text: SUBMITTED TO IEEE TRANSACTIONS ON PATTERN ANALYSIS AND MACHINE INTELLIGENCE 14 Fig. 14. The Hall 3 reconstruction: ten of the input images (top) and ﬁve views of the quasi-dense 3D reconstruction (bottom). Reviewer 1 for his/her helpful and detailed comments [29] T. P. Wu, S. K. Yeung, J. Jia, and C. K. Tang. Quasi-dense 3D throughout the review cycle. reconstruction using tensor-based multiview stereo. In CVPR10, 2010. REFERENCES [1] S. Arya and D. M. Mount. Approximate nearest neighbor search- ing. ACM-SIAM SODA’93, pages 271–280. [2] J. Bilmes. A gentle tutorial on the EM algorithm and its appli- cation to parameter estimation for Gaussian mixture and hidden Markov models. Technical Report ICSI-TR-97-021, ICSI, 1997. Tai-Pang Wu received the PhD degree in com- [3] H. Chen and P. Meer. Robust regression with projection based puter science from the Hong Kong University of m-estimators. In ICCV03, pages II: 878–885, 2003. [4] O. Chum and J. Matas. Optimal randomized ransac. PAMI, Science and Technology (HKUST) in 2007. He 30(8):1472–1482, August 2008. was awarded the Microsoft Fellowship in 2006. [5] D. Comaniciu and P. Meer. Mean shift: A robust approach toward After graduation, he has been employed as a feature space analysis. PAMI, 24:603–619, May 2002. postdoctoral fellow in Microsoft Research Asia [6] R. Dahyot. Statistical hough transform. PAMI, 31(8):1502–1509, Beijing (2007–2008) and the Chinese University August 2009. of Hong Kong (2008–2010). He is currently a Se- [7] M. A. Fischler and R. C. Bolles. Random sample consensus: A nior Research at the Enterprise and Consumer paradigm for model ﬁtting with applications to image analysis Electronics Group of the Hong Kong Applied and automated cartography. Comm. of the ACM, 24:381–395, 1981. Science and Technology Research Institute (AS- [8] D. Forsyth and J. Ponce. Computer Vision: A Modern Approach. TRI). His research interests include computer vision and computer Prentice Hall, 2003. graphics. He is a member of the IEEE and the IEEE Computer Society. [9] E. Franken, M. van Almsick, P. Rongen, L. Florack, and B. ter Haar Romeny. An efﬁcient method for tensor voting using steerable ﬁlters. In ECCV06, pages IV: 228–240, 2006. [10] Y. Furukawa and J. Ponce. Accurate, dense, and robust multiview stereopsis. PAMI, 32(8):1362–1376, August 2010. [11] B. Georgescu, I. Shimshoni, and P. Meer. Mean shift based clustering in high dimensions: A texture classiﬁcation example. In ICCV03, pages 456–463, 2003. Sai-Kit Yeung received his PhD degree in elec- [12] R. Hartley. In defense of the eight-point algorithm. PAMI, tronic and computer engineering from the Hong 19(6):580–593, June 1997. Kong University of Science and Technology [13] P. Hough. Machine analysis of bubble chamber pictures. Centre (HKUST) in 2009. He received his BEng de- Europe´enne pour la Recherch Nucle´aire (CERN), 1959. gree (First Class Honors) in computer engineer- [14] P. J. Huber. Robust Statistics. John Wiley & Sons, 1981. ing and MPhil degree in bioengineering from [15] K. Lee, P. Meer, and R. Park. Robust adaptive segmentation of HKUST in 2003 and 2005 respectively. He is cur- range images. PAMI, 20(2):200–205, February 1998. rently an Assistant Professor at the Singapore [16] M. Lourakis and A. Argyros. The design and implementation of University of Technology and Design (SUTD). a generic sparse bundle adjustment software package based on Prior to joining SUTD, he was a Postdoctoral the levenberg-marquardt algorithm. In Technical Report 340, 2004. Scholar in the Department of Mathematics, Uni- [17] D. Lowe. Distinctive image features from scale-invariant key- versity of California, Los Angeles (UCLA) in 2010. His research interests points. IJCV, 60(2):91–110, November 2004. include computer vision, computer graphics, computational photogra- [18] G. J. McLachlan and T. Krishnan. EM Algorithms and Extension. phy, image/video processing, and medical imaging. He is a member of Elsevier, 1997. [19] G. Medioni, M. S. Lee, and C. K. Tang. A Computational Framework IEEE and IEEE Computer Society. for Segmentation and Grouping. Elsevier, 2000. [20] P. Meer. Robust techniques for computer vision. In Emerging Topics in Computer Vision, page Chapter 4, 2004. [21] P. Mordohai and G. Medioni. Dimensionality estimation, man- ifold learning and function approximation using tensor voting. JMLR, 11:411–450, January 2010. [22] P. Rousseeuw. Least median of squares regression. Journal of American Statistics Assoc., 79:871–880, 1984. [23] P. Rousseeuw. Robust Regression and Outlier Detection. Wiley, 1987. [24] K. Sim and R. Hartley. Removing outliers using the l-inf norm. In CVPR06, pages I: 485–494, 2006. Jiaya Jia Jiaya Jia received the PhD degree in [25] N. Snavely, S. Seitz, and R. Szeliski. Modeling the world from computer science from Hong Kong University of internet photo collections. IJCV, 80(2), November 2008. [26] R. Subarao and P. Meer. Beyond ransac: User independent robust Science and Technology in 2004 and is currently regression. In Workshop on 25 Years of RANSAC, June 2006. an associate professor in Department of Com- [27] W. S. Tong, C. K. Tang, and G. Medioni. Simultaneous two-view puter Science and Engineering at the Chinese epipolar geometry estimation and motion segmentation by 4d University of Hong Kong (CUHK). He was a tensor voting. PAMI, 26(9):1167–1184, September 2004. visiting scholar at Microsoft Research Asia from [28] H. Wang and D. Suter. Robust adaptive-scale parametric model March 2004 to August 2005 and conducted col- estimation for computer vision. PAMI, 26(11):1459–1474, Nov laborative research at Adobe Systems in 2007. 2004. He leads the research group in CUHK, focusing on computational photography, 3D reconstruc- tion, practical optimization, and motion estimation. He serves as an associate editor for TPAMI and as an area chair for ICCV 2011. He was on the program committees of several major conferences, including ICCV, ECCV, and CVPR, and co-chaired the Workshop on Interactive Computer Vision in conjunction with ICCV 2007. He received the Young Researcher Award 2008 and Research Excellence Award 2009 from CUHK. He is a senior member of the IEEE. ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure##

