text: 22 Liu et al. TABLE 1 — Continued (cid:82) (cid:82) IRAS HCN (4-3) CS (7-6) ∗ ∗ ∗ ∗ T dV V FWHM T Proﬁle T dV V FWHM T Proﬁle A − ls−r − A A − ls−r − A (K km s 1) (km s 1) (km s 1) (K) (K km s 1) (km s 1) (km s 1) (K) 16362-4639 2.68(0.13) -37.38(0.12) 5.07(0.31) 0.50 G 0.53(0.07) -37.93(0.12) 1.85(0.30) 0.27 G 16372-4545 10.64(0.33) -57.54(0.06) 5.69(0.18) 1.76 R 8.72(0.26) -58.44(0.06) 4.48(0.11) 1.83 R 16385-4619 21.37(0.30) -117.85(0.07) 10.59(0.26) 1.90 B 19.40(0.16) -118.02(0.03) 6.45(0.07) 2.82 G 16424-4531 17.39(0.30) -31.06(0.05) 7.89(0.19) 2.07 G 3.58(0.18) -34.57(0.07) 2.96(0.20) 1.13 G 16445-4459 10.37(0.23) -121.43(0.07) 6.92(0.22) 1.41 G 4.55(0.15) -122.31(0.08) 5.42(0.24) 0.79 G 16458-4512 17.40(0.36) -49.92(0.07) 7.22(0.24) 2.27 R 4.57(0.24) -50.75(0.09) 4.18(0.29) 1.03 G 16484-4603 25.05(0.62) -31.58(0.08) 7.53(0.32) 3.13 R 16.30(0.23) -32.19(0.03) 4.76(0.11) 3.21 G 16487-4423 6.76(0.17) -43.24(0.06) 5.75(0.19) 1.11 R 2.35(0.11) -43.92(0.08) 3.45(0.19) 0.64 G 16489-4431 6.20(0.18) -40.40(0.07) 5.06(0.19) 1.15 G 2.75(0.11) -40.74(0.06) 3.18(0.16) 0.81 G 16506-4512 19.81(0.39) -25.80(0.03) 4.31(0.07) 4.31 R 12.18(0.21) -25.84(0.02) 2.84(0.04) 4.02 R 16524-4300 19.11(0.38) -40.56(0.05) 5.38(0.07) 3.34 R 9.87(0.11) -41.30(0.02) 3.49(0.05) 2.66 G 16547-4247 117.07(1.30) -29.73(0.08) 17.16(0.31) 6.41 B 53.42(0.37) -30.27(0.03) 9.95(0.11) 5.05 B 16562-3959 71.50(0.48) -11.90(0.02) 6.85(0.06) 9.80 R 38.38(0.22) -12.53(0.01) 4.55(0.03) 7.93 G 16571-4029 20.79(1.13) -14.69(0.12) 5.65(0.41) 3.46 R 14.24(0.27) -15.03(0.03) 3.12(0.07) 4.29 G 17006-4215 15.92(0.26) -24.22(0.05) 5.71(0.11) 2.62 G 13.40(0.13) -24.69(0.01) 3.06(0.04) 4.12 G 17008-4040 32.02(0.67) -15.93(0.07) 6.67(0.12) 4.51 R 20.00(0.24) -16.67(0.03) 5.01(0.07) 3.75 G 17016-4124 83.23(1.44) -26.48(0.11) 12.89(0.45) 6.07 R 32.83(0.42) -26.80(0.05) 9.12(0.15) 3.38 G 17136-3617 27.31(0.16) -10.27(0.01) 4.99(0.04) 5.14 G 20.77(0.11) -10.99(0.01) 3.47(0.02) 5.62 G 17143-3700 11.09(0.26) -31.32(0.07) 7.94(0.24) 1.31 B 7.38(0.13) -31.81(0.04) 4.56(0.10) 1.52 G 17158-3901 35.28(0.66) -15.85(0.07) 10.25(0.29) 3.23 R 20.34(0.14) -16.19(0.00) 6.08(0.05) 3.14 G 17160-3707 18.45(0.43) -69.65(0.06) 7.47(0.19) 2.32 B 10.52(0.14) -69.85(0.03) 4.53(0.07) 2.18 G 17175-3544 91.17(1.20) -5.74(0.06) 11.00(0.25) 7.79 R 84.85(0.19) -6.54(0.01) 6.55(0.02) 12.17 G 17204-3636 14.48(0.82) -17.76(0.10) 5.66(0.45) 2.40 B 6.05(0.16) -17.94(0.05) 3.66(0.12) 1.55 G 17220-3609 41.77(0.40) -95.04(0.06) 12.73(0.23) 3.08 B 35.20(0.16) -94.67(0.02) 6.65(0.04) 4.97 G 17233-3606 182.37(0.81) -4.15(0.04) 21.04(0.15) 8.14 R 113.92(0.84) -3.16(0.03) 10.33(0.14) 10.36 R 17244-3536 5.39(0.12) -9.36(0.04) 3.87(0.10) 1.31 G 2.63(0.10) -10.50(0.05) 2.84(0.13) 0.87 G 17258-3637 46.49(1.51) -11.68(0.06) 7.64(0.19) 5.72 R 35.66(0.29) -12.11(0.02) 5.38(0.04) 6.22 R 17269-3312 3.36(0.22) -20.70(0.20) 5.57(0.82) 0.57 G 0.69(0.08) -21.60(0.17) 3.06(0.40) 0.21 G 17271-3439 30.21(2.53) -15.86(0.43) 10.27(3.36) 2.76 B 24.24(0.33) -15.40(0.03) 4.55(0.07) 5.01 G 17278-3541 7.40(0.23) 0.73(0.07) 4.71(0.21) 1.48 G 3.19(0.14) 0.35(0.08) 3.58(0.15) 0.84 G 17439-2845 4.54(0.15) 18.76(0.07) 4.32(0.17) 0.99 G 1.56(0.13) 18.15(0.11) 3.90(0.55) 0.38 G 17441-2822 186.66(40.2) 51.49(1.53) 15.12(1.81) 11.60 B 262.61(1.23) 60.05(0.03) 22.95(0.10) 10.75 B 17455-2800 17.20(0.15) -14.19(0.02) 5.62(0.06) 2.88 G 10.28(0.13) -14.79(0.02) 3.87(0.06) 2.50 G 17545-2357 5.85(0.11) 9.16(0.04) 3.98(0.09) 1.38 