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---
author:
- 'G.C. Van de Steene'
- 'G.H. Jacoby [^1]'
date: 'received / accepted'
title: Radio observations of new galactic bulge planetary nebulae
---
Introduction
============
Planetary Nebulae (PNe) are bright emission line objects, observable throughout the Galaxy. They are excellent probes of abundance gradients, the chemical enrichment history of the interstellar medium, the effects of metallicity on stellar evolution, and kinematics.
Most small PNe ($\sim$90$\%$) within 10 of the galactic center are physically close to it (Pottasch & Acker 1989). Since they can be assumed to be at the same known distance of $\sim$7.8 kpc, their distance-dependent parameters, such as luminosity and size, can be determined. These parameters are needed to define the underlying population. The chemical composition and the central star parameters are computed via self-consistent photo-ionization modelling of the nebula. Because luminosity correlates with the central star mass, which correlates with the progenitor mass, which, in turn, correlates with stellar age, the relationship between age and composition can be deduced (Dopita et al. [@Dopita97], Walsh et al. [@Walsh00]). The chemical enrichment history of the Bulge could be tracked using PNe.
We surveyed a 4 x 4 degree field centered on the galactic center in \[S[iii]{}\]$\lambda$9532 and | 0 | non_member_972 |
a continuum band at KPNO with the 60-cm Schmidt telescope and a 2048 x 2048 pixels thick STIS CCD in July 1994 and June 1995. The field of view was 65x65 and the pixel size 2. This survey has uncovered 95 new PN candidates in addition to the 34 previously known in this region (Acker et al. 1992, Kohoutek 1994). 45 PNe were confirmed via optical spectroscopy with the 1.52-m ESO telescope and the Boller & Chivens spectrograph, while 19 fainter ones were confirmed at the CTIO 4-m with the RC spectrograph (Van de Steene & Jacoby 2001, in preparation). Accurate radio flux densities and angular diameters are crucial to obtain a good photo-ionization model of the PNe (van Hoof & Van de Steene [@vHoof99]). The very high extinction causes the H$\beta$ line to be faint or even undetected in the optical spectra. Hence the radio flux density is needed to determine the extinction and the total ionizing flux.
In this article we present the radio continuum observations of 64 PNe confirmed spectroscopically with the ESO 1.52-m and CTIO 4-m telescopes. We describe the observations in Sect. \[observations\] and the data reduction in Sect. \[reduction\]. The results are presented in | 0 | non_member_972 |
Sect. \[results\]. The improved method for determining the distances, based on a relationship between radii and radio surface brightness (Van de Steene & Zijlstra [@VdSteene95]) is used in Sect. \[distances\] to determine the distances of the PNe and discuss their distribution in the galactic bulge. In Sect. \[extinction\] we determine the extinction values of these new bulge PNe.
Observations
============
Radio continuum observations for 64 new PNe which had been confirmed spectroscopically at the ESO 1.52-m and CTIO 4-m telescopes were obtained with the Australia Telescope Compact Array. The array was in configuration $\#$ 6A. The shortest baseline was 337 m and the longest 5939 m. The bandwidth was 128 MHz centered at 4800 MHz and 8640 MHz, corresponding to 6 and 3 cm respectively. The synthesized beam for the 6 km array at 6 cm is $\sim$ 2 and $\sim$1 at 3 cm. According to the ACA manual, the largest well imaged structure in a 60 min observation is 30 at 6 cm and 15 at 3 cm. Consequently we expected all our sources to be well imaged at 6 cm and virtually all at 3 cm. Forty PNe were observed at 3 and 6 cm simultaneously for 12 | 0 | non_member_972 |
h on each of 12, 13, 16 & 17 February 1997 and 36 PNe were observed during 10 h on each of 24, 25, and 26 May 1999. In 1997 each subsample contained 10 PNe and each PN was observed for 5 min every hour. In 1999 each subsample contained 6 PNe and each PN was observed for 8 min every hour. The total quality integration time per PN was at least 50 min in 1997 and 80 min in 1999. To avoid artifacts at the center of the field we offsetted the positions 30 in declination. At the beginning and the end of each 12 h shift the primary flux density calibrator 1934-638 was observed for 5 to 10 min. The phase calibrator VLA 1730-130 and VLA 1748-253 were observed about every 20 min for 2 min.
Data Reduction {#reduction}
==============
The data were reduced using the package [Miriad]{} following standard reduction steps as described in the reference guide by Bob Sault and Neil Killeen (Sault et al. 1995) . The data were loaded, bad points flagged, and then split into single source files. The bandpass function and the antenna gains were determined as a function of time to | 0 | non_member_972 |
calibrate the data. From the visibilities images were made using the multi-frequency synthesis technique and robust weighting with a robustness parameter of 0.5, which gives nearly the same sensitivity as natural weighting but with a significantly better beam. First a low resolution image of the primary beam was made to identify confusing sources. Next a high resolution image was made including all sources. These maps were [CLEAN]{}ed. Because we had offsetted the PNe from the field center, we applied the primary beam correction which amounted to about 1% at 6 cm and 2% at 3 cm. Self-calibration was not applied because the flux values of the PNe are too low. Contour plots of the detected PNe are presented in the Appendix.
Analysis and Results {#results}
====================
Positions
---------
In Table \[tabpos\] we list the position of the peak flux density per beam of the PN at 6 cm together with the optical positions as determined from the H$\alpha$ or \[S III\]$\lambda$9532 images (Jacoby & Van de Steene, 2001, in preparation). The peak of the radio emission is adopted as the PN position. If the PN is extended in the radio, the radio position may be off center. The radio position | 0 | non_member_972 |
will be better determined for higher peak flux values and smaller PNe. It is also for this reason that we chose to use the value at 6 cm and not at 3 cm, where the resolution is twice as high and the signal to noise per beam lower for extended sources.
We note that the radio positions have a tendency to be offset towards the west of the optical position. In declination there is no clear tendency noticeable in the offsets.
The optical and radio positions agree very well. PNe for which the radio position differs more than 2 in RA or DEC from the optical position are extended and the peak in the radio is usually not centered.
Of the 64 PNe observed, 7 were not detected: JaSt 7, 21, 45, 80, 88, 92, and 96. Most likely they are very extended and have a surface brightness that is too low to be detected in the radio. They are also very faint in the H$\alpha$ images and their H$\alpha$ flux values are very uncertain (Jacoby & Van de Steene 2001, in preparation). All but JaSt 96 were also faint and extended in the \[S III\]$\lambda$9532 images.
