22 11.07     ~ 9 0
86 TYC 3556-3655-1 dS* 19 40 21.3859398576 +46 16 50.066664600   11.84 11.41     F0IV 15 0
87 TYC 3140-1868-1 Pe* 19 40 29.9136530544 +40 44 15.149060556   11.94 11.51     kA2hA5mA8Si 8 0
88 TYC 3144-2314-1 * 19 40 32.0869080576 +42 39 47.753540100   11.85 11.54     B9III 7 0
89 TYC 3144-2053-1 * 19 40 42.8081741086 +42 28 18.696442830   12.17 12.22     A9V 7 0
90 BD+38 3705 V* 19 40 50.1171472464 +38 32 27.086715312   10.52 10.11     F0m 10 0
91 TYC 3564-3085-1 * 19 41 14.4450533736 +49 06 20.450378592   11.05 10.79     A9V 12 0
92 TYC 3140-1536-1 Pe* 19 41 50.3675103720 +39 42 58.074332652   12.67 13.01     kA4hA7mA9Sr 7 0
93 TYC 3140-2923-1 * 19 42 09.5139131112 +40 25 04.715490000   11.44 11.16     ~ 7 0
94 TYC 3144-2204-1 * 19 42 16.4522760456 +42 55 46.009918740   11.69 11.36     A5IV 7 0
95 2MASS J19422599+4526172 * 19 42 25.9970880840 +45 26 17.233960956           ~ 6 0
96 HD 225504 gD* 19 43 07.5734028192 +38 56 49.890057792   10.64 10.44     A5V 13 0
97 HD 225515 * 19 43 13.2126231840 +39 32 32.103826116   11.31 11.16     A2V 14 0
98 2MASS J19431805+4436313 * 19 43 18.0593404104 +44 36 31.327293888           ~ 6 0
99 HD 225570 * 19 44 01.9185476592 +38 52 58.510011216   11.23 11.25     A0V 11 0
100 BD+42 3433 * 19 44 12.2665075512 +43 14 31.887399804   10.26 10.21     B9IV 15 0
101 UCAC3 268-174697 dS* 19 44 57.9433777128 +43 57 15.699396996       13.99   F1V 8 0
102 2MASS J19445949+4707407 * 19 44 59.4873884424 +47 07 40.734979284           ~ 4 0
103 TYC 3144-1956-1 * 19 45 34.5616209792 +42 16 03.602364708   13.03 12.88     A5IV 7 0
104 TYC 3561-1671-1 dS* 19 46 12.9724123056 +47 50 36.277362816   11.24 11.05     A8III 11 0
105 TYC 3148-1286-1 * 19 46 15.4825468704 +43 53 09.259647360   11.67 11.40     B7V 6 0
106 HD 225753 * 19 46 19.6588194744 +41 33 42.194840832   10.059 9.88     A4V 11 0
107 2MASS J19462596+4113224 * 19 46 25.9568344920 +41 13 22.318157856           B9IV 8 0
108 TYC 3561-1825-1 * 19 46 31.2718454352 +48 07 09.029329320   10.60 10.40     A8III 10 0
109 2MASS J19465544+4743406 * 19 46 55.4372823216 +47 43 40.615236228           ~ 4 0
110 TYC 3565-301-1 Er* 19 47 20.9477041056 +50 30 34.475566176   12.51 12.44     A9V 9 0
111 HD 225856 * 19 47 47.1686670264 +40 00 57.330197352   10.123 9.906     A5IV 11 0
112 HD 225863 * 19 47 55.1055092784 +39 58 24.043875096   10.36 10.32     B9.5Vn 17 0
113 TYC 3557-1516-1 * 19 48 14.0602947216 +45 55 43.009239072   12.70 12.03     ~ 5 0
114 TYC 3561-3-1 * 19 48 16.9204035408 +47 08 51.433533240   12.60 12.34     A9V 8 0
115 HD 225912 * 19 48 24.2943135120 +41 16 51.275166312   10.30 10.05     A5V 12 0