G 2.58(0.08) 8.57(0.04) 2.99(0.10) 0.81 G 17589-2312 6.92(0.49) 21.64(0.18) 6.07(0.25) 1.07 R 1.79(0.07) 20.81(0.08) 4.05(0.17) 0.42 G 17599-2148 12.69(0.42) 19.75(0.06) 5.34(0.12) 2.23 R 6.28(0.09) 19.16(0.04) 4.79(0.08) 1.23 G 18032-2032 47.73(0.78) 5.86(0.10) 12.55(0.26) 3.57 R 32.22(0.46) 4.49(0.05) 8.00(0.17) 3.78 R 18056-1952 51.66(1.12) 66.70(0.10) 11.03(0.33) 4.40 B 26.98(0.43) 66.75(0.07) 9.34(0.17) 2.71 B 18075-2040 – – – – – – – – – – 18079-1756 14.09(0.16) 18.39(0.02) 4.58(0.08) 2.89 G 8.67(0.10) 17.68(0.01) 2.70(0.04) 3.02 G 18089-1732 26.06(0.96) 33.52(0.06) 6.39(0.23) 3.83 B 16.48(0.14) 33.13(0.02) 4.65(0.06) 3.33 B 18110-1854 18.81(0.29) 39.04(0.03) 5.90(0.14) 2.99 R 8.88(0.08) 38.55(0.02) 3.72(0.04) 2.24 G 18116-1646 23.90(0.53) 50.25(0.03) 6.71(0.12) 3.35 B 21.86(1.22) 49.47(0.06) 4.02(0.13) 5.11 B 18117-1753 34.21(0.87) 38.00(0.09) 9.14(0.29) 3.52 R 20.85(0.28) 36.46(0.04) 6.47(0.06) 3.03 G 18134-1942 10.81(0.12) 10.74(0.02) 3.50(0.05) 2.90 G 5.15(0.07) 10.14(0.02) 2.44(0.04) 1.99 G 18139-1842 13.05(0.84) 39.98(0.05) 4.44(0.21) 2.76 R 7.44(0.12) 39.61(0.03) 3.39(0.06) 2.06 G 18159-1648 39.77(0.77) 23.88(0.07) 7.66(0.26) 4.88 R 28.10(0.13) 22.38(0.01) 5.42(0.03) 4.87 G 18182-1433 14.95(0.48) 60.11(0.10) 7.39(0.29) 1.90 B 6.99(0.25) 59.51(0.10) 5.47(0.18) 1.20 G 18223-1243 3.85(0.14) 45.97(0.07) 4.29(0.19) 0.84 G 1.67(0.09) 45.15(0.06) 2.38(0.18) 0.66 G 18228-1312 11.00(0.42) 33.64(0.08) 5.44(0.16) 1.90 B 4.62(0.33) 33.69(0.12) 3.86(0.17) 1.13 R 18236-1205 17.34(0.47) 27.81(0.10) 8.51(0.47) 1.92 R 6.17(0.25) 26.81(0.09) 5.37(0.29) 1.08 G 18264-1152 32.26(0.59) 45.38(0.07) 8.46(0.20) 3.58 G 12.06(0.21) 43.95(0.03) 4.49(0.10) 2.52 G 18290-0924 – – – – – – – – – – 18308-0503 2.89(0.14) 43.56(0.07) 3.09(0.19) 0.88 G 0.71(0.07) 43.01(0.08) 1.59(0.20) 0.42 G 18311-0809 13.71(0.48) 113.50(0.10) 5.62(0.24) 2.29 G 4.46(0.11) 112.99(0.04) 3.81(0.11) 1.10 G 18314-0720 3.75(0.17) 102.14(0.17) 7.81(0.44) 0.45 G 1.38(0.12) 100.96(0.11) 2.91(0.33) 0.45 G 18316-0602 17.96(0.31) 45.28(0.05) 6.65(0.15) 2.54 G 13.20(0.16) 42.80(0.03) 6.19(0.10) 2.00 G 18317-0513 8.52(0.14) 42.21(0.03) 3.67(0.07) 2.18 G 4.26(0.09) 41.58(0.03) 2.45(0.06) 1.63 G 18317-0757 10.86(0.20) 80.52(0.05) 5.32(0.12) 1.92 G 7.00(0.08) 80.07(0.02) 3.09(0.04) 2.13 G 18341-0727 8.93(0.25) 113.49(0.09) 6.19(0.27) 1.36 G 2.30(0.11) 112.92(0.09) 3.57(0.20) 0.61 G 18411-0338 7.39(0.35) 103.68(0.10) 4.84(0.31) 1.44 G 7.58(0.51) 103.66(0.12) 5.02(0.46) 1.42 G 18434-0242 28.42(0.17) 98.06(0.01) 5.08(0.04) 5.25 G 50.68(0.76) 98.77(0.05) 7.17(0.14) 6.64 G 18440-0148 2.82(0.16) 98.28(0.12) 4.12(0.27) 0.64 G – – – – – 18445-0222 5.79(0.17) 87.21(0.06) 4.48(0.15) 1.22 G 3.44(0.11) 87.05(0.05) 3.19(0.12) 1.01 G 18461-0113 14.34(0.37) 97.10(0.09) 7.61(0.37) 1.77 R 8.50(0.14) 96.25(0.04) 4.40(0.09) 1.81 G 18469-0132 12.31(0.26) 86.87(0.09) 8.21(0.20) 1.41 G 7.03(0.12) 86.51(0.04) 5.07(0.10) 1.30 G 18479-0005 25.37(0.69) 15.10(0.08) 10.22(0.35) 2.33 R 17.67(0.20) 14.62(0.05) 8.16(0.10) 2.03 G 18502+0051 – – – – – – – – – – 18507+0110 74.45(0.82) 58.46(0.03) 8.30(0.09) 8.43 B 48.73(0.51) 58.47(0.03) 6.29(0.06) 7.28 B 18507+0121 14.51(0.30) 58.19(0.06) 6.20(0.20) 2.20 R 3.17(0.15) 57.89(0.08) 3.77(0.31) 0.79 G 18517+0437 11.55(0.36) 44.08(0.06) 5.31(0.18) 2.04 G 5.54(0.12) 43.70(0.04) 3.33(0.09) 1.56 G 18530+0215 7.62(0.27) 77.78(0.08) 4.89(0.21) 1.46 G 4.38(0.13) 77.26(0.04) 2.97(0.10) 1.38 G ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: Liu