------ ------------- -------------- | 0 | non_member_972 |
-------------- --------------- ------- --------- --- --------------
JaSt $\Delta$RA $\Delta$DEC run comment
RA(h m s) DEC ( ) RA(h m s) DEC ( )
1 17 34 43.64 -29 47 05.03 17 34 43.60 -29 47 06.03 -0.60 1.00 2
2 17 35 00.96 -29 22 15.72 17 35 00.96 -29 22 15.85 0.00 0.13 1
3 17 35 22.90 -29 22 17.58 17 35 22.79 -29 22 17.08 -1.65 -0.50 1
4 17 35 37.47 -29 13 17.67 17 35 37.39 -29 13 17.81 -1.20 0.14 1
5 17 35 52.51 -28 58 27.95 17 35 52.44 -28 58 27.45 -1.05 -0.50 1
7 17 38 26.69 -28 47 06.48 2 not detected
8 17 38 27.74 -28 52 01.31 17 38 27.58 -28 52 01.36 -2.40 0.05 2
9 17 38 45.64 -29 08 59.27 17 38 45.54 -29 08 55.65 -1.50 -3.62 2
11 17 39 00.55 -30 11 35.23 17 39 00.48 -30 11 32.23 -1.05 -3.00 2
16 17 39 22.70 -29 41 46.08 17 39 22.64 -29 41 45.35 -0.90 -0.70 1
17 17 39 31.32 -27 27 46.78 17 39 31.21 -27 27 47.28 -1.65 -0.50 1
19 17 39 39.38 -27 47 22.58 | 0 | non_member_972 |
17 39 39.31 -27 47 22.21 -1.05 -0.37 1
21 17 39 52.92 -27 44 20.54 2 not detected
23 17 40 23.17 -27 49 12.04 17 40 23.08 -27 49 12.29 -1.35 0.25 1
24 17 40 28.23 -30 13 51.30 17 40 28.19 -30 13 51.00 -0.60 -0.30 1
26 17 40 33.52 -29 46 14.98 17 40 33.25 -29 46 12.48 -4.05 -2.50 2
27 17 40 42.34 -28 12 31.90 17 40 42.16 -28 12 30.81 -2.7 -1.1 1
31 17 41 27.93 -28 52 51.61 17 41 27.89 -28 52 50.61 -0.60 -1.00 1
34 17 41 54.80 -27 03 20.33 17 41 54.69 -27 03 18.33 -1.65 -2.00 2
36 17 42 25.20 -27 55 36.36 17 42 25.13 -27 55 36.36 -1.05 0.00 1
37 17 42 28.60 -30 09 34.93 17 42 28.52 -30 09 32.93 -2.00 -2.00 1
38 17 42 32.41 -27 33 15.18 17 42 32.29 -27 33 16.26 -1.80 1.08 1
41 17 42 49.96 -27 21 19.68 17 42 49.85 -27 21 19.31 -1.65 -0.37 1
42 17 43 17.06 -26 44 17.67 17 43 16.98 -26 44 18.25 -1.20 0.58 1
44 17 43 23.48 -27 | 0 | non_member_972 |
34 06.03 17 43 23.52 -27 34 06.55 0.60 0.52 1
45 17 43 23.44 -27 11 16.91 2 not detected
46 17 43 30.43 -26 47 32.33 17 43 30.36 -26 47 31.83 -1.05 -0.50 1
49 17 44 04.34 -28 15 57.86 17 44 04.23 -28 15 56.99 -1.65 -0.87 1
52 17 44 37.30 -26 47 25.23 17 44 37.23 -26 47 25.23 -1.05 0.00 1
54 17 45 11.06 -27 32 36.80 17 45 11.02 -27 32 36.80 -0.60 0.00 2
55 17 45 37.36 -27 01 18.44 17 45 37.15 -27 01 18.16 -3.15 -0.28 1
56 17 45 47.05 -27 30 42.03 17 45 46.96 -27 30 42.01 -1.35 -0.02 2
58 17 46 52.20 -30 37 42.83 17 46 52.32 -30 37 42.33 1.80 -0.50 2
60 17 47 53.91 -29 36 49.67 17 47 53.91 -29 36 49.67 0.00 0.00 2
63 17 48 46.27 -27 25 37.22 17 48 46.27 -27 25 36.72 0.00 -0.50 1
64 17 48 56.04 -31 06 41.95 17 48 56.04 -31 06 42.45 0.00 0.50 1
65 17 49 20.02 -30 36 05.57 17 49 20.02 -30 36 05.08 0.00 0.23 1
66 17 49 | 0 | non_member_972 |
22.15 -29 59 27.02 17 49 22.11 -29 59 27.02 -0.60 0.00 2
67 17 49 28.10 -29 20 47.56 17 49 28.02 -29 20 47.56 -1.20 0.00 2
68 17 49 50.87 -30 03 10.47 17 49 50.83 -30 03 10.97 -0.60 0.50 2
69 17 50 10.04 -29 19 05.14 17 50 10.11: -29 19 08.82: 1.05 3.68 1
70 17 50 21.07 -28 39 02.46 17 50 21.03 -28 39 01.46 -0.60 1.0 2
71 17 50 23.32 -28 33 10.95 17 50 23.21 -28 33 10.45 -1.65 -0.50 1
73 17 50 47.74 -29 53 16.01 17 50 47.82 -29 53 14.01 1.20 -1.99 2
74 17 50 46.85 -28 44 35.42 17 50 46.77 -28 44 34.92 -1.20 -0.50 1
75 17 50 48.08 -29 24 43.60 17 50 48.00 -29 24 43.18 -1.20 -0.42 1
76 17 50 56.47 -28 31 24.63 17 50 56.33 -28 31 24.13 -2.10 -0.50 2
77 17 51 11.65 -28 56 27.20 17 51 11.58 -28 56 27.20 -1.20 0.00 1
78 17 51 24.68 -28 35 40.34 17 51 24.79 -28 35 39.44 1.65 -0.90 1
79 17 51 53.63 -29 30 53.41 17 51 53.55 -29 | 0 | non_member_972 |
30 53.41 -1.20 0.00 1
80 17 51 55.54 -27 48 02.46 1 not detected
81 17 52 04.35 -27 36 39.28 17 52 04.28 -27 36 38.28 -1.05 1.00 1
83 17 52 45.17 -29 51 05.21 17 52 45.17 -29 51 03.30 0.00 -1.91 1
85 17 52 49.05 -29 41 54.92 17 52 48.97 -29 41 55.92 -1.20 1.0 2
------ ------------- -------------- -------------- --------------- ------- --------- --- --------------
\[tabpos\]
------ ------------- -------------- -------------- --------------- ------- --------- --- --------------
JaSt $\Delta$RA $\Delta$DEC run comment
RA(h m s) DEC ( ) RA(h m s) DEC ( )
86 17 52 52.20 -29 30 00.07 17 52 52.16 -29 30 01.08 -0.60 1.08 1
88 17 53 00.89 -29 05 44.08 2 not detected
89 17 53 06.67 -28 18 07.91 17 53 06.79 -28 18 10.47 1.80 2.56 1
90 17 53 17.77 -28 04 33.20 17 53 17.73 -28 04 32.67 -0.60 -0.53 2
92 17 53 19.81 -28 27 14.67 2 not detected
93 17 53 24.14 -29 49 48.45 17 53 24.05: -29 49 51.15: -1.35 2.70 1
95 17 53 35.38 -28 28 51.02 17 53 35.51 -28 28 51.96 1.95 0.94 1
96 | 0 | non_member_972 |
17 53 57.16 -29 20 14.97 2 not detected
97 17 54 13.36 -28 05 16.82 17 54 13.28 -28 05 16.82 -1.20 0.00 1
98 17 55 46.39 -27 53 38.91 17 55 46.28 -27 53 38.41 -1.65 -0.50 1
------ ------------- -------------- -------------- --------------- ------- --------- --- --------------
------ ------ -------- ------- ------- ------ -------- ------- ------- --------------------------------- ------------ --------- ------------ ---------- --
Name 6cm 3cm size beam FWHM method $\theta$