116 TYC 3140-486-1 * 19 48 31.7080405320 +40 15 02.525588892   11.78 11.44     F0III-IV 4 0
117 TYC 3140-53-1 * 19 48 38.9216069976 +39 57 51.262228296   12.28 12.59     A0V 6 0
118 GSC 03144-00882 * 19 48 54.2819367216 +43 05 49.867679544           A7III 4 0
119 HD 225971 * 19 49 03.7778890728 +39 32 54.088603260   11.75 11.63     A4V 10 0
120 HD 225986 Ro* 19 49 12.0070414992 +41 19 36.627626100   10.25 10.15     B9.5V 15 0
121 TYC 3561-1719-1 * 19 49 12.7021685544 +47 25 02.082081972   11.08 10.74     A2V 10 0
122 HD 226008 * 19 49 25.5898596864 +40 52 03.765569952   11.33 11.43     A1V 9 0
123 2MASS J19492964+3950126 V* 19 49 29.6406976056 +39 50 12.629487516           Am 4 0
124 HD 226049 * 19 49 47.1478572576 +41 08 34.614414672   11.58 11.55     ~ 8 0
125 2MASS J19503294+4646117 V* 19 50 32.9408128272 +46 46 11.682390648           F0IV 4 0
126 TYC 3561-991-1 Pe* 19 51 11.0492686872 +48 25 35.400733500   11.68 11.47     kA2hA8mA8 6 0
127 KOI-1062 EB* 19 51 55.5003724872 +41 26 41.551015128   11.82 11.19     ~ 20 0
128 HD 226295 * 19 52 25.7440192872 +40 32 05.668740636   10.194 10.08     A2V 12 0
129 HD 226305 * 19 52 32.0186141712 +40 57 31.597211160   10.56 10.49     ~ 8 0
130 KIC 10357965 V* 19 52 33.9402244226 +47 28 11.718112255           ~ 3 0
131 TYC 3149-510-1 * 19 52 40.6594502040 +44 23 48.548892984   11.60 11.36     ~ 6 0
132 HD 226335 * 19 52 47.3178363744 +40 23 43.606721760   11.27 11.33     ~ 8 0
133 BD+42 3509 * 19 53 13.8386533512 +42 24 27.104721408   10.64 10.45     A2 9 0
134 TYC 3141-2114-1 dS* 19 53 19.3829454720 +40 26 13.300249812   12.38 12.63     A3V 3 0
135 TYC 3562-219-1 * 19 53 21.1674746064 +48 08 30.720362460   11.59 11.48     ~ 6 0
136 2MASS J19532162+4806039 Pe* 19 53 21.6267609576 +48 06 03.941193744           kA2hA5mA8 5 0
137 TYC 3141-2421-1 * 19 53 24.5773117201 +39 57 06.468482744   11.56 11.69     B9IVn 9 0
138 TYC 3141-853-1 dS* 19 53 27.1374891048 +40 09 40.448274912   12.07 11.84     ~ 5 0
139 TYC 3558-2397-1 dS* 19 53 27.2866605768 +45 43 15.588107508   12.13 11.97     A2V 9 0
140 2MASS J19533975+4518417 * 19 53 39.7502217384 +45 18 41.722977996           B9IV 4 0
141 HD 188610 * 19 53 48.5315761296 +48 17 24.216397548   8.23 8.12     A0 15 0
142 HD 226442 Pe* 19 53 52.9626752808 +42 14 24.579110580   10.99 10.86     A2III 11 0
143 2MASS J19540343+4538551 * 19 54 03.4305577080 +45 38 55.192852932           ~ 10 0
144 TYC 3145-2669-1 * 19 54 29.4664708800 +41 50 44.449170972   11.46 11.07     ~ 6 0
145 TYC 3149-461-1 * 19 54 38.0403179664 +44 39 53.677683036   12.26 11.85     A0V 6 0