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text: 4 Liu et al. 3. RESULTS 3.2. Line luminosities 3.1. Observational results of Molecular lines Assuming a Gaussian brightness distribution for the source and a Gaussian beam, molecular line luminosities We detected HCN (4-3) toward 141 clumps and CS (7- (cid:48) L can be derived following Wu et al. (2010) 6) toward 137 clumps. In Figure 1, we show the HCN (4- mol 3) and CS (7-6) spectra of two sources for example. For × IRAS 18317-0757, both HCN (4-3) and CS (7-6) spec- (cid:48) × − × × π θ2 × θ(cid:90)2 + θ2 tra can be well ﬁtted with single Gaussian proﬁles. For L mol = 23.5 10 6 D2 ( 4ln2s ) ( s θ2beam ) IRAS 09018-4816, CS (7-6) has a Gaussian proﬁle, while s × ∗ HCN (4-3) shows asy∼mmetric proﬁles with an absorption T dV/η dip. There are 74 ( 52%) clumps showing asymmetric A proﬁles in HCN (4-3) like IRAS 09018-4816. While there (1) are only 22 (16%) clumps showing asymmetric proﬁles in CS (7-6). The line proﬁles were classiﬁed into three Here D is the distance in kpc obtained from Fau´ndez categories as shown in the 6th and 11th columns in Ta- et al. (2004), and θs and θbeam are the size of the line ble 1. Those denoted with “G” have symmetric proﬁles emission source and of the beam in arcsecond, and η which can be well ﬁtted with Gaussian proﬁles. Those is the main beam eﬃciency. Since we only carried out denoted with “B” or “R” have asymmetric proﬁles with single pointing observations and thus can not determine their observed emission peaks blueshifted or redshifted the exact line emission area, we here assume that the with respect to the peak velocities derived from Gaus- source sizes of HCN (4-3) and CS (7-6) are the same as sian ﬁts, respectively. In this work, we will not discuss that of 1.2 mm continuum emission in Fau´ndez et al. the line proﬁles in details. The line proﬁles will be mod- (2004). The caveats in estimating line luminosity will be elled and discussed in another paper for kinematics (e.g., discussed in section 4.1. (cid:48) (cid:48) infall and outﬂow) studies. The derived HCN (4-3) (L ) and CS (7-6) (L ) HCN CS Many lines especially HCN (4-3) lines with asymmet- luminosities are listed in the 4th(cid:48) and 5th co(cid:48)lumns o∼f Ta- ric proﬁles show absorption dips due to self-absorption. ble 2. The median values of L and L are 16.4 ∼ − HCN CS Previous works like Wu et al. (2005, 2010) and Stephens and 9.2 K km s 1 pc2, respectively. The mean values (cid:48) (cid:48) ∼ ∼ − et al. (2016) simply used Moment 0 method to calculat- of L and L are 62.5 and 56.6 K km s 1 pc2, HCN CS ing the integrated intensity of such optically thick line, respectively. which may severely underestimate the total emission in- 3.3. Gravitational Stability of dense clumps cluding the blocked radiation due to self-absorption (van To examine the gravitational stability of those dense Kempen et al. 2009; Ossenkopf et al. 2010). Gaussian clumps, we derive one dim(cid:115)ensional virial velocity disper- ﬁts were usually performed to obtain a rough correction sions (σ ): for self-absorption and to compare with radiation trans- vir fer models (van (cid:82)Kempen et al. 2009; Ossenkopf et al. γM G 2010). In this work, we ﬁtted all the detected spectra σ = clump (2) with Gaussian funct∗ions and present the results (inte- vir 5Rclump grated intensity T dV , systemic velocity V , the full A lsr width of half maximum (FWHM) and peak antenna tem- here M is the clump mass, G is the gravita- ∗ clump perature T ) in Table 1. Due to self-absorption, most tional constant, and R is the radius of the dense A clump HCN (4-3) lines have larger errors in the integrated in- clump. M and R were taken from Fau´ndez et clump clump tensities from Gaussian ﬁt than CS (7-6) lines. In Figure al. (2004). Parameter γ is the geometric factor equal to 2, we compare the integrated intensities inferred from unity for a uniform density proﬁle and 5/3 for an inverse Gaussian ﬁts with the values simply obtained from inte- square proﬁle (Williams, de Geus,& Blitz 1994). Here grating the lines (Moment 0). The integrated intensities we take γ equal to 5/3. The derived one dimensional obtained from diﬀerent methods are very consistent and virial velocity dispersions (σ ) are listed in the 6th vir roughly linearly correlated. Therefore, we claim that the column of Table 2. integrated intensities obtained from Gaussian ﬁts are re- Assuming the line proﬁles are Gaussian, the one di- liable. mensional velocity dis(cid:112)persions of molecular lines are The median line widths for HCN (4-3) lines with sym- ∼ metric proﬁles and with asymmetric proﬁles are 5.3 FWHM FWHM ∼ − σ = = . (3) and 8.1 km s 1, respectively. The median line widths line 2 2ln(2) 2.355 for CS (7-6) lines with symmetric proﬁles and with asym- ∼ ∼ − metric proﬁles are 4.0 and 7.5 km s 1, respectively. We derived σ and σ for HCN (4-3) and CS (7- HCN CS In general, lines with asymmetric proﬁles have much 6), respectively. The dense clumps may be under gravi- larger line widths than lines with symmetric proﬁles, in- tational collapse if σ is larger than σ . The median vir ∼ line dicating that those clumps with asymmetric line proﬁles ratio of σ to σ is 1.2, and the median ratio of HCN ∼ vir may be more turbulent and more likely be aﬀected by σ to σ is 0.8, indicating that most sources should CS vir stellar feedback or bulk motions (e.g. infall, outﬂows). be in virial equilibrium. There are 45 dense clumps hav- CS (7-6) lines usually have smaller line widths than HCN ing both σ and σ smaller than σ , indicating HCN CS vir (4-3). T∼he median line width ratio of CS (7-6) to HCN that the gas motions may not provide suﬃcient support (4-3) is 0.67. against the local gravitational collapse in these clumps. In contrast, there are 37 dense clumps that have both σ and σ larger than σ , indicating that these HCN CS vir ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