JaSt Peak Flux error noise Peak Flux error noise $\Delta\delta$ x $\Delta\alpha$ maj x min diam
mJy/ mJy mJy mJy/ mJy/ mJy mJy mJy/ x x
beam beam beam beam
1 0.85 1.6 0.2 0.10 0.47 1.4 0.6 0.12 4.5 x 2.6 4.3 x 1.9 3.4 G 5.5
2 1.8 4.3 0.5 0.22 0.82 4.2 1.4 0.12 5.0 x 4.0 4.7 x 1.8 4.5 C 6.3
3 4.0 12.4 1.2 0.17 1.6 10.0 1.0 0.13 5.8 x 4.4 4.8 x 2.0 5.1 C 9.2
4 1.2 4.1 0.4 0.20 0.65 3.5 1.2 0.15 7.7 x 7.1 4.7 x 1.8 7.4 C 11.2
5 4.0 11.3 1.1 0.15 1.8 10.5 1.0 0.15 6.2 x 5.1 4.9 x 2.0 5.6 C 8.4
8 1.2 3.8 0.4 0.08 0.5 2.6 | 0 | non_member_972 |
0.6 0.04 3.6 x 5.8 4.4 x 1.9 4.6 C 7.2
9 0.96 4.0 0.4 0.08 0.38 2.5 0.8 0.08 5.6 x 6.2 4.7 x 1.8 5.9 C 8.1
11 0.76 3.3 0.3 0.07 0.41 $>$2.0 0.12 6.9 x 5.9 4.6 x 1.8 6.4 C 10.9
16 13.6 27.0 2.8 0.27 7.2 24.7 2.5 0.30 3.4 x 2.4 4.8 x 2.0 2.9 G 4.8
17 2.5 9.4 1.0 0.15 1.2 8.2 0.9 0.13 5.4 x 5.3 5.7 x 1.8 5.3 C 7.9
19 2.1 6.8 0.8 0.29 1.0 5.5 0.6 0.14 3.6 x 3.7 5.3 x 1.6 3.6 C 6.8
23 3.3 3.7 0.4 0.25 2.6 3.9 0.4 0.21 PS 3.1 x 1.1 $<$1.8 G$_{3cm}$
24 6.3 16.0 1.7 0.27 2.7 14.1 1.4 0.23 4.8 x 2.7 4.9 x 1.7 3.6 G 5.8
26 2.4 14.1 1.4 0.09 1.0 9.0 0.9 0.08 9.4 x 9.3 5.1 x 1.7 8.5 C 11.1
27 0.8 2.5 0.3 0.14 0.5 0.15 9.9 x 5.3 5.3 x 1.7 8.2 C 8.7
31 3.3 11.5 1.2 0.22 2.4 $>$3.0 0.61 5.0 x 5.1 5.1 x 1.7 5.0 C 9.0
34 0.58 1.7 0.2 0.07 0.34 $>$1.5 0.09 3.2 x 6.0 5.1 x 1.7 4.4 C | 0 | non_member_972 |
7.1
36 18.6 31.1 3.1 0.33 11.7 31.1 3.1 0.35 2.6 x 1.9 3.1 x 1.1 2.2 G$_{3cm}$ 3.6
37 5.2 16.0 1.7 0.70 2.5 $>$5.5 0.35 5.7 x 3.8 4.6 x 1.5 4.6 C 7.4
38 1.2 3.0 0.6 0.28 0.9 2.0 0.7 0.15 6.8 x 2.2 5.3 x 1.7 3.9 C 7.0
41 8.0 16.7 1.7 0.20 4.6 17.6 1.8 0.18 3.4 x 3.2 5.6 x 1.8 3.3 C 6.3
42 5.6 12.9 1.3 0.21 3.3 15.5 1.6 0.26 4.0 x 2.5 5.5 x 1.6 3.1 G 5.0
44 1.3 3.8 1.0 0.27 0.9 $>$1.8 0.27 12.2: x 6.0: 5.4 x 1.6 8.6: C 8.6:
46 14.6 20.8 2.1 0.40 12.3 20.5 2.1 0.30 2.0 x 1.1 3.3 x 1.1 1.5 G 2.5
49 3.1 8.2 0.9 0.27 2.1 10.8 1.1 0.21 4.6 x 3.6 5.2 x 1.6 4.0 C 6.6
52 20.8 24.0 2.4 0.34 16.9 22.8 2.3 0.40 2.5 x 0.4 3.3 x 1.1 1.0 G$_{3cm}$ 1.7
54 21.9 82.3 8.2 0.08 7.5 75.2 7.5 0.09 4.7 x 4.3 5.1 x 1.8 4.5 G 7.2
55 2.0 9.2 0.9 0.19 0.9 10.0 2.0 0.17 5.5 x 6.4 5.4 x 1.6 6.0 C 10.7
56 5.8 14.1 | 0 | non_member_972 |
1.4 0.09 2.4 12.2 1.2 0.07 3.8 x 2.7 2.8 x .95 3.2 G$_{3cm}$ 5.0
58 3.5 29.5 2.9 0.06 1.3 21.3 2.1 0.07 8.9 x 8.0 4.6 x 1.7 8.4 C 15.5
60 6.1 6.5 0.7 0.06 7.0 7.9 0.8 0.07 .85 x .28 2.6 x 1.0 0.5 G$_{3cm}$ 0.9
63 1.1 3.9 0.4 0.11 0.7 3.6 1.2 0.13 6.1 x 6.0 5.5 x 1.8 6.0 C 8.4
64 23.8 28.3 2.9 0.42 15.8 25.6 2.6 0.58 2.0 x 0.7 5.0 x 1.9 1.4 G 2.4
65 3.3 3.4 0.4 0.23 6.3 6.6 0.7 0.30 PS 2.7 x 1.0 $<$ 1.6 G$_{3cm}$
66 45.9 65.3 6.5 0.15 26.0 60.9 6.1 0.10 2.0 x 1.5 2.6 x 1.00 1.7 G$_{3cm}$ 2.8
67 24.6 27.8 2.8 0.08 17.7 26.0 2.6 0.10 1.3 x .9 2.5 x 1.0 1.1 G $_{3cm}$ 1.8
68 5.4 5.9 0.6 0.06 7.7 8.8 0.9 0.10 .8 x .3 2.5 x 1.0 1.5 G$_{3cm}$ 0.9
69 0.9 $>$2.0 0.26 0.5 $>$2.0 0.15 4.9 x 1.7
70 4.0 13.0 1.3 0.11 1.9 11.5 1.2 0.06 4.3 x 5.3 4.6 x 1.7 4.8 C 8.5
71 20.2 34.0 3.5 0.50 10.0 36.0 3.6 0.55 4.6 x 1.6 5.4 x | 0 | non_member_972 |
1.9 2.7 G 4.5
73 11.7 13.7 1.4 0.09 9.4 13.0 1.3 0.11 1.3 x .7 2.6 x 1.0 0.95 G$_{3cm}$ 1.6
74 12.7 22.7 2.3 0.28 6.1 22.2 2.2 0.25 4.5 x 2.2 3.3 x 1.0 3.1 G$_{3cm}$ 4.9
75 16.8 22.5 2.3 0.53 11.0 19.0 2.0 0.55 3.0 x 0.9 5.0 x 1.6 1.6 G 2.7
76 5.4 7.4 0.7 0.07 2.5 6.7 0.7 0.08 3.9 x 1.8 2.4 x 1.0 2.6 G$_{3cm}$ 4.7
77 39.4 49.0 5.0 0.85 34.0 51.2 5.1 0.98 PS 3.1 x 1.1 $<$1.5 G$_{3cm}$
78 1.5 12.3 1.2 0.18 0.9 9.0 2.3 0.15 12.0 x 9.0 5.1 x 1.6 10.4 C 14.6
79 4.1 4.1 0.4 0.14 6.7 7.0 0.7 0.23 PS 2.7 x 1.2 $<$1.5 G$_{3cm}$
------ ------ -------- ------- ------- ------ -------- ------- ------- --------------------------------- ------------ --------- ------------ ---------- --
\[tabflux\]
------ ------ ------ ------- ------- ------ -------- ------- ------- --------------------------------- ----------- --------- ----------- ---------- --
Name 6cm 3cm size beam FWHM method $\theta$
JaSt Peak Flux error noise Peak Flux error noise $\Delta\delta$ x $\Delta\alpha$ maj x min diam
mJy/ mJy mJy mJy/ mJy/ mJy mJy mJy/ x x
beam beam beam beam
81 17.8 20.8 2.0 0.25 14.1 20.8 | 0 | non_member_972 |
2.1 0.29 1.8 x 0.7 5.1 x 2.0 1.3 G 2.2
83 1.7 5.0: 2.5 0.40 0.6 $>$0.6 0.16 5.3: x 5.7: 4.5 x 1.8 $>$5.5: C $>$6.3
85 0.6 4.4 1.2 0.09 13.1: x 13.1: 4.2 x 1.9 13.1: C 17.5:
86 1.7 8.5 0.9 0.17 0.9 $>$2.5 0.17 7.5 x 7.7 4.7 x 1.8 7.6 C 11.4
89 1.8 8.5 0.9 0.27 0.8 $>$2.0 0.19 9.5 x 5.5 4.8 x 1.7 7.2 C 8.6
90 0.51 1.0 0.2 0.07 0.13 4.5 x 2.5 4.4 x 1.9 3.4 G 5.5
93 0.8 6.3 0.9 0.15 0.6 $>$2.1 0.13 20.2 x 8.0 4.8 x 2.0 12.7: C 17.8:
95 1.1 5.5 1.0 0.25 0.7 4.5: 1.5 0.18 14.6 x 7.5 4.8 x 1.8 10.5 C 12.6
97 2.7 11.5 1.2 0.16 1.2 11.8 1.2 0.18 8.0 x 5.9 5.2 x 1.9 6.8 C 11.4
98 21.0 21.8 2.2 0.38 20.3 24.0 2.4 0.46 PS 2.9 x 1.1 $<$1.5 G$_{3cm}$
------ ------ ------ ------- ------- ------ -------- ------- ------- --------------------------------- ----------- --------- ----------- ---------- --
Radio flux densities
--------------------
Table \[tabflux\] gives the radio flux densities at 6 and 3 cm.