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text: ACCEPTED FOR JOURNAL PUBLICATION 5 (cid:18) (cid:19) [7] H. Yu, and R. Zhang, ”Throughput maximization in wireless powered ∂L (cid:18) α x2 − α(cid:19)x2 communication networks,” IEEE Trans. Wireless Commun., vol 13, no. = log 1 + i i i i 1, pp. 418-428, Jan 2014 ∂θ θ θ + α x2 [8] H. Ju, and R. Zhang, ”Optimal resource allocation in full-duplex i i i i i wireless-powered communication network,” IEEE Trans. Commun., vol. − p − p λ c q c + P N + υ = 0 (26) 62, no. 10, pp. 3528 - 3540, Oct. 2014 x i x max 0 i [9] X. Kang, C. Keong Ho, and S. Sun, ”Optimal time allocation for i i dynamic-TDMA-based wireless powered communication networks,” Proc. IEEE Globecom 2014, Signal Processing for Communications According to KKT conditions, the complementary Symposium, Austin, TX, USA, Dec. 2014 ∀ slackness conditions should be satisﬁed, i: µ α = [10] R. Morsi, D. S. Michalopoulos, and R. Schober, ”Performance analysis q [α + θ (p /x + N P ) − N P ] = υ θ = 0i, wi here of wireless powered communication with ﬁnite/inﬁnite energy storage,” i ≥i i≥ c i 0≥max 0 max i i arXiv:1410.1805v2, Oct 2014. µi 0, qi 0, and υi 0. We consider three cases: [11] He Chen, et al., ”Harvest-then-cooperate: wireless-powered cooperative Case 1: If θ = 0, then α = 0 and no power is allocated to communications”, IEEE Trans. Signal Processing, vol. 63, no. 7, pp. epoch i, i.e., pi∗ = 0. i 1700 - 1711, Apr. 2015 i − [12] O. Orhan, D. Gunduz, E. Erkip, ”Throughput maximization for an energy Case 2: L−et us assume 0 < θi < 1 and αi∗ = (1 harvesting communication system with processing cost”, Proc. 2012 θ )N P p θ /x . This case corresponds to p = P . IEEE Information Theory Workshop, Lausanne, Sept. 2012, pp. 84-88 Tihe 0slamckanxess cconiditiions require q > 0 and µ =i υ =max0. [13] S. Boyd, EE364b Lecture Notes. Stanford, CA: Stanford Univ. Available: i i i http://stanford.edu/class/ee364b/lectures/ellipsoid method notes.pdf From (25), we obtain the condition x2 − qi = 1 − xipc + Pmiax(1−θiθi)N0x2i λ > 0. (27) − − Introducing (27) and α = (1 θ )N P p θ /x into (cid:112) i i 0 max c i i (26), we obtain (11). Based upon (11) and (27), it can be shown that the sufﬁcient condition for the occurrence of this case is − given by λp /2 + (λp /2)2 + λ < x < 1/p . c c i c − Case 3: Let us assume 0 < θ < 1 and 0 < α < (1 − i i θ )N P p θ /x . Then, the slackness conditions require i 0 max c i i µ = q = υ = 0. From (25), we obtain the equality α = i i− i i θ (1/λ 1/x2). Introducing this equality into (26) leads to i i − the equality condition λ(1 p x )/x2 + log(x2/λ) = 1. It c i i i implies that this case occurs only for one speciﬁc value of x , i but its occurrence probability is zero since x is a continuous ∗ i random variable. Thus, the optimal p can be either 0 (Case i 1) or P (Case 2). max REFERENCES [1] D. Gunduz, K. Stamatiou, N. Michelusi, and M. Zorzi, ”Designing intelligent energy harvesting communication systems,” IEEE Commun. Magazine, vol. 52, no. 1, pp. 210 - 216, Jan. 2014 [2] C. K. Ho, and R. Zhang, ”Optimal energy allocation for wireless communications with energy harvesting constraints,” IEEE Trans. Signal Processing, vol. 60, no. 9, pp. 4808 - 4818, Sept. 2012 [3] I. Krikidis, T. Charalambous, and J. S. Thompson, ”Stability analysis and power optimization for energy harvesting cooperative networks,” IEEE Signal Processing Letters, vol. 19, no. 1, pp. 20 - 23, Nov. 2011 [4] Z. Ding, and H. V. Poor, ”Cooperative energy harvesting networks with spatially random users,” IEEE Signal Processing Letters, vol. 20, no. 12, pp. 1211 - 1214, Oct. 2013 [5] P. Grover, and A. Sahai, ”Shannon meets Tesla: wireless information and power transfer,” Proc. IEEE ISIT 2010, Austin, TX, USA, June 2010, pp. 2363 - 2367 [6] R. Zhang, and C. K. Ho, ”MIMO broadcasting for simultaneous wireless information and power transfer,” IEEE Trans. Wireless Commun., vol. 12, no. 5, pp. 1989 - 2001, May 2013 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: ACCEPTED

label: paragraph

text: FOR

text: JOURNAL

text: PUBLICATION

label: equation

text: H.