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text: and

text: virial

text: velocity

text: dispersions

text: (σ

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text: listed

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text: the

text: vir

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text: Therefore,

text: we

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text: are

text: Gaussian,

text: the

text: one

text: di-

text: liable.

text: mensional

text: velocity

text: of

text: molecular

text: lines

text: are

text: The

text: median

text: line

text: widths

text: for

text: HCN

text: lines

text: with

text: sym-

text: metric

text: proﬁles

text: and

text: with

text: asymmetric

text: proﬁles

text: are

text: FWHM

text: FWHM

text: σ

text: and

text: km

text: s

text: respectively.

text: The

text: median

text: line

text: widths

text: line

text: for

text: CS

text: lines

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text: symmetric

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text: and

text: with

text: asym-

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text: proﬁles

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text: km

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text: We

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text: σ

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text: σ

text: for

text: HCN

text: and

text: CS

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text: CS

text: In

text: general,

text: lines

text: with

text: asymmetric

text: proﬁles

text: have

text: much

text: respectively.

text: The

text: dense

text: clumps

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text: under

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text: larger

text: line

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text: lines

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text: symmetric

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text: in-

text: tational

text: collapse

text: if

text: σ

text: is

text: larger

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text: σ

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text: median

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text: line

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text: that

text: those

text: clumps

text: with

text: asymmetric

text: line

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text: ratio

text: of

text: σ

text: to

text: σ

text: is

text: and

text: the

text: median

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text: HCN

text: vir

text: may

text: be

text: more

text: turbulent

text: and

text: more

text: likely

text: be

text: aﬀected

text: by

text: σ

text: to

text: σ

text: is

text: indicating

text: that

text: most

text: sources

text: should

text: CS

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text: feedback

text: or

text: bulk

text: motions

text: (e.g.

text: infall,

text: outﬂows).

text: be

text: in

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text: There

text: are

text: dense

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text: than

text: HCN

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text: σ

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text: HCN

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text: provide

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text: support

text: is

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text: local

text: gravitational

text: collapse

text: in

text: these

text: clumps.