If a Gaussian model provided a satisfactory fit to the | 0 | non_member_972 |
surface brightness the total Gaussian flux density was adopted. This was mostly the case for small, unresolved PNe. If the PN was extended the intensities within the 2 or 3 $\sigma$ level contour were summed. This value was compared with the statistics over a larger region across the nebula to obtain an error estimate (Fomalont, 1989).
For JaSt 69 only a small 3 $\sigma$ blob at the right position indicated the presence of a PN. No flux or size values could be determined. In some cases several blobs indicated the presence of a PN at 3 cm and hence their flux is a lower limit. Some objects detected at 6 cm, but having low peak flux density per beam, could not be seen at 3 cm.
PNe are normally optically thin at 6 cm in which case its 3 cm flux density is about 95% of the 6 cm flux density. In our case this means that the flux densities are similar within the error-bars. However when a PN is already well resolved at 6 cm, more flux may have been missed at 3 cm where the beam is half the size, especially if the nebula is extended and of | 0 | non_member_972 |
low surface brightness. It is clear that due to these factors the flux density at 3 cm is generally less well-determined than at 6 cm, except for the bright and compact PNe. When a PN is optically thick at 6 cm, its 3 cm flux density is expected to be three times the flux density at 6 cm. There are some PNe like JaSt 65 and JaSt 79 for which the 3 cm flux density is clearly higher than the 6 cm flux density and which are point sources. In this case the PNe may not yet be completely optically thin at 6 cm and should be quite young.
In Fig. \[fluxhist\] we plotted the histogram of the radio flux of the known and new galactic bulge PNe. We selected a sample of known galactic bulge PNe for which radio flux densities and angular diameters are available, as in Van de Steene & Zijlstra ([@VdSteene95]). The new bulge PNe are within 2 degrees of the galactic center. None has a radio flux larger than 100 mJy and their angular sizes are smaller than 20. Hence they fulfill the same selection criteria as these previously known bulge PNe.
There is a | 0 | non_member_972 |
larger number of PNe with low flux densities among the new bulge PNe than among the known ones. 67 % of the new PNe have a radio flux less than 15 mJy, while this is only 45 % for the known ones. Of the 7 known ones within 2 degrees of the galactic center only 2 have a radio flux below 15 mJy. The median flux for the new PNe is 11.3 mJy, while the median for the known bulge PNe is 17.0 mJy. Our rms noise level in the maps is similar to the 1 $\sigma$ noise of 0.1 mJy in the 6 cm maps of Zijlstra et al. ([@Zijlstra89]). Apparently these faint and small PNe have just been missed in optical surveys done to date.
\[fluxhist\]
Angular Size
------------
Table \[tabflux\] also gives the angular size of the detected PNe. We chose to determine the angular size at 6 cm because at this wavelength the resolution was lower and thus gives the best signal to noise ratio for the extended nebulae. The diameter at 3 cm is given if it is better determined than at 6 cm, such as for very small and bright PNe.
Diameters may differ | 0 | non_member_972 |
considerably depending upon how they are calculated. The diameter was derived by one of several ways depending on the structure of the brightness distribution. If a two-dimensional Gaussian fit provided a satisfactory model to the observed structureless surface distribution, its deconvolved FWHM major and minor axis are given. The equivalent diameter is the square root of their product. To obtain the full diameter, this value must be multiplied with a conversion factor which is a function of the beam FWHM and depends upon the intrinsic surface distribution of the source. We assumed a spherical constant emissivity shell of 0.5 and used formula 5 and Table 1 from van Hoof ([@vHoof00]) to estimate the true radii. For small objects, if the Gaussian deconvolution was well determined and similar at 3 and 6 cm, the FWHM at 3 cm is given. If the deconvolution produced a point source at 6 cm, the source size at 3 cm is given. If the source was still a point source at 3 cm, the beam-size is an upper limit. If the source was extended, a Gaussian model was usually not a good representation of the radio source. The diameter of the PN was measured on | 0 | non_member_972 |
the contour at 50 % of the peak and deconvolved with the beam size. To determine the full diameter we determined the ratio of the flux density within the 3 $\sigma$ contour with the flux density within the 50 % contour. Hence, we assumed that the flux decreased linearly with radius outside the 50 % contour. We checked that this procedure seemed to give very good agreement with the size measured based on the 3 $\sigma$ contour. We didn’t use the contour level at 10% of the peak (Zijlstra et al. [@Zijlstra89], Aaquist & Kwok [@Aaquist90]), because this was often below noise level.
It was noticed in the review paper by Pottasch ([@Pottasch92]) that there is a selection against discovering both large and small PNe in the galactic bulge. 36 % of the new PNe have a diameter smaller than 5 , while this is 71 % for the known ones. The median is 66 for the new PNe and only 32 for the known bulge PNe with radio data. We seem to identify relatively more larger PNe than in previous surveys. In regions with large extinction the \[SIII\]$\lambda$9532 line appears efficient in picking out the larger, low surface brightness | 0 | non_member_972 |
PNe, and not only the small and dusty ones. Obviously these are the PNe which may have been missed in optical surveys.
\[diamhist\]
Distances
=========
\[disthist\]
\[distsch\]
Based on the flux densities and the angular equivalent diameters we calculated the statistical distances to the PNe as established in Van de Steene & Zijlstra ([@VdSteene95]). The relation is not strictly valid for optically thick and very small nebulae which may have their distance overestimated. The distances will tend to be overestimated for PNe with lower surface brightness than average, while PNe with a higher surface brightness than average will tend to have their distances underestimated. However, statistically the distance distribution will be representative. The distances are presented in Table \[quantities\]. A histogram of the distance distribution is presented in Fig. \[disthist\]. The median distance is 7.2 kpc, about 8% closer than the galactic center; presumably, the more distant PNe suffer greater extinction and fell below our detection limit in our survey.
In Fig. \[distsch\] we plotted the scale height versus the distance. It seems that we observe PNe at the edge of the galactic bulge. As we look further inward, the extinction probably becomes too large. The PNe further away are | 0 | non_member_972 |
at larger scale height where, apparently, the extinction is less. The known PNe are generally further away from the galactic center, surrounding our new PNe. The median absolute scale height is 136 pc.