text: Yu,

text: and

text: R.

text: Zhang,

text: ”Throughput

text: maximization

text: in

text: wireless

text: powered

text: ∂L

text: α

text: communication

text: networks,”

text: IEEE

text: Trans.

text: Wireless

text: Commun.,

text: vol

text: no.

text: log

text: i

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text: i

text: i

text: pp.

text: Jan

text: ∂θ

text: θ

text: θ

text: α

text: H.

text: Ju,

text: and

text: R.

text: Zhang,

text: ”Optimal

text: resource

text: allocation

text: in

text: full-duplex

text: i

text: i

text: i

text: i

text: i

text: wireless-powered

text: communication

text: network,”

text: IEEE

text: Trans.

text: Commun.,

text: vol.

text: p

text: p

text: λ

text: c

text: q

text: c

text: P

text: N

text: υ

text: no.

text: pp.

text: Oct.

text: x

text: i

text: x

text: max

text: i

text: X.

text: Kang,

text: C.

text: Keong

text: Ho,

text: and

text: S.

text: Sun,

text: ”Optimal

text: time

text: allocation

text: for

text: i

text: i

text: dynamic-TDMA-based

text: wireless

text: powered

text: communication

text: networks,”

text: Proc.

text: IEEE

text: Globecom

text: Signal

text: Processing

text: for

text: Communications

text: According

text: to

text: KKT

text: conditions,

text: the

text: complementary

text: Symposium,

text: Austin,

text: TX,

text: USA,

text: Dec.

text: slackness

text: conditions

text: should

text: be

text: satisﬁed,

text: i:

text: µ

text: α

text: R.

text: Morsi,

text: D.

text: S.

text: Michalopoulos,

text: and

text: R.

text: Schober,

text: ”Performance

text: analysis

text: q

text: [α

text: θ

text: (p

text: /x

text: N

text: P

text: N

text: P

text: υ

text: θ

text: wi

text: here

text: of

text: wireless

text: powered

text: communication

text: with

text: ﬁnite/inﬁnite

text: energy

text: storage,”

text: i

text: ≥i

text: i≥

text: c

text: i

text: max

text: i

text: i

text: Oct

text: µi

text: qi

text: and

text: υi

text: We

text: consider

text: three

text: cases:

text: He

text: Chen,

text: et

text: al.,

text: ”Harvest-then-cooperate:

text: wireless-powered

text: cooperative

text: Case

text: If

text: θ

text: then

text: α

text: and

text: no

text: power

text: is

text: allocated

text: to

text: communications”,

text: IEEE

text: Trans.

text: Signal

text: Processing,

text: vol.

text: no.

text: pp.

text: epoch

text: i,

text: i.e.,

text: pi∗

text: i

text: Apr.

text: i

text: O.

text: Orhan,

text: D.

text: Gunduz,

text: E.

text: Erkip,

text: ”Throughput

text: maximization

text: for

text: an

text: energy

text: Case

text: L−et

text: us

text: assume

text: θi

text: and

text: αi∗

text: harvesting

text: communication

text: system

text: with

text: processing

text: cost”,

text: Proc.

text: θ

text: )N

text: P

text: p

text: θ

text: /x

text: This

text: case

text: corresponds

text: to

text: p

text: P

text: IEEE

text: Information

text: Theory

text: Workshop,

text: Lausanne,

text: Sept.

text: pp.

text: Tihe

text: cconiditiions

text: require

text: q

text: and

text: µ

text: =i

text: υ

text: S.

text: Boyd,

text: Lecture

text: Notes.

text: Stanford,

text: CA:

text: Stanford

text: Univ.

text: Available:

text: i

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text: i

text: method

text: notes.pdf

text: From

text: we

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text: the

text: condition

text: qi

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text: λp

text: (λp

text: λ

text: x

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text: Case

text: Let

text: us

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text: θ

text: and

text: α

text: i

text: i

text: θ

text: )N

text: P

text: p

text: θ

text: /x

text: Then,

text: the

text: slackness

text: conditions

text: require

text: i

text: max

text: c

text: i

text: i

text: µ

text: q

text: υ

text: From

text: we

text: obtain

text: the

text: equality

text: α

text: i

text: i−

text: i

text: i

text: θ

text: Introducing

text: this

text: equality

text: into

text: leads

text: to

text: i

text: i

text: the

text: equality

text: condition

text: p

text: x

text: It

text: c

text: i

text: i

text: i

text: implies

text: that

text: this

text: case

text: occurs

text: only

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text: speciﬁc

text: value

text: of

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text: probability

text: is

text: zero

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text: x

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text: optimal

text: p

text: can

text: be

text: either

text: (Case

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text: P

text: (Case

text: max

text: REFERENCES

text: D.

text: Gunduz,

text: K.

text: Stamatiou,

text: N.

text: Michelusi,

text: and

text: M.

text: Zorzi,

text: ”Designing

text: intelligent

text: energy

text: harvesting

text: communication

text: systems,”

text: IEEE

text: Commun.

text: Magazine,

text: vol.

text: no.

text: pp.

text: Jan.

text: C.

text: K.

text: Ho,

text: and

text: R.

text: Zhang,

text: ”Optimal

text: energy

text: allocation

text: for

text: wireless

text: communications

text: with

text: energy

text: harvesting

text: constraints,”

text: IEEE

text: Trans.

text: Signal

text: Processing,

text: vol.

text: no.

text: pp.

text: Sept.

text: I.

text: Krikidis,

text: T.

text: Charalambous,

text: and

text: J.

text: S.