text: In

text: contrast,

text: there

text: are

text: dense

text: clumps

text: that

text: have

text: both

text: σ

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text: ##LTLine##

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text: Star formation laws in both Galactic massive clumps and external galaxies 5 Fig. 1.— Spectra of IRAS 09018-4816 (left panels) and IRAS 18317-0757 (right panels). The upper panels show CS (7-6) lines. The lower panels show HCN (4-3) lines. The green lines are Gaussian ﬁts. clumps may be more turbulent and unbound. However, a broad line proﬁle can also be caused by infall. Among LTIR = 101.80±0.15 Mclump (4) the 37 sources, 11 show “blue proﬁle” with blueshifted L(cid:12) M(cid:12) emission peak stronger than redshifted one, a signature for collapse (Zhou et al. 1993), indicating that these 11 Following Kennicutt & Evans (2012), we can convert sources may be also bound and probably be in global L to a star formation rate (SFR; M˙ ∗) as: TIR collapse. Further mapping studies will tell whether they are really in collapse or not. M˙ ∗ L − log( − ) = log( TIR− ) 43.41 (5) M(cid:12) yr 1 ergs s 1 3.4. The Kennicutt-Schmidt law revealed by 1.2 mm continuum The use of extragalactic relation between L and TIR In panel (a) of Figure 3, we ﬁrst examine the cor- SFR may signiﬁcantly underestimate the SFR for low- relation between total infrared luminosities (LTIR) and mass molecular clumps, where the initial mass func- Clump masses (Mclump) for Galactic clumps. LTIR was tion (IMF) is not fully sampled (Wu et al. 2005; Vuti- obtained from ﬁt of the SED (consisting of the 1.2 mm salchavakul & Evans 2013). While L can well trace TIR ﬂux and the ﬂuxes in the four IRAS bands), and Mclump SFR for dense clumps having LTIR exceed 104.5 L(cid:12) (Wu was derived from 1.2 mm continuum (Fau´ndez et al. et al. 2005; Vutisalchavakul & Evans 2013). In our 2004). In section 4.1, we will show that the LTIR ob- sample, 71% sources have L exceed 104.5 L(cid:12) and tained from integrating the SED in our work are con- 92% have L exceed 104 L(cid:12)TI.RIn addition, we do not sistent with the total infrared luminosities derived only TIR see clear breakup in the L and M correlation. from the four IRAS bands as used in other works(Wu TIR clump Therefore, the applying of extragalactic relation on our et al. 2005, 2010). We use L for the total infrared IR sample may not severely underestimate the SFR. luminosities derived only from the four IRAS bands to From Equation (4) and (5), we can derive the relation distinguish from the total infrared luminosities derived between M˙ ∗ and M: from SED ﬁt (L ). The least squares ﬁt to all the TIR data gave a super-linear slope due to the contamination M˙ ∗ × − M of several low luminosity sources which are located obvi- − = (0.93+−0.38) 10 8 clump (6) ously below the ﬁt. The clumps with luminosities lower M(cid:12) yr 1 0.27 M(cid:12) than 103 L(cid:12) are likely low-mass star forming regions. If we only consider the clumps with LTIR larger than 103 Then, the gas depletion time τdep is: L(cid:12), the least squares ﬁt indicates that the relation be- M tLweenaLnTdIRMand Mcaisnlibneeaerx.pTrehseseldineaas:r relation between τdep = Mcl˙u∗mp = 107+−4341 Myr (7) TIR clump ##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## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##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: Star

text: formation

text: laws

text: in

text: both

text: Galactic

text: massive

text: clumps

text: and

text: external

text: galaxies

text: Fig.

text: Spectra

label: caption

text: of

text: IRAS

text: (left

text: panels)

text: and

text: IRAS

text: (right

text: panels).

text: The

text: upper

text: panels

text: show

text: CS

text: lines.

text: The

text: lower

text: panels

text: show

text: HCN

text: lines.

text: The

text: green

text: lines

text: are

text: Gaussian

text: ﬁts.

text: clumps

text: may

text: be

text: more

text: turbulent

text: and

text: unbound.

text: However,

text: a

text: broad

text: line

text: proﬁle

text: can

text: also

text: be

text: caused

text: by

text: infall.

text: Among

text: LTIR

text: Mclump

text: the

text: sources,

text: show

text: “blue

text: proﬁle”

text: with

text: blueshifted

text: emission

text: peak

text: stronger

text: than

text: redshifted

text: one,

text: a

text: signature

text: for

text: collapse

text: (Zhou

text: et

text: al.

text: indicating

text: that

text: these

text: Following

text: Kennicutt

text: Evans

text: we

text: can

text: convert

text: sources

text: may

text: be

text: also

text: bound

text: and

text: probably

text: be

text: in

text: global

text: L

text: to

text: a

text: star

text: formation

text: rate

text: (SFR;

text: M˙

text: as:

text: TIR

text: collapse.

text: Further

text: mapping

text: studies

text: will

text: tell

text: whether

text: they

text: are

text: really

text: in

text: collapse

text: or

text: not.

text: M˙

text: L

text: log(

text: log(

text: TIR−

text: yr

text: ergs

text: s

text: The

text: Kennicutt-Schmidt

text: law

text: revealed

text: by

text: mm

text: continuum

text: The

text: use

text: of

text: extragalactic

text: relation

text: between

text: L

text: and

text: TIR

text: In

text: panel

text: (a)

text: of

text: Figure

text: we

text: ﬁrst

text: examine

text: the

text: cor-

text: SFR

text: may

text: signiﬁcantly

text: underestimate

text: the

text: SFR

text: for

text: low-

text: relation

text: between

text: total

text: infrared

text: luminosities

text: (LTIR)

text: and

text: mass

text: molecular

text: clumps,

text: where

text: the

text: initial

text: mass

text: func-

text: Clump

text: masses

text: (Mclump)

text: for

text: Galactic

text: clumps.