146 TYC 3562-1071-1 * 19 54 48.0303921984 +48 10 57.265584780   12.18 12.50     ~ 5 0
147 HD 226527 * 19 54 48.6846633288 +41 08 38.823617184   10.74 10.69     ~ 13 0
148 2MASS J19545705+4004094 Em* 19 54 57.0525817679 +40 04 09.481353151           B9.5Vn 14 0
149 HD 188774 dS* 19 55 10.0375808184 +41 17 10.271172480   9.05 8.81     A9IV 43 0
150 2MASS J19551369+4356195 * 19 55 13.6986634464 +43 56 19.474756836           ~ 5 0
151 HD 188889 * 19 55 31.5379258248 +44 27 51.837194664   11.00 10.69     A2III 10 0
152 TYC 3141-1968-1 * 19 55 59.1438063648 +40 35 49.099182900   12.09 11.94     ~ 5 0
153 2MASS J19560978+4015304 * 19 56 09.7839665088 +40 15 30.388489272           ~ 5 0
154 HD 226697 * 19 56 36.7580821248 +40 48 32.069002392   10.021 9.922     B9III 11 0
155 HD 226712 * 19 56 41.9389637592 +41 06 08.823975444   10.69 10.58     ~ 14 0
156 KOI-706 EB* 19 57 07.9343169576 +45 57 51.944311584           A9V 20 0
157 2MASS J19574621+4401227 * 19 57 46.2086639016 +44 01 22.784711808           A2III 7 0
158 2MASS J19575113+4431550 Pe* 19 57 51.1360191144 +44 31 54.981620256           A7III 4 0
159 HD 226874 * 19 58 05.9200805592 +40 35 47.922976896   8.723 8.70     A0 12 0
160 HD 226893 * 19 58 20.1725560392 +40 43 58.294755732   10.745 10.654 10.696   ~ 9 0
161 TYC 3149-1074-1 * 19 58 48.4985141448 +44 18 11.843890272   11.10 10.92     B9III 11 0
162 2MASS J19591467+4545148 * 19 59 14.6741691552 +45 45 14.830424172           ~ 6 0
163 TYC 3149-605-1 Pe* 19 59 46.6913426664 +44 51 55.573389828   12.20 11.68     A9V 4 0
164 TYC 3149-415-1 * 19 59 58.6921767672 +44 39 00.912104064   11.67 11.38     A2III 6 0
165 TYC 3558-2110-1 dS* 20 00 31.0999914384 +45 59 25.447602696   11.85 11.84     A2V 7 0
166 HD 190165 gD* 20 01 46.7156874888 +45 28 35.844023964 7.940 7.840 7.530     kA3hF0.5mF3(III)Am 38 0
167 TYC 3558-2045-1 * 20 01 54.9339639120 +45 35 33.229492440   11.79 11.54     A2V 7 0
168 TYC 3162-731-1 * 20 02 54.4202058936 +44 34 03.476265708   11.90 11.80     ~ 7 0
169 TYC 3162-72-1 * 20 03 39.1724765269 +44 14 19.086066484 12.594 12.382 12.071 11.892 11.686 A3V 17 0
170 2MASS J20040888+4420159 * 20 04 08.8852012464 +44 20 15.954073476   13.72 13.29     ~ 9 0
171 UCAC4 672-080945 dS* 20 04 14.0284259064 +44 17 37.168233288 13.581 13.427 13.085 12.869 12.671 ~ 12 0
172 TYC 3162-1069-1 * 20 04 14.6830887610 +44 22 10.902662942   12.41 12.15     A3V 13 0
173 TYC 3162-1081-1 * 20 04 50.1457530576 +44 21 57.607681392   12.19 11.93     B9IV 7 0
174 TYC 3162-281-1 * 20 04 55.9023841656 +44 43 29.937081900   11.87 11.64     kA1hA2mA4 8 0

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