text: Thompson,

text: ”Stability

text: analysis

text: and

text: power

text: optimization

text: for

text: energy

text: harvesting

text: cooperative

text: networks,”

text: IEEE

text: Signal

text: Processing

text: Letters,

text: vol.

text: no.

text: pp.

text: Nov.

text: Z.

text: Ding,

text: and

text: H.

text: V.

text: Poor,

text: ”Cooperative

text: energy

text: harvesting

text: networks

text: with

text: spatially

text: random

text: users,”

text: IEEE

text: Signal

text: Processing

text: Letters,

text: vol.

text: no.

text: pp.

text: Oct.

text: P.

text: Grover,

text: and

text: A.

text: Sahai,

text: ”Shannon

text: meets

text: Tesla:

text: wireless

text: information

text: and

text: power

text: transfer,”

text: Proc.

text: IEEE

text: ISIT

text: Austin,

text: TX,

text: USA,

text: June

text: pp.

text: R.

text: Zhang,

text: and

text: C.

text: K.

text: Ho,

text: ”MIMO

text: broadcasting

text: for

text: simultaneous

text: wireless

text: information

text: and

text: power

text: transfer,”

text: IEEE

text: Trans.

text: Wireless

text: Commun.,

text: vol.

text: no.

text: pp.

text: May

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: Publications of the Astronomical Society of Japan, (2014), Vol. 00, No. 0 3 Table 1. X-ray ﬂux, period and period derivative obtained in past observations of 4U 1626–67. Observation Flux Pulsation ∗ † ‡ Date Period Band Obs. Bolometric Epoch P P˙ Ref. − − − − − (MJD) (keV) (10 10 erg s 1 cm 2) (MJD) (s) (10 11 s s 1) ± ± − § 1978 Mar 43596 0.7–60 24 3 25 3 43596.7 7.679190(26) 4.55 1 ± ± − ± § 1979 Feb 43928–43946 2–10 5.1 0.3 19 1 43946.0 7.677632(13) 4.9 0.1 2 ± ± − ± 1983 May 45457–45459 2–20 10.1 0.2 19.0 0.4 45458.0 7.671350(1) 13 5 3 ± ± − ± 1983 Aug 45576 2–10 5.6 0.5 20.3 1.8 45576.9 7.67077(1) 5.65 0.10l 4 ± ± − ± § 1986 Mar 46519–46520 1–20 14 0.4# 22 0.7 46520.0 7.6664220(5) 4.96 0.06 5 ± ± 1987 Mar 46855–48012 1–20 8.89 0.56 14.0 0.9 6 1988 Aug 47400.3 7.6625685(30) 7 ∼ ∼ 1990 Apr 47999–48002 2–60 20.6 24.7 48001.1 7.660069(2) 8 ± ± 1990 Jun 48043–48530 1–20 6.67 0.89 10.9 1.5 6 1990 Aug 48133.5 7.66001(4) 4 ∼ ∼ 1993 Aug 49210–49211 0.5–10 2.8 9.9 9 ∼ ∼ 1996 Aug 50301–50306 2–60 6.1 6.5 10 ∗∗ 2000 Sep 51803 0.3–10 2.4 8.4 51803.6 7.6726(2) 11 ∗∗ ± 2001 Aug 52145 0.3–10 1.8 6.3 52145.1 7.6736(2) 3.39 0.96l 11 ∗∗ ± 2003 Jun 52793 0.3–10 1.8 6.3 52795.1 7.67514(5) 2.74 0.37l 11 ∗∗ ± 2003 Aug 52871 0.3–10 1.6 5.6 52871.2 7.67544(6) 4.6 1.2l 11 ± ± ∼ 2007 Jun 54280 15–50 2.8 0.1 5.6 0.2 54280 7.6793 2.9 12 ± ± ∼ 2007 Sep 54370 15–50 2.6 0.1 5.2 0.2 54370 7.6793 2.7 12 ± ± ∼ 2007 Dec 54450 15–50 4.0 0.1 8.0 0.3 54450 7.6793 1.9 12 ± ± ∼ 2008 Jan 54480 15–50 4.6 0.2 9.3 0.4 54480 7.6793 0.18 12 ± ± ∼ − 2008 Feb 54510 15–50 4.6 0.1 11.5 0.3 54510 7.6793 0.53 12 ± ± 2008 Mar 54530 2–100 10.1 0.8 11.7 0.9 13 ± ± ∼ − 2008 Mar 54550 15–50 5.8 0.1 14.5 0.3 54550 7.6793 2.3 12 ± ± ∼ − 2008 Jun 54620 15–50 5.9 0.1 14.8 0.2 54620 7.6793 2.7 12 All errors represent 1-σ uncertainties. ∗ Converted 0.5–100 keV ﬂux, assuming the spectral models in Camero-Arranz et al. (2012). † Values in parentheses are 1 σ error in the last digit(s). ‡ (1) Pravdo et al. (1979) (HEAO 1/A-2), (2) Elsner et al. (1983) (Einstein/MPC), (3) Kii et al. (1986) (Tenma), (4) Mavromatakis (1994) (EXOSAT/GSPC, ROSAT), (5) Levine et al. (1988) (EXOSAT/ME), (6) Vaughan & Kitamoto (1997) (Ginga/ASM), (7) Shinoda et al. (1990) (Ginga), (8) Mihara (1995) (Ginga), (9) Angelini et al. (1995) (ASCA), (10) Orlandini et al. (1998) (BeppoSAX), (11) Krauss et al. (2007) (Chandra, XMM-Newton), (12) The data were read from ﬁgure 4 in Camero-Arranz et al. (2010) (Swift/BAT), (13) Camero-Arranz et al. (2010) (RXTE/PCA spectra). § Averaged value. l Averaged P˙ calculated from P of the observation and the previous one in this table. # The error was estimated by the count rate with ME, using the values of the count rate and the error with GSPC. ∗∗ Absorption-corrected ﬂux. ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: Publications

font: ZDDFZZ+NimbusSanL-ReguItal

text: of

text: the

text: Astronomical

text: Society

text: of

text: Japan,

font: VVNBOV+NimbusSanL-Regu

text: Vol.