text: LTIR

text: was

text: tion

text: (IMF)

text: is

text: not

text: fully

text: sampled

text: (Wu

text: et

text: al.

text: Vuti-

text: obtained

text: from

text: ﬁt

text: of

text: the

text: SED

text: (consisting

text: of

text: the

text: mm

text: salchavakul

text: Evans

text: While

text: L

text: can

text: well

text: trace

text: TIR

text: ﬂux

text: and

text: the

text: ﬂuxes

text: in

text: the

text: four

text: IRAS

text: bands),

text: and

text: Mclump

text: SFR

text: for

text: dense

text: clumps

text: having

text: LTIR

text: exceed

text: (Wu

text: was

text: derived

text: from

text: mm

text: continuum

text: (Fau´ndez

text: et

text: al.

text: et

text: al.

text: Vutisalchavakul

text: Evans

text: In

text: our

text: In

text: section

text: we

text: will

text: show

text: that

text: the

text: LTIR

text: ob-

text: sample,

text: sources

text: have

text: L

text: exceed

text: and

text: tained

text: from

text: integrating

text: the

text: SED

text: in

text: our

text: work

text: are

text: con-

text: have

text: L

text: exceed

text: addition,

text: we

text: do

text: not

text: sistent

text: with

text: the

text: total

text: infrared

text: luminosities

text: derived

text: only

text: TIR

text: see

text: clear

text: breakup

text: in

text: the

text: L

text: and

text: M

text: correlation.

text: from

text: the

text: four

text: IRAS

text: bands

text: as

text: used

text: in

text: other

text: works(Wu

text: TIR

text: clump

text: Therefore,

text: the

text: applying

text: of

text: extragalactic

text: relation

text: on

text: our

text: et

text: al.

text: We

text: use

text: L

text: for

text: the

text: total

text: infrared

text: IR

text: sample

text: may

text: not

text: severely

text: underestimate

text: the

text: SFR.

text: luminosities

text: derived

text: only

text: from

text: the

text: four

text: IRAS

text: bands

text: to

text: From

text: Equation

text: and

text: we

text: can

text: derive

text: the

text: relation

text: distinguish

text: from

text: the

text: total

text: infrared

text: luminosities

text: derived

text: between

text: M˙

text: and

text: M:

text: from

text: SED

text: ﬁt

text: (L

text: The

text: least

text: squares

text: ﬁt

text: to

text: all

text: the

text: TIR

text: data

text: gave

text: a

text: super-linear

text: slope

text: due

text: to

text: the

text: contamination

text: M˙

text: M

text: of

text: several

text: low

text: luminosity

text: sources

text: which

text: are

text: located

text: obvi-

text: clump

text: ously

text: below

text: the

text: ﬁt.

text: The

text: clumps

text: with

text: luminosities

text: lower

text: yr

text: than

text: are

text: likely

text: low-mass

text: star

text: forming

text: regions.

text: If

text: we

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text: Synchronization failure caused by interplay between noise and network heterogeneity Synchronization failure caused by interplay between noise and network heterogeneitya) Y. Kobayashi1, b) and H. Kori1, c) (Dated: 29 May 2020) We investigate synchronization in complex networks of noisy phase oscillators. We ﬁnd that, while too weak a coupling is not suﬃcient for the whole system to synchronize, too strong a coupling induces a nontrivial type of phase slip among oscillators, resulting in synchronization failure. Thus, an intermediate coupling range for synchronization exists, which becomes narrower when the network is more heterogeneous. Analyses of two noisy oscillators reveal that nontrivial phase slip is a generic phenomenon when noise is present and coupling is strong. Therefore, the low synchronizability of heterogeneous networks can be understood as a result of the diﬀerence in eﬀective coupling strength among oscillators with diﬀerent degrees; oscillators with high degrees tend to undergo phase slip while those with low degrees have weak coupling strengths that are insuﬃcient for synchronization. Keywords: synchronization, heterogeneous network, noise Synchronization phenomena are found exist both lower and upper thresholds of the coupling in various systems, where the maintenance strength13, and thus, global synchronization tends to fail of synchronization is quite often crucial for for heterogeneous networks14. However, only a few ex- proper functioning. Such systems are rep- amples of such chaotic oscillators with a similar property resented by mutually interacting oscillators, are known. Moreover, for periodic oscillators, little has and their collective dynamics depend both been reported about the potential negative eﬀect of net- on the interaction function between a pair work heterogeneity on synchronization. of connected oscillators and on the network We have recently investigated a phase oscillator that is structure among the oscillators. Here, we unilaterally inﬂuenced by a pacemaker and is also under show that when such coupled oscillators are noise15 and found nontrivial phase slip in the strong cou- under the inﬂuence of noise, too strong a cou- pling regime; this implies that synchronization is possi- pling induces nontrivial phase slip. Moreover, ble only for intermediate coupling strengths. It has been we show that heterogeneous networks, which shown that this reentrant transition is observed for gen- have a wide dispersion of network connec- eral interaction functions. Although it is not at all ob- tions among individual oscillators, are more vious whether dynamical behavior obtained in the phase strongly aﬀected by the nontrivial phase slip. model with strong coupling is also reproduced in limit- While synchronization failure is known to cycle oscillators, it has been demonstrated that the Brus- occur in heterogeneous networks of a par- selator model, a typical system of limit cycle oscillators, ticular class of chaotic oscillators, our study does show the reentrant transition. Such nontrivial phase demonstrates the diﬃculty of synchroniza- slip can be another source of instability for a population tion in heterogeneous networks of periodic of network-coupled oscillators, and thus it is important oscillators under the inﬂuence of noise. to understand how diﬀerent networks respond to this new type of instability. In this study, we show that heterogeneous networks I. INTRODUCTION are more prone to phase slip. The essence of this be- havior can be understood from the phase diagram of two mutually coupled phase oscillators. Mutually coupled os- Synchronized oscillation of active elements can be ob- cillators fail to synchronize for too weak and too strong served in various ﬁelds, including biology, engineering, ecosystem, and chemical systems1–5. In many cases, syn- coupling strengths when they are subjected to noise. Be- cause of this property, synchronization is easily violated chronization of the entire system is required for proper in heterogeneous networks. functioning under various types of noise and heterogene- ity. Important examples include the heart (a population This paper is organized as follows: in Sec. , we inves- of cardiac cells)4,6, the circadian pacemaker (a popula- tigate a system of two mutually interacting oscillators tion of clock cells)7,8, and power grids9–11. under noise and compare it with two unilaterally con- Oscillators are often connected through a complex net- nected oscillators, which have been studied previously; work, where heterogeneity in network connectivity may two main causes of frequency drop, nontrivial phase slip critically hamper synchronization. Such a case is actu- and oscillation death, are discussed in Sec. and Sec. , ally observed in a special type of chaotic oscillators. For respectively. Then, in Sec. we present the results for the synchronization of these chaotic oscillators12, there network-connected oscillators.