------ ------- -------- ------- ------------ -------- ---------
JaSt Dist z R M$_{ion}$ E(B-V) Comment
kpc pc pc M$_{\sol}$
1 14.6: 401.6: 0.20: 0.25: 2.5:
2 10.0 305.7 0.15 0.19 2.0
3 6.2 183.0 0.14 0.17 2.3
4 8.5 258.9 0.23 0.29 1.9
5 6.6 210.4 0.13 0.17 2.3
8 10.0 241.9 0.17 0.22 2.1
9 9.5 194.9 0.19 0.23 2.9
11 9.2 91.6 0.24 0.31 3.7
16 5.9 77.6 0.07 0.08 3.5
17 7.1 242.5 0.14 0.17 1.8
19 8.4 264.7 0.14 0.17 1.8
23 2.4 PS
24 6.6 31.4 0.09 0.11 X
26 5.6 49.7 0.15 0.19 X
27 10.8 246.2 0.23 0.29 2.3
31 6.4 91.2 0.14 0.17 3.7
34 13.2 392.9 0.23 0.29 2.1
36 6.1 120.2 0.05 0.06 2.7
37 6.1 -6.3 0.11 0.13 X
38 10.9: 251.7: 0.19: 0.23: 2.0:
41 6.3 150.4 0.10 0.12 2.1
42 7.4 207.0 0.09 0.11 2.1
44 9.6: 190.2: 0.20: 0.26: 1.9:
46 7.9 207.1 0.05 0.06 1.9
49 7.9 89.4 0.13 0.16 X
52 8.5 | 0 | non_member_972 |
190.6 0.03 0.04 2.1
54 3.5 50.2 0.06 0.07 X
55 6.5 115.9 0.17 0.21 2.5
56 7.2 89.9 0.09 0.11 X
58 3.9 -77.8 0.15 0.18 4.1
60 16.2 -213.6 0.04 0.04 X
63 9.4 31.5 0.19 0.25 2.6
64 7.2 -216.3 0.04 0.05 2.6
65 1.3 PS
66 5.5 -111.1 0.03 0.04 2.8
67 7.9 -127.0 0.03 0.04 4.6: Z
68 16.7 -388.0 0.04 0.04 2.8
70 6.2 -82.0 0.13 0.16 X
71 5.5 -67.4 0.06 0.07 3.6
73 10.5 -261.7 0.04 0.05 1.9
74 6.2 -93.9 0.07 0.09 3.6
75 7.5 -157.5 0.05 0.06 2.7
76 16.8 -223.1 0.03 0.03 5.0: Z
77 2.4 PS
78 5.4 -87.5 0.19 0.24 3.4: Z
79 1.5 PS
81 8.2 -75.8 0.04 0.05 X
85 7.2: -224.1 0.30 0.39: 1.2
------ ------- -------- ------- ------------ -------- ---------
: In this table we present the statistical distance in column 2, the scale height, radius, and ionized mass calculated using this distance in columns 3, 4, and 5 respectively, and E(B-V) $=$ c$_{H_{\beta}}$ / 1.46 in column 6. If the object was not detected in H$\alpha,$ E(B-V) is marked with X. If no H$\alpha$ image was obtained E(B-V) is marked with Z. | 0 | non_member_972 |
---
abstract: 'Layers of transition metal dichalcogenides (TMDs) combine the enhanced effects of correlations associated with the two-dimensional limit with electrostatic control over their phase transitions by means of an electric field. Several semiconducting TMDs, such as MoS$_2$, develop superconductivity (SC) at their surface when doped with an electrostatic field, but the mechanism is still debated. It is often assumed that Cooper pairs reside only in the two electron pockets at the K/K’ points of the Brillouin Zone. However, experimental and theoretical results suggest that a multi-valley Fermi surface (FS) is associated with the SC state, involving 6 electron pockets at the Q/Q’ points. Here, we perform low-temperature transport measurements in ion-gated MoS$_2$ flakes. We show that a fully multi-valley FS is associated with the SC onset. The Q/Q’ valleys fill for doping$\gtrsim2\cdot10^{13}$cm$^{-2}$, and the SC transition does not appear until the Fermi level crosses both spin-orbit split sub-bands Q$_1$ and Q$_2$. The SC state is associated with the FS connectivity and promoted by a Lifshitz transition due to the simultaneous population of multiple electron pockets. This FS topology will serve as a guideline in the quest for new superconductors.'
author:
- Erik Piatti
- Domenico De Fazio
- Dario | 0 | non_member_973 |
Daghero
- Srinivasa Reddy Tamalampudi
- Duhee Yoon
- 'Andrea C. Ferrari'
- 'Renato S. Gonnelli'
title: 'Multi-Valley Superconductivity In Ion-Gated MoS$_2$ Layers'
---
[^1]
[^2]
Transition metal dichalcogenides (TMDs) are layered materials with a range of electronic properties. Depending on chemical composition, crystalline structure, number of layers (N), doping, and strain, different TMDs can be semiconducting, metallic and superconducting[@FerrN2015]. Amongst semiconducting TMDs, MoS$_2$, MoSe$_2$, WS$_2$ and WSe$_2$ have sizeable bandgaps in the range$\sim$1-2eV[@WangNN2012]. When exfoliated from bulk to single layer (1L), they undergo an indirect-to-direct gap transition[@MakPRL2010; @SpleNL2010; @WangNN2012], offering a platform for electronic and optoelectronic applications[@FerrN2015; @WangNN2012; @MakNP2016], such as transistors[@PodzAPL2004; @RadiNN2011; @FangNL2012], photodetectors[@GourSEMSC1997; @LeeNL2012; @YinACS2012; @KoppNN2014], modulators[@SunNP2016] and electroluminescent devices[@CarlPRB2002; @SundNL2013].
For all TMDs with 2H crystal structure, the hexagonal Brillouin Zone (BZ) features high-symmetry points $\Gamma$, M, K and K’[@SpleNL2010; @BrummePRB2015], Fig.\[figure:bands\]a. The minima of the conduction band fall at K, K’, as well as at Q, Q’, approximately half-way along the $\Gamma$-K(K’) directions[@SpleNL2010; @BrummePRB2015], Fig.\[figure:bands\]a. In absence of an out-of-plane electric field, the relative position of Q and Q’ depends on N and strain[@SpleNL2010; @LambrechtPRB2012; @BrummePRB2015]. The global minimum of the conduction band sits at K/K’ in 1L-MoS$_2$ and at Q/Q’ in few layer (FL)-MoS$_2$ | 0 | non_member_973 |
with N$\geq$4[@SpleNL2010]. When an electric field is applied perpendicular to the MoS$_2$ plane, inversion symmetry is broken and the global minimum of the conduction band is shifted to K/K’ in any FL-MoS$_2$[@BrummePRB2015], Figs.\[figure:bands\]b-d. The valleys at K/K’ and at Q/Q’ are characterized by a different electron-phonon coupling (EPC)[@GePRB2013] and, when inversion symmetry is broken, by a different spin-orbit coupling (SOC)[@KadantsevSSC2012]. In particular, both EPC and SOC are larger in the Q/Q’ valleys[@GePRB2013; @KadantsevSSC2012].
{width="80.00000%"}
The field-effect transistor (FET) architecture is ideally suited to control the electronic properties of 1L flakes, as it simultaneously provides an electrostatic control of the transverse electric field and the carrier density. In the electric-double-layer (EDL) technique[@FujimotoReview2013], the standard solid gate dielectric is replaced by an ionic medium, such as an ionic liquid or electrolyte. In this configuration, the EDL that forms at the ionic liquid/electrode interfaces supports electric fields in excess of$\sim$10MV/cm[@UenoReview2014], corresponding to surface carrier densities $n_{2d}\gtrsim10^{14}$cm$^{-2}$[@UenoReview2014]. Ionic-liquid gating has been used to tune the Fermi level, $E_F$, in TMDs and explore transport at different carrier concentrations[@SaitoReview2016; @YeScience2012; @BragNL2012; @YuNN2015; @XiPRL2016]. The vibrational properties of TMDs can also be controlled by means of the EDL technique, as suggested by gate-induced softening of Raman-active modes | 0 | non_member_973 |
in 1L-MoS$_2$[@ChakrabortyPRB2012], while the opposite is observed in gated 1L[@DasNatureNanotech2008] and two-layer (2L)[@DasPRB2009] graphene. Ref. reported a gate-induced superconducting state at the surface of liquid-gated MoS$_2$ flakes with N$\gtrsim$25[@YeScience2012], while Ref. detected this down to N=1.