text: No.

font: ZHFUES+NimbusSanL-Bold

text: Table

text: X-ray

label: caption

text: ﬂux,

text: period

text: and

text: period

text: derivative

text: obtained

text: in

text: past

text: observations

text: of

text: Observation

label: table

text: Flux

text: Pulsation

text: Date

text: Period

text: Band

text: Obs.

text: Bolometric

text: Epoch

text: P

text: P˙

text: Ref.

text: (MJD)

text: (keV)

text: erg

text: s

text: cm

text: (MJD)

text: (s)

text: s

text: s

text: Mar

text: Feb

text: May

text: Aug

text: Mar

text: Mar

text: Aug

text: Apr

text: Jun

text: Aug

text: Aug

text: Aug

text: Sep

text: Aug

text: Jun

text: Aug

text: Jun

text: Sep

text: Dec

text: Jan

text: Feb

text: Mar

text: Mar

text: Jun

text: All

text: errors

text: represent

text: uncertainties.

text: Converted

text: keV

text: ﬂux,

text: assuming

text: the

text: spectral

text: models

text: in

text: Camero-Arranz

text: et

text: al.

text: Values

text: in

text: parentheses

text: are

text: σ

text: error

text: in

text: the

text: last

text: digit(s).

text: Pravdo

text: et

text: al.

text: (HEAO

text: Elsner

text: et

text: al.

text: (Einstein/MPC),

text: Kii

text: et

text: al.

text: (Tenma),

text: Mavromatakis

text: (EXOSAT/GSPC,

text: ROSAT),

text: Levine

text: et

text: al.

text: (EXOSAT/ME),

text: Vaughan

text: Kitamoto

text: (Ginga/ASM),

text: Shinoda

text: et

text: al.

text: (Ginga),

text: Mihara

text: (Ginga),

text: Angelini

text: et

text: al.

text: (ASCA),

text: Orlandini

text: et

text: al.

text: (BeppoSAX),

text: Krauss

text: et

text: al.

text: (Chandra,

text: XMM-Newton),

text: The

text: data

text: were

text: read

text: from

text: ﬁgure

text: in

text: Camero-Arranz

text: et

text: al.

text: (Swift/BAT),

text: Camero-Arranz

text: et

text: al.

text: (RXTE/PCA

text: spectra).

text: Averaged

text: value.

text: l

text: Averaged

text: P˙

text: calculated

text: from

text: P

text: of

text: the

text: observation

text: and

text: the

text: previous

text: one

text: in

text: this

text: table.

text: The

text: error

text: was

text: estimated

text: by

text: the

text: count

text: rate

text: with

text: ME,

text: using

text: the

text: values

text: of

text: the

text: count

text: rate

text: and

text: the

text: error

text: with

text: GSPC.

text: Absorption-corrected

text: ﬂux.

text: ##LTLine##

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text: Publications of the Astronomical Society of Japan, (2014), Vol. 00, No. 0 5 3.1 Model equations relating P˙ to the ﬂux According to GL79, P˙ is expressed by P˙ = − 5.0 × 10−5µ30 27 n(ωs)S1(M)P2L37 67 s yr−1, (3) where µ is the magnetic dipole moment µ in units of 30 − 1030 G cm3, and L is L in units of 1037 erg s 1. The func- 37 tions n(ω ) and S (M) are given in Appendix 1, where ω is s 1 s the fastness parameter deﬁned by equation (A6). If ω is in the − s range of 0 0.9, n(ω ) is approximated by equation (A4) within s 5%. Since n(ω ) changes from positive to negative depending s on ω (ﬁgure 7), P˙ can become both positive (ω > 0.349; spin s s down) and negative (ω < 0.349; spin up). s As shown in equation (A5), S (M) contains the effective 1 moment of inertia I. It is expressed as a function of M and R, dependent on the EOS. We utilize its approximation given by Fig. 3. Distribution of χ2 of the folded pulse proﬁles, shown as a function Lattimer & Schutz (2005), of the trial P and P˙, obtained from the 2–10 keV MAXI/GSC event data in MJD 55290–55350. The right bar indicates the χ2 values. The maximum (cid:34) (cid:18) (cid:19)(cid:16) (cid:17) (cid:18) (cid:19) (cid:16) (cid:17) (cid:35) − × (cid:39) ± χ2 is 108 for 31 degrees of freedom at P = 7.6777282 s and P˙ = 2.64 I (0.237 0.008)MR2 − − 10 11 s s 1. × M R −1 M 4 R −4 1 + 0.42 + 0.009 , M(cid:12) 10 km M(cid:12) 10 km Camero-Arranz et al. (2012), the spectra in both phases can be (4) ﬁtted with a model composed of a blackbody and a cutoff power law, and their difference is in the blackbody component, whose ∼ ∼ temperature is 0.5 keV in the spin-up phase and 0.2 keV which is applicable in most of the major EOS models if in the spin-down phase. We hence employed these respective M/R >∼ 0.07M(cid:12) km−1. spectral models for the spin-up and spin-down phases, and con- In 4U 1626–67, the surface magnetic ﬁeld is known from the × verted the 2–20 keV MAXI/GSC ﬂux to those in the 0.5–100 cyclotron resonance scattering feature as B = 3.2 1012 (1 + c keV band, which we identify with F . Since the power-law z ) G (section 1). It is considered to represent the ﬁeld strength ∼ bol ∼ g continuum is ﬂat (photon index 1) below 20 keV and ex- near the magnetic poles. Assuming that the magnetic axis is ∼ ponentially cuts off above 20 keV, the ﬂuxes in the energy aligned to the pulsar rotation axis, µ at the equator in the GL79 band below 0.5 keV and above 100 keV are negligible. For ex- model is represented by ample, the conversion factor from the 2–20 keV ﬂux observed 1 µ = B R3. (5) by the MAXI/GSC (in the spin-up phase) to the 0.5–100 keV 2 c ﬂux is 1.88. In a similar way, we calculated Fbol in the past If the source emission is isotropic, L is calculated from Fbol observations, and present the results in table 1. and the distance D as In ﬁgure 4, we plot the relation between the observed P˙ and L = 4πD2F . (6) bol the calculated F , including the past data. It clearly shows bol Since the pulsar emission is anisotropic, it is not exactly cor- their negative correlation, expected from the pulsar spin up due rect. We employ this approximation and then discuss the effect to the accretion torque. Furthermore, the data points in the spin- later. up and spin-down phases apparently deﬁnes a well-deﬁned sin- Combining equations (3), (4), (5), and (6), as well as the gle dependence on F . 3 Application obfolthe Ghosh & Lamb relation expression for zg [equation (1)], we obtain a theoretical model equation to describe the observed P˙ –L relation, including three As reviewed in section 1, GL79 derived a relation between unknown parameters, D, M and R. P˙ and L in accreting X-ray pulsars, assuming that the accret- ing matter transfers its angular momentum to the pulsar at the 3.2 Comparison between the data and theory “outer transition zone”, r [equation (A2)]. The equations we 0 used are summarized in Appendix 1. Rappaport & Joss (1977) In order to determine the possible parameter ranges of D, M also proposed an almost equivalent equation. Since their model and R, we ﬁtted the observed P˙ –F relation in ﬁgure 4 with bol equation includes an unknown parameter (ξv /v ), of which the the model prepared as above. When errors associated with some r ﬀ relation to n(ω ) in the GL79 relation is calculable. Therefore, past measurements of P˙ are unavailable, we assumed an arbi- s × − − we employ the GL79 relation. trarily small value (∆P˙ = 6 10 16 s s 1), because the overall ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: Publications