text: Synchronization

text: failure

text: caused

text: by

text: interplay

text: between

text: noise

text: and

text: network

text: heterogeneity

text: Synchronization

label: title

text: failure

text: caused

text: by

text: interplay

text: between

text: noise

text: and

text: network

text: heterogeneitya)

text: Y.

label: author

text: b)

text: and

text: H.

text: c)

text: (Dated:

label: date

text: May

text: We

label: abstract

text: investigate

text: synchronization

text: in

text: complex

text: networks

text: of

text: noisy

text: phase

text: oscillators.

text: We

text: ﬁnd

text: that,

text: while

text: too

text: weak

text: a

text: coupling

text: is

text: not

text: suﬃcient

text: for

text: the

text: whole

text: system

text: to

text: synchronize,

text: too

text: strong

text: a

text: coupling

text: induces

text: a

text: nontrivial

text: type

text: of

text: phase

text: slip

text: among

text: oscillators,

text: resulting

text: in

text: synchronization

text: failure.

text: Thus,

text: an

text: intermediate

text: coupling

text: range

text: for

text: synchronization

text: exists,

text: which

text: becomes

text: narrower

text: when

text: the

text: network

text: is

text: more

text: heterogeneous.

text: Analyses

text: of

text: two

text: noisy

text: oscillators

text: reveal

text: that

text: nontrivial

text: phase

text: slip

text: is

text: a

text: generic

text: phenomenon

text: when

text: noise

text: is

text: present

text: and

text: coupling

text: is

text: strong.

text: Therefore,

text: the

text: low

text: synchronizability

text: of

text: heterogeneous

text: networks

text: can

text: be

text: understood

text: as

text: a

text: result

text: of

text: the

text: diﬀerence

text: in

text: eﬀective

text: coupling

text: strength

text: among

text: oscillators

text: with

text: diﬀerent

text: degrees;

text: oscillators

text: with

text: high

text: degrees

text: tend

text: to

text: undergo

text: phase

text: slip

text: while

text: those

text: with

text: low

text: degrees

text: have

text: weak

text: coupling

text: strengths

text: that

text: are

text: insuﬃcient

text: for

text: synchronization.

text: Keywords:

text: synchronization,

text: heterogeneous

text: network,

text: noise

text: Synchronization

text: phenomena

text: are

text: found

text: exist

text: both

text: lower

text: and

text: upper

text: thresholds

text: of

text: the

text: coupling

text: in

text: various

text: systems,

text: where

text: the

text: maintenance

text: and

text: thus,

text: global

text: synchronization

text: tends

text: to

text: fail

text: of

text: synchronization

text: is

text: quite

text: often

text: crucial

text: for

text: for

text: heterogeneous

text: However,

text: only

text: a

text: few

text: ex-

text: proper

text: functioning.

text: Such

text: systems

text: are

text: rep-

text: amples

text: of

text: such

text: chaotic

text: oscillators

text: with

text: a

text: similar

text: property

text: resented

text: by

text: mutually

text: interacting

text: oscillators,

text: are

text: known.

text: Moreover,

text: for

text: periodic

text: oscillators,

text: little

text: has

text: and

text: their

text: collective

text: dynamics

text: depend

text: both

text: been

text: reported

text: about

text: the

text: potential

text: negative

text: eﬀect

text: of

text: net-

text: on

text: the

text: interaction

text: function

text: between

text: a

text: pair

text: work

text: heterogeneity

text: on

text: synchronization.

text: of

text: connected

text: oscillators

text: and

text: on

text: the

text: network

text: We

text: have

text: recently

text: investigated

text: a

text: phase

text: oscillator

text: that

text: is

text: structure

text: among

text: the

text: oscillators.