Most of these results have been interpreted in terms of the population of the conduction band minima at K/K’[@YeScience2012; @LuScience2015; @SaitoNatPhys2016; @ChenPRL2017], which are global minima in both 1L-MoS$_2$[@MakPRL2010; @GePRB2013] and electrostatically-doped FL-MoS$_2$[@YeScience2012; @LuScience2015; @SaitoNatPhys2016; @BrummePRB2015], Fig.\[figure:bands\]b. Theoretical investigations however suggested that the population of the high-energy minima at Q/Q’ may have an important role in determining the properties of gated MoS$_2$ flakes, by providing contributions both to EPC[@GePRB2013; @BrummePRB2015] and SOC[@KadantsevSSC2012; @WuNatCommun2016]. Ref. predicted that when the Q/Q’ valleys of 1L-MoS$_2$ are populated (Fig.\[figure:bands\]c,d), EPC strongly increases (from$\sim0.1$ to$\sim18$), leading to a superconducting transition temperature $T_c\sim20$K for a doping level $x=0.18$ electrons(e$^-$)/unit cell (corresponding to $E_F=0.18\pm0.02$eV at K/K’ and $0.08\pm0.02$eV at Q/Q’)[@BrummePRB2015]). However, Ref. measured $T_c\sim2$K for $x\sim0.09\div0.17$ e$^-$/unit cell in e$^-$-doped 1L-MoS$_2$. This mismatch may be associated with the contribution of e$^-$–e$^-$ interactions, whose role in the determination of $T_c$ is still under debate[@RoldanPRB2013; @DasPRB2015]. Overall, the agreement between the model of Ref. and the trend of $T_c$ with e$^-$ doping in | 0 | non_member_973 |
Ref. suggests that the mechanism of Ref. for EPC enhancement when the Q/Q’ valleys are crossed may also hold for FL-MoS$_2$.
Inversion symmetry can be broken in MoS$_2$ either by going to the 1L limit[@GePRB2013], or by applying a transverse electric field[@KormanyosPRB2013; @YuanPRL2014]. This leads to a finite SOC[@KormanyosPRB2013; @YuanPRL2014], which lifts the spin degeneracy in the conduction band and gives rise to two spin-orbit-split sub-bands in each valley[@KormanyosPRB2013; @BrummePRB2015], as shown in Fig.\[figure:bands\]b-d for FL-MoS$_2$. When the system is field-effect doped, the inversion symmetry breaking increases with increasing transverse electric field [@LuScience2015; @ChenPRL2017], due to the fact that induced e$^-$ tend to become more localized within the first layer [@LuScience2015; @ChenPRL2017; @RoldanPRB2013]. Hence, the SOC and the spin-orbit splitting between the bands increase as well, as was calculated in Ref..
When combined to the gate-induced SC state[@YuanPRL2014], this can give rise to interesting physics, such as spin-valley locking of the Cooper pairs[@SaitoNatPhys2016] and 2d Ising superconductivity (SC)[@LuScience2015] with a non-BCS-like energy gap[@CostanzoNatNano2018], suggested to host topologically non-trivial SC states[@RoldanPRB2013; @HsuNatComms2017; @NakamuraPRB2017]. Refs. predicted SOC and spin-orbit splitting between sub-bands to be significantly stronger for the Q/Q’ valleys than for K/K’, thus supporting spin-valley locking at Q/Q’ as well[@WuNatCommun2016]. A | 0 | non_member_973 |
dominant contribution of the Q/Q’ valleys in the development of the SC state would be consistent with the high ($\gtrsim50$T) in-plane upper critical field, $H_{c2}^{||}$, observed in ion-gated MoS$_2$ [@LuScience2015; @SaitoNatPhys2016] and WS$_2$ [@LuPNAS2018]. The $H_{c2}^{||}$ enhancement is caused by locking of the spin of the Cooper pairs in the out-of-plane direction in a 2d superconductor in the presence of finite SOC, and is therefore promoted by increasing the SOC. However, $H_{c2}^{||}$ for MoS$_2$ and WS$_2$ is higher than in metallic TMD Ising superconductors (such as NbSe$_2$ and TaS$_2$), where $H_{c2}^{||}\lesssim30$T [@delaBarreraNatCommun2018], despite the SOC in the K/K’ valleys being much smaller ($\sim3$meV for MoS$_2$[@KormanyosPRB2013]). Spin-valley locking in the Q/Q’ valleys may thus explain this apparent inconsistency in the physics of ion-gated semiconducting TMDs under magnetic field.
From the experimental point of view, the possible multi-valley character of transport in gated TMDs is currently debated. Refs. measured the Landau-level degeneracy at moderate $n_{2d}\sim10^{12}-10^{13}$cm$^{-2}$, finding it compatible with a carrier population in the Q/Q’ valleys. However, Ref. argued that this would be suppressed for larger $n_{2d}\gtrsim10^{13}$cm$^{-2}$, typical of ion-gated devices and mandatory for the emergence of SC) due to stronger confinement within the first layer[@ChenPRL2017]. In contrast, angle-resolved photoemission spectroscopy in | 0 | non_member_973 |
surface-Rb-doped TMDs[@KangNanoLett2017] highlighted the presence of a non-negligible spectral weight at the Q/Q’ valleys only for $n_{2d}\gtrsim8\cdot10^{13}$cm$^{-2}$ in the case of MoS$_2$. Thus, which valleys and sub-bands are involved in the gate-induced SC state still demands a satisfactory answer.
Here we report multi-valley transport and SC at the surface of liquid-gated FL-MoS$_2$. We use a dual-gate geometry to tune doping across a wide range of $n_{2d}\sim5\cdot10^{12}-1\cdot10^{14}$cm$^{-2}$, induce SC, and detect characteristic “kinks” in the transconductance. These are non-monotonic features that emerge in the $n_{2d}$-dependence of the low-temperature ($T$) conductivity when $E_F$ crosses the high-energy sub-bands[@BrummePRB2016], irrespectively of their specific effective masses, Fig.\[figure:bands\]e. We show that the population of the Q/Q’ valleys is fundamental for the emergence of SC. The crossing of the first sub-band Q$_1$ (Fig.\[figure:bands\]c) occurs at small $n_{2d}\lesssim2\cdot10^{13}$cm$^{-2}$, implying that multi-valley transport already occurs in the metallic phase over a wide range of $n_{2d}\sim2-6\cdot10^{13}$cm$^{-2}$. We also show that the crossing of the second sub-band Q$_2$ occurs after a finite $T_c$ is observed, while a full population of both spin-orbit-split sub-bands (Fig.\[figure:bands\]d) in the Q/Q’ valleys is required to reach the maximum $T_c$. These results highlight how SC can be enhanced in MoS$_2$ by optimizing the connectivity of its | 0 | non_member_973 |
Fermi Surface (FS), i.e. by adding extra FSs in different BZ regions to provide coupling to further phonon branches[@PickettBook]. Since the evolution of the band structure of MoS$_2$ with field-effect doping is analogous to that of other semiconducting TMDs[@BrummePRB2015; @BrummePRB2016; @DasPRB2015; @WuNatCommun2016; @KangNanoLett2017], a similar mechanism is likely associated with the emergence of SC in TMDs in general. Thus, optimization of the FS connectivity can be a viable strategy in the search of new superconductors.