text: of

text: the

text: Astronomical

text: Society

text: of

text: Japan,

text: Vol.

text: No.

label: section

text: Model

text: equations

text: relating

text: P˙

text: to

text: the

text: ﬂux

text: According

text: to

text: P˙

text: is

text: expressed

text: by

text: P˙

text: s

text: where

text: µ

text: is

text: the

text: magnetic

text: dipole

text: moment

text: µ

text: in

text: units

text: of

text: G

text: and

text: L

text: is

text: L

text: in

text: units

text: of

text: erg

text: s

text: The

text: func-

text: tions

text: n(ω

text: and

text: S

text: (M)

text: are

text: given

text: in

text: Appendix

text: where

text: ω

text: is

text: s

text: s

text: the

text: fastness

text: parameter

text: deﬁned

text: by

text: equation

text: If

text: ω

text: is

text: in

text: the

text: s

text: range

text: of

text: n(ω

text: is

text: approximated

text: by

text: equation

text: within

text: s

text: Since

text: n(ω

text: changes

text: from

text: positive

text: to

text: negative

text: depending

text: s

text: on

text: ω

text: (ﬁgure

text: P˙

text: can

text: become

text: both

text: positive

text: (ω

text: spin

text: s

text: s

text: down)

text: and

text: negative

text: (ω

text: spin

text: up).

text: s

text: As

text: shown

text: in

text: equation

text: S

text: (M)

text: contains

text: the

text: effective

text: moment

text: of

text: inertia

text: I.

text: It

text: is

text: expressed

text: as

text: a

text: function

text: of

text: M

text: and

text: R,

text: dependent

text: on

text: the

text: EOS.

text: We

text: utilize

text: its

text: approximation

text: given

text: by

text: Fig.

text: Distribution

text: of

text: χ2

text: of

text: the

text: folded

text: pulse

text: proﬁles,

text: shown

text: as

text: a

text: function

text: Lattimer

text: Schutz

text: of

text: the

text: trial

text: P

text: and

text: P˙,

text: obtained

text: from

text: the

text: keV

text: MAXI/GSC

text: event

text: data

text: in

text: MJD

text: The

text: right

text: bar

text: indicates

text: the

text: χ2

text: values.

text: The

text: maximum

text: χ2

text: is

text: for

text: degrees

text: of

text: freedom

text: at

text: P

text: s

text: and

text: P˙

text: I

text: s

text: s

text: M

text: R

text: −1

text: M

text: R

text: −4

text: km

text: km

text: Camero-Arranz

text: et

text: al.

text: the

text: spectra

text: in

text: both

text: phases

text: can

text: be

text: ﬁtted

text: with

text: a

text: model

text: composed

text: of

text: a

text: blackbody

text: and

text: a

text: cutoff

text: power

text: law,

text: and

text: their

text: difference

text: is

text: in

text: the

text: blackbody

text: component,

text: whose

text: temperature

text: is

text: keV

text: in

text: the

text: spin-up

text: phase

text: and

text: keV

text: which

text: is

text: applicable

text: in

text: most

text: of

text: the

text: major

text: EOS

text: models

text: if

text: in

text: the

text: spin-down

text: phase.