text: Here,

text: we

text: unilaterally

text: inﬂuenced

text: by

text: a

text: pacemaker

text: and

text: is

text: also

text: under

text: show

text: that

text: when

text: such

text: coupled

text: oscillators

text: are

text: and

text: found

text: nontrivial

text: phase

text: slip

text: in

text: the

text: strong

text: cou-

text: under

text: the

text: inﬂuence

text: of

text: noise,

text: too

text: strong

text: a

text: cou-

text: pling

text: regime;

text: this

text: implies

text: that

text: synchronization

text: is

text: possi-

text: pling

text: induces

text: nontrivial

text: phase

text: slip.

text: Moreover,

text: ble

text: only

text: for

text: intermediate

text: coupling

text: strengths.

text: It

text: has

text: been

text: we

text: show

text: that

text: heterogeneous

text: networks,

text: which

text: shown

text: that

text: this

text: reentrant

text: transition

text: is

text: observed

text: for

text: gen-

text: have

text: a

text: wide

text: dispersion

text: of

text: network

text: connec-

text: eral

text: interaction

text: functions.

text: Although

text: it

text: is

text: not

text: at

text: all

text: ob-

text: tions

text: among

text: individual

text: oscillators,

text: are

text: more

text: vious

text: whether

text: dynamical

text: behavior

text: obtained

text: in

text: the

text: phase

text: strongly

text: aﬀected

text: by

text: the

text: nontrivial

text: phase

text: slip.

text: model

text: with

text: strong

text: coupling

text: is

text: also

text: reproduced

text: in

text: limit-

text: While

text: synchronization

text: failure

text: is

text: known

text: to

text: cycle

text: oscillators,

text: it

text: has

text: been

text: demonstrated

text: that

text: the

text: Brus-

text: occur

text: in

text: heterogeneous

text: networks

text: of

text: a

text: par-

text: selator

text: model,

text: a

text: typical

text: system

text: of

text: limit

text: cycle

text: oscillators,

text: ticular

text: class

text: of

text: chaotic

text: oscillators,

text: our

text: study

text: does

text: show

text: the

text: reentrant

text: transition.

text: Such

text: nontrivial

text: phase

text: demonstrates

text: the

text: diﬃculty

text: of

text: synchroniza-

text: slip

text: can

text: be

text: another

text: source

text: of

text: instability

text: for

text: a

text: population

text: tion

text: in

text: heterogeneous

text: networks

text: of

text: periodic

text: of

text: network-coupled

text: oscillators,

text: and

text: thus

text: it

text: is

text: important

text: oscillators

text: under

text: the

text: inﬂuence

text: of

text: noise.

text: to

text: understand

text: how

text: diﬀerent

text: networks

text: respond

text: to

text: this

text: new

text: type

text: of

text: instability.

text: In

text: this

text: study,

text: we

text: show

text: that

text: heterogeneous

text: networks

text: I.

text: INTRODUCTION

text: are

text: more

text: prone

text: to

text: phase

text: slip.

text: The

text: essence

text: of

text: this

text: be-

text: havior

text: can

text: be

text: understood

text: from

text: the

text: phase

text: diagram

text: of

text: two

text: mutually

text: coupled

text: phase

text: oscillators.

text: Mutually

text: coupled

text: os-

text: Synchronized

text: oscillation

text: of

text: active

text: elements

text: can

text: be

text: ob-

text: cillators

text: fail

text: to

text: synchronize

text: for

text: too

text: weak

text: and

text: too

text: strong

text: served

text: in

text: various

text: ﬁelds,

text: including

text: biology,

text: engineering,

text: ecosystem,

text: and

text: chemical

text: In

text: many

text: cases,

text: syn-

text: coupling

text: strengths

text: when

text: they

text: are

text: subjected

text: to

text: noise.

text: Be-

text: cause

text: of

text: this

text: property,

text: synchronization

text: is

text: easily

text: violated

text: chronization

text: of

text: the

text: entire

text: system

text: is

text: required

text: for

text: proper

text: in

text: heterogeneous

text: networks.

text: functioning

text: under

text: various

text: types

text: of

text: noise

text: and

text: heterogene-

text: ity.

text: Important

text: examples

text: include

text: the

text: heart

text: (a

text: population

text: This

text: paper

text: is

text: organized

text: as

text: follows:

text: in

text: Sec.

text: we

text: inves-

text: of

text: cardiac

text: the

text: circadian

text: pacemaker

text: (a

text: popula-

text: tigate

text: a

text: system

text: of

text: two

text: mutually

text: interacting

text: oscillators

text: tion

text: of

text: clock

text: and

text: power

text: under

text: noise

text: and

text: compare

text: it

text: with

text: two

text: unilaterally

text: con-

text: Oscillators

text: are

text: often

text: connected

text: through

text: a

text: complex

text: net-

text: nected

text: oscillators,

text: which

text: have

text: been

text: studied

text: previously;

text: work,

text: where

text: heterogeneity

text: in

text: network

text: connectivity

text: may

text: two

text: main

text: causes

text: of

text: frequency

text: drop,

text: nontrivial

text: phase

text: slip

text: critically

text: hamper

text: synchronization.

text: Such

text: a

text: case

text: is

text: actu-

text: and

text: oscillation

text: death,

text: are

text: discussed

text: in

text: Sec.

text: and

text: Sec.

text: ally

text: observed

text: in

text: a

text: special

text: type

text: of

text: chaotic

text: oscillators.

text: For

text: respectively.

text: Then,

text: in

text: Sec.

text: we

text: present

text: the

text: results

text: for

text: the

text: synchronization

text: of

text: these

text: chaotic

text: there

text: network-connected

text: oscillators.