![a) Hall bar FL-MoS$_2$ flake with voltage probes ($V_i$), source (S), drain (D) and liquid-gate (LG) electrodes. A ionic liquid droplet covers the flake and part of the LG electrode. The sample is biased with a source-drain voltage ($V_{SD}$) and dual gate control is enabled by a voltage applied on the liquid gate ($V_{LG}$) and on the solid back gate ($V_{BG}$). b) Optical image of Hall bar with six voltage probes. The LG electrode is on the upper-right corner. c) AFM scan of the MoS$_2$ Hall bar after ionic liquid removal.[]{data-label="figure:device"}](Figure_device_6.eps){width="0.8\columnwidth"}
We study flakes with N=4-10, as Refs. predicted that flakes with N$\geq$4 are representative of the bulk electronic structure, and Ref. experimentally observed that both $T_c$ and the critical magnetic field $H_{c2}$ in 4L | 0 | non_member_973 |
flakes are similar to those of 6L and bulk flakes. Our devices are thus comparable with those in literature[@YeScience2012; @LuScience2015; @SaitoNatPhys2016; @CostanzoNatNano2016; @FuQuantMater2017]. We do not consider 1L flakes as they exhibit a lower $T_c$ and their mobility is suppressed due to disorder[@CostanzoNatNano2016; @FuQuantMater2017].
FL-MoS$_2$ flakes are prepared by micro-mechanical cleavage[@NovoPNAS2005] of 2H-MoS$_2$ crystals from SPI Supplies. The 2H phase is selected to match that in previous reports of gate-induced SC[@YeScience2012; @CostanzoNatNano2016]. Low resistivity ($<0.005\Omega\cdot$cm) Si coated with a thermal oxide SiO$_2$ is chosen as a substrate. We tested both 90 or 285nm SiO$_2$ obtaining identical SC results. Thus, 90nm SiO$_2$ is used to minimize the back gate voltage $V_{BG}$ , while 285nm is used to minimize leakage currents through the back gate $I_{BG}$. Both SiO$_2$ thicknesses provide optical contrast at visible wavelengths[@CasiraghiNanoLett2007]. A combination of optical contrast, Raman spectroscopy and atomic force microscopy (AFM) is used to select the flakes and determine N.
Electrodes are then defined by patterning the contacts area by e-beam lithography, followed by Ti:$10$nm/Au:$50$nm evaporation and lift-off. Ti is used as an adhesion layer[@Kutz2002], while the thicker Au layer provides the electrical contact. Flakes with irregular shapes are further patterned in the shape of Hall | 0 | non_member_973 |
bars by using polymethyl methacrylate (PMMA) as a mask and removing the unprotected MoS$_2$ with reactive ion etching (RIE) in a $150$mTorr atmosphere of CF$_4$:O$_2$=5:1, as shown in Figs.\[figure:device\]a,b. A droplet of 1-Butyl-1-methylpiperidinium bis(trifluoromethylsulfonyl)imide (BMPPD-TFSI) is used to cover the FL-MoS$_2$ surface and part of the side electrode for liquid gate operation (LG), as sketched in Fig.\[figure:device\]a.
AFM analysis is performed with a Bruker Dimension Icon in tapping mode. The scan in Fig.\[figure:device\]c is done after the low-T experiments and removal of the ionic liquid, and confirms that the FL-MoS$_2$ sample does not show topographic damage after the measurement cycle.
![Representative Raman spectra at 514nm of a 4L-MoS$_2$ flake before (blue) and after (red) device fabrication, deposition of the ionic liquid droplet and low-T transport measurements.[]{data-label="figure:Raman"}](Figure_Raman_1.eps){width="\columnwidth"}
We use Raman spectroscopy to characterize the devices both before and after fabrication and BMPPD-TFSI deposition. Raman measurements are performed with a Horiba LabRAM Evolution at 514nm, with a 1800grooves/mm grating and a spectral resolution$\sim0.45$cm$^{-1}$. The power is kept below 300$\mu$W to avoid any damage. A representative Raman spectrum of 4L-MoS$_2$ is shown in Fig.\[figure:Raman\] (blue curve). The peak at$\sim$455cm$^{-1}$ is due to a second-order longitudinal acoustic mode at the M point[@StacJPCS1985]. The E$_{2g}^1$ | 0 | non_member_973 |
peak at$\sim$385cm$^{-1}$ and the A$_{1g}$ at$\sim$409cm$^{-1}$ correspond to in-plane and out-of plane vibrations of Mo and S atoms[@VerbPRL1970; @WietPRB1971]. Their difference, Pos(E$_{2g}^1$)-Pos(A$_{1g}$), is often used to monitor N[@LeeACS2010]. However, for N$\geq$4, the variation in Pos(E$_{2g}^1$)-Pos(A$_{1g}$) between N and N$+$1 approaches the instrument resolution[@LeeACS2010] and this method is no longer reliable. Thus, we use the low frequency modes ($<100$cm$^{-1}$) to monitor N[@ZhanPRB2013; @TanNM2012]. The shear (C) and layer breathing modes (LBM) are due to the relative motions of the atomic planes, either perpendicular or parallel to their normal[@ZhanPRB2013]. Pos(C) and Pos(LBM) change with N as[@ZhanPRB2013; @TanNM2012]: $$\label{Eq1}
\mathrm{Pos(C)_N}=\frac{1}{\sqrt{2}\pi c}\sqrt{\frac{\alpha_{\parallel}}{\mu_m}}\sqrt{1+\cos\left(\frac{\pi}{N}\right)}$$ $$\label{Eq2}
\mathrm{Pos(LBM)_N}=\frac{1}{\sqrt{2}\pi c}\sqrt{\frac{\alpha_{\perp}}{\mu_m}}\sqrt{1-\cos\left(\frac{\pi}{N}\right)}$$ where $\alpha_{\parallel}\sim$2.82$\cdot$10$^{19}$N/m$^3$ and $\alpha_{\perp}\sim$8.90$\cdot$10$^{19}$N/m$^3$ are spring constants for C and LBM modes, respectively, $c$ is the speed of light in vacuum, $\mu_m\sim$3$\cdot$10$^{-6}$Kg/m$^2$ is the 1L mass per unit area[@ZhanPRB2013; @TanNM2012]. Fig.\[figure:Raman\] shows a C mode at$\sim$30cm$^{-1}$ and an LBM at$\sim$22cm$^{-1}$. These correspond to N$=$4 using Eqs.\[Eq1\],\[Eq2\]. Fig.\[figure:Raman\] also plots the Raman measurements after device fabrication, deposition of the ionic liquid, low-T measurements, $V_{LG}$ removal and warm-up to room T (red curve). We still find Pos(C)$\sim$30cm$^{-1}$ and Pos(LBM)$\sim$22cm$^{-1}$, the same as those of the pristine flake, suggesting no damage nor residual doping.
Four-probe resistance and Hall measurements are then performed | 0 | non_member_973 |
in the vacuum chambers of either a Cryomech pulse-tube cryocooler, $T_{min}$=2.7K, or a Lakeshore cryogenic probe-station, $T_{min}$=8K, equipped with a 2T superconducting magnet. A small ($\sim1\mu$A) constant current is applied between S and D (Fig.\[figure:device\]a) by using a two-channel Agilent B2912A source-measure unit (SMU). The longitudinal and transverse voltage drops are measured with an Agilent 34420 low-noise nanovoltmeter. Thermoelectrical and other offset voltages are eliminated by measuring each resistance value and inverting the source current in each measurement[@DagheroPRL2012]. Gate biases are applied between the corresponding G and D with the same two-channel SMU (liquid gate) or a Keithley 2410 SMU (back gate). Samples are allowed to degas in vacuum ($<10^{-5}$mbar) at room $T$ for at least$\sim1$h before measurements, in order to remove residual water traces in the electrolyte.