text: We

text: hence

text: employed

text: these

text: respective

text: M/R

text: spectral

text: models

text: for

text: the

text: spin-up

text: and

text: spin-down

text: phases,

text: and

text: con-

text: In

text: the

text: surface

text: magnetic

text: ﬁeld

text: is

text: known

text: from

text: the

text: verted

text: the

text: keV

text: MAXI/GSC

text: ﬂux

text: to

text: those

text: in

text: the

text: cyclotron

text: resonance

text: scattering

text: feature

text: as

text: B

text: c

text: keV

text: band,

text: which

text: we

text: identify

text: with

text: F

text: Since

text: the

text: power-law

text: z

text: G

text: (section

text: It

text: is

text: considered

text: to

text: represent

text: the

text: ﬁeld

text: strength

text: bol

text: g

text: continuum

text: is

text: ﬂat

text: (photon

text: index

text: below

text: keV

text: and

text: ex-

text: near

text: the

text: magnetic

text: poles.

text: Assuming

text: that

text: the

text: magnetic

text: axis

text: is

text: ponentially

text: cuts

text: off

text: above

text: keV,

text: the

text: ﬂuxes

text: in

text: the

text: energy

text: aligned

text: to

text: the

text: pulsar

text: rotation

text: axis,

text: µ

text: at

text: the

text: equator

text: in

text: the

text: band

text: below

text: keV

text: and

text: above

text: keV

text: are

text: negligible.

text: For

text: ex-

text: model

text: is

text: represented

text: by

text: ample,

text: the

text: conversion

text: factor

text: from

text: the

text: keV

text: ﬂux

text: observed

text: µ

text: B

text: by

text: the

text: MAXI/GSC

text: (in

text: the

text: spin-up

text: phase)

text: to

text: the

text: keV

text: c

text: ﬂux

text: is

text: In

text: a

text: similar

text: way,

text: we

text: calculated

text: Fbol

text: in

text: the

text: past

text: If

text: the

text: source

text: emission

text: is

text: isotropic,

text: L

text: is

text: calculated

text: from

text: Fbol

text: observations,

text: and

text: present

text: the

text: results

text: in

text: table

text: and

text: the

text: distance

text: D

text: as

text: In

text: ﬁgure

text: we

text: plot

text: the

text: relation

text: between

text: the

text: observed

text: P˙

text: and

text: L

text: bol

text: the

text: calculated

text: F

text: including

text: the

text: past

text: data.

text: It

text: clearly

text: shows

text: bol

text: Since

text: the

text: pulsar

text: emission

text: is

text: anisotropic,

text: it

text: is

text: not

text: exactly

text: cor-

text: their

text: negative

text: correlation,

text: expected

text: from

text: the

text: pulsar

text: spin

text: up

text: due

text: rect.

text: We

text: employ

text: this

text: approximation

text: and

text: then

text: discuss

text: the

text: effect

text: to

text: the

text: accretion

text: torque.

text: Furthermore,

text: the

text: data

text: points

text: in

text: the

text: spin-

text: later.

text: up

text: and

text: spin-down

text: phases

text: apparently

text: deﬁnes

text: a

text: well-deﬁned

text: sin-

text: Combining

text: equations

text: and

text: as

text: well

text: as

text: the

text: gle

text: dependence

text: on

text: F

text: Application

text: obfolthe

text: Ghosh

text: Lamb

text: relation

text: expression

text: for

text: zg

text: [equation

text: we

text: obtain

text: a

text: theoretical

text: model

text: equation

text: to

text: describe

text: the

text: observed

text: P˙

text: –L

text: relation,

text: including

text: three

text: As

text: reviewed

text: in

text: section

text: derived

text: a

text: relation

text: between

text: unknown

text: parameters,

text: D,

text: M

text: and

text: R.

text: P˙

text: and

text: L

text: in

text: accreting

text: X-ray

text: pulsars,

text: assuming

text: that

text: the

text: accret-

text: ing

text: matter

text: transfers

text: its

text: angular

text: momentum

text: to

text: the

text: pulsar

text: at

text: the

text: Comparison

text: between

text: the

text: data

text: and

text: theory

text: “outer

text: transition

text: zone”,

text: r

text: [equation

text: The

text: equations

text: we

text: used

text: are

text: summarized

text: in

text: Appendix

text: Rappaport

text: Joss

text: In

text: order

text: to

text: determine

text: the

text: possible

text: parameter

text: ranges

text: of

text: D,

text: M

text: also

text: proposed

text: an

text: almost

text: equivalent

text: equation.

text: Since

text: their

text: model

text: and

text: R,

text: we

text: ﬁtted

text: the

text: observed

text: P˙

text: –F

text: relation

text: in

text: ﬁgure

text: with

text: bol

text: equation

text: includes

text: an

text: unknown

text: parameter

text: (ξv

text: /v

text: of

text: which

text: the

text: the

text: model

text: prepared

text: as

text: above.

text: When

text: errors

text: associated

text: with

text: some

text: r

text: ﬀ

text: relation

text: to

text: n(ω

text: in

text: the

text: relation

text: is

text: calculable.

text: Therefore,

text: past

text: measurements

text: of

text: P˙

text: are

text: unavailable,

text: we

text: assumed

text: an

text: arbi-

text: s

text: we

text: employ

text: the

text: relation.

text: trarily

text: small

text: value

text: (∆P˙

text: s

text: s

text: because

text: the

text: overall

text: ##LTFigure##

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text: 6 Publications of the Astronomical Society of Japan, (2014), Vol. 00, No. 0 Fig. 4. A relation between P˙ and Fbol by the MAXI/GSC and the past observations. Filled circles, open circles and squares represent the MAXI/GSC data, the Swift/BAT data and the others in table 1, respectively. The dashed-dotted horizontal line indicates P˙ = 0. The solid line is the best ﬁt model calculated by equation (3), assuming a distance of 10 kpc±. The parameters are M = 1.83M(cid:12) and R = 11.4 km, with χν2 of 2.9 for 37 deg±rees of freedom. Dashed two lines show the case when R is changed by 1 km with M kept unchanged, while dotted two lines those when M is varied by 0.3 M(cid:12) with R ﬁxed at 11.4 km. errors are dominated by those in F . This treatment was con- point and the slope, and their joint use allows us to simultane- bol ﬁrmed to little affect the ﬁtting results. Since it is difﬁcult to ously constrain two (in the present case M and R) out of the constrain all the three parameters simultaneously from the P˙ – three model parameters: the other one (D in the present case) F relation alone, we ﬁrst assumed the source distance D to be remains unconstrained. Below, let us consider physical mean- bol some values from 3 to 20 kpc, and then determined the allowed ings behind this model behavior. M–R regions as a function of the assumed distance. As an ex- In ﬁgure 4, the zero-cross point at the spin-up/down thresh- ample, the ﬁtting result assuming D = 10 kpc is shown in ﬁgure old represents a torque-equilibrium condition, wherein so- 4, where the best ﬁt values were obtained as M = 1.83 M(cid:12) and called co-rotation radius, which is almost uniquely determined R = 11.4 km (errors are considered later). by the observed P (with some dependence on M), can be To understand how the model curve depends on M and R, equated with magnetospheric radius, or the outer-transition ra- we show in ﬁgure 4 some predictions by equation (3) when ei- dius r in the GL79 model (Appendix 1). At this radius, the 0 ∝ ther M or R is slightly changed. Thus, changes in R (with D gravitational pressure calculated from L D2F should bal- bol and M ﬁxed) causes parallel displacements of the model with ance the magnetic pressure, and this condition speciﬁes the little changes in its slope, while those in M (with D and R ﬁxed) value of µ. By further comparing this µ with the accurately appears mainly in slope changes with the “zero-cross” point not measured B , we can constrain R via equation (5). As a result, c much affected. In other words, the observed P˙ –F relation the zero-cross point becomes more sensitively to R rather than bol has essentially two degrees of freedom, namely, the zero-cross to M. More quantitatively, the torque-equilibrium condition in ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: Publications

text: of

text: the

text: Astronomical

text: Society

text: of

text: Japan,

text: Vol.

text: No.

text: Fig.

text: A

text: relation

text: between

text: P˙

text: and

text: Fbol

text: by

text: the

text: MAXI/GSC

text: and

text: the

text: past

text: observations.

text: Filled

text: circles,

text: open

text: circles

text: and

text: squares

text: represent

text: the

text: MAXI/GSC

text: data,

text: the

text: Swift/BAT

text: data

text: and

text: the

text: others

text: in

text: table

text: respectively.

text: The

text: dashed-dotted

text: horizontal

text: line

text: indicates

text: P˙

text: The

text: solid

text: line

text: is

text: the

text: best

text: ﬁt

text: model

text: calculated

text: by

text: equation

text: assuming

text: a

text: distance

text: of

text: kpc±.

text: The

text: parameters

text: are

text: M

text: and

text: R

text: km,

text: with

text: χν2

text: of

text: for

text: deg±rees

text: of

text: freedom.

text: Dashed

text: two

text: lines

text: show

text: the

text: case

text: when

text: R

text: is

text: changed

text: by

text: km

text: with

text: M

text: kept

text: unchanged,

text: while

text: dotted

text: two

text: lines

text: those

text: when

text: M

text: is

text: varied

text: by

text: with

text: R

text: ﬁxed

text: at

text: km.

text: errors

text: are

text: dominated

text: by

text: those

text: in

text: F

text: This

text: treatment

text: was

text: con-

text: point

text: and

text: the

text: slope,

text: and

text: their

text: joint

text: use

text: allows

text: us

text: to

text: simultane-

text: bol

text: ﬁrmed

text: to

text: little

text: affect

text: the

text: ﬁtting

text: results.

text: Since

text: it

text: is

text: difﬁcult

text: to

text: ously

text: constrain

text: two

text: (in

text: the

text: present

text: case

text: M

text: and

text: R)

text: out

text: of

text: the

text: constrain

text: all

text: the

text: three

text: parameters

text: simultaneously

text: from

text: the

text: P˙

text: three

text: model

text: parameters:

text: the

text: other

text: one

text: (D

text: in

text: the

text: present

text: case)

text: F

text: relation

text: alone,

text: we

text: ﬁrst

text: assumed

text: the

text: source

text: distance

text: D

text: to

text: be

text: remains

text: unconstrained.

text: Below,

text: let

text: us

text: consider

text: physical

text: mean-

text: bol

text: some

text: values

text: from

text: to

text: kpc,

text: and

text: then

text: determined

text: the

text: allowed

text: ings

text: behind

text: this

text: model

text: behavior.

text: M–R

text: regions

text: as

text: a

text: function

text: of

text: the

text: assumed

text: distance.

text: As

text: an

text: ex-

text: In

text: ﬁgure

text: the

text: zero-cross

text: point

text: at

text: the

text: spin-up/down

text: thresh-

text: ample,

text: the

text: ﬁtting

text: result

text: assuming

text: D

text: kpc

text: is

text: shown

text: in

text: ﬁgure

text: old

text: represents

text: a

text: torque-equilibrium

text: condition,

text: wherein

text: so-

text: where

text: the

text: best

text: ﬁt

text: values

text: were

text: obtained

text: as

text: M

text: and

text: called

text: co-rotation

text: radius,

text: which

text: is

text: almost

text: uniquely

text: determined

text: R

text: km

text: (errors

text: are

text: considered

text: later).

text: by

text: the

text: observed

text: P

text: (with

text: some

text: dependence

text: on

text: M),

text: can

text: be

text: To

text: understand

text: how

text: the

text: model

text: curve

text: depends

text: on

text: M

text: and

text: R,

text: equated

text: with

text: magnetospheric

text: radius,

text: or

text: the

text: outer-transition

text: ra-

text: we

text: show

text: in

text: ﬁgure

text: some

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