{width="80.00000%"}
We first characterize the T dependence of the sheet resistance, $R_s$, under the effect of the liquid top gate. We apply the liquid gate voltage, $V_{LG}$, at 240K, where the electrolyte is still liquid, and under high-vacuum ($<10^{-5}$mbar) to minimize unwanted electrochemical interactions and extend the stability window of the ionic liquid[@UenoReview2014]. After $V_{LG}$ is applied, we allow the ion dynamics to settle for$\sim$10min before cooling to a base T=2.7K.
Fig.\[figure:transport\]a | 0 | non_member_973 |
plots the T dependence of $R_s$ measured in a four-probe configuration, for different $V_{LG}$ and induced carrier density $n_{2d}$. Our devices behave similarly to Ref., undergoing first an insulator-to-metal transition near $R_s\sim h/e^2$ at low $n_{2d}<1\cdot 10^{13}$cm$^{-2}$, followed by a metal-to-superconductor transition at high $n_{2d}>6\cdot 10^{13}$cm$^{-2}$. The saturating behavior in the $R_s$ vs T curves in Fig.\[figure:transport\]a for $\mathrm{T}\lesssim50$K, close to the insulator-to-metal transition, is typically observed in systems at low $n_{2d}$ characterized by a fluctuating electrostatic potential, such as that due to charged impurities[@ZabrodskiiJETP1984]. This applies to ion-gated crystalline systems at low $V_{LG}$, since the doping is provided by a low density of ions in close proximity to the active channel. These ions induce a perturbation of the local electrostatic potential, locally inducing charge carriers, but are otherwise far apart. The resulting potential landscape is thus inhomogeneous. This low-doping ($\lesssim1\times10^{13}$cm$^{-2}$) density inhomogeneity is a known issue in ion-gated crystalline systems, but becomes less and less relevant at higher ionic densities[@RenNanoLett2015]. We employ Hall effect measurements to determine $n_{2d}$ as a function of $V_{LG}$ (see Fig.\[figure:transport\]b), and, consequently, the liquid gate capacitance $C_{LG}$. $C_{LG}$ for the BMPPD-TFSI/MoS$_2$ interface ($\sim3.4\pm0.6\mu$F/cm$^{2}$) is of the same order of magnitude as for DEME-TFSI/MoS$_2$ in | 0 | non_member_973 |
Ref. ($\sim8.6\pm4.1\mu$F/cm$^2$), where DEME-TFSI is the N,N-Diethyl-N-methyl-N-(2-methoxyethyl)ammonium bis(trifluoromethanesulfonyl)imide ionic liquid[@ShiSciRep2015].
Fig.\[figure:transport\]a shows that, while for T$\gtrsim$100K $R_s$ is a monotonically decreasing function of $n_{2d}$, the same does not hold for T$\lesssim$100K, where the various curves cross. In particular, the residual $R_s$ in the normal state $R_s^0$ (measured just above $T_c$ when the flake is superconducting) varies non-monotonically as a function of $n_{2d}$. This implies the existence of multiple local maxima in the $R_s^0 (n_{2d})$ curve. Consistently with the theoretical predictions of Ref., we find two local maxima. The first and more pronounced occurs when the flake is superconducting, i.e. for $n_{2d}>6\cdot 10^{13}$cm$^{-2}$. This feature was also reported in Refs., but not discussed. The second, less pronounced kink, is observed for $1\cdot 10^{13} \lesssim n_{2d} \lesssim 2\cdot 10^{13}$cm$^{-2}$, not previously shown. Both kinks can be seen only for T$\lesssim$70K and they are smeared for T$\gtrsim$150K.
The kink that emerges in the same range of $n_{2d}$ as the superconducting dome extends across a wide range of $V_{LG}$ ($3 \lesssim V_{LG} \lesssim 6$V) for $n_{2d}\gtrsim6\cdot10^{13}$cm$^{-2}$, and can be accessed only by LG biasing, due to the small capacitance of the solid BG. This prevents a continuous characterization of its behavior, as $n_{2d}$ induced | 0 | non_member_973 |
by LG cannot be altered for $T\lesssim 220$K, as the ions are locked when the electrolyte is frozen. The kink that appears early in the metallic state, on the other hand, extends across a small range of $n_{2d}$ ($1\lesssim n_{2d} \lesssim 2\cdot10^{13}$cm$^{-2}$), and is ideally suited to be explored continuously by exploiting the dual-gate configuration.
We thus bias our samples in the low-density range of the metallic state ($n_{2d}\sim7\cdot 10^{12}$cm$^{-2}$) by applying V$_{LG}=0.9$V, and cool the system to 2.7K. We then apply $V_{BG}$ and fine-tune $n_{2d}$ across the kink. We constantly monitor $I_{BG}$ to avoid dielectric breakdown. Fig.\[figure:transport\]c plots $\sigma_{2d}$ of a representative device subject to multiple $V_{BG}$ sweeps, as $n_{2d}$ is tuned across the kink. This reproduces well the behavior observed for low $V_{LG}$ ($1\lesssim n_{2d} \lesssim 2\cdot10^{13}$cm$^{-2}$). The hysteresis between increasing and decreasing $V_{BG}$ is minimal. This kink is suppressed by increasing T, similar to LG gating.
$V_{BG}$ provides us an independent tool to estimate $n_{2d}$: If $V_{LG}$ is small enough ($V_{LG}\lesssim 1$V) so that conduction in the channel can be switched off by sufficiently large negative $V_{BG}$ ($V_{BG}\lesssim -25$V), we can write $n_{2d}=C_{ox}/e \cdot (V_{BG}-V_{th})$. Here, $C_{ox}=\epsilon_{ox}/d_{ox}$ is the back gate oxide specific capacitance, $e=1.602\cdot10^{-19}$C is the | 0 | non_member_973 |
elementary charge and $V_{th}$ is the threshold voltage required to observe a finite conductivity in the device. We neglect the quantum capacitance $C_q$ of MoS$_2$, since $C_q\gtrsim$100$\mu$F/cm$^2\gg C_{ox}$[@BrummePRB2015]. By using the dielectric constant of SiO$_2\,\epsilon_{ox}$=3.9[@ElKarehBook1995] and an oxide thickness $t_{ox}=90$nm (or $t_{ox}=285$nm, depending on the experiment) we obtain the $n_{2d}$ scale in the top axis of Fig.\[figure:transport\]c, in good agreement with the corresponding values in Fig.\[figure:transport\]a, estimated from the Hall effect measurements in Fig.\[figure:transport\]b.
The bandstructure of field-effect doped NL-MoS$_2$ depends on N[@BrummePRB2015] and strain[@BrummePRB2016]. A fully relaxed N-layer flake, with N$\geq$3, has been predicted to behave as follows[@BrummePRB2015; @BrummePRB2016]: For small doping ($x\lesssim 0.05$e$^-$/unit cell, Figs.\[figure:bands\]b and \[figure:kinks\]a) only the two spin-orbit split sub-bands at K/K’ are populated. At intermediate doping ($0.05\lesssim x \lesssim 0.1$ e$^-$/unit cell, Figs.\[figure:bands\]c and \[figure:kinks\]b), $E_F$ crosses the first spin-orbit split sub-band at Q/Q’ (labeled Q$_1$). For large doping ($x\gtrsim 0.1$ e$^-$/unit cell, Figs.\[figure:bands\]d and \[figure:kinks\]c) $E_F$ crosses the second sub-band (Q$_2$) and both valleys become highly populated[@BrummePRB2015]. Even larger doping ($x\gtrsim 0.35$ e$^-$/unit cell) eventually shifts the K/K’ valleys above $E_F$[@BrummePRB2015].
{width="80.00000%"}
When $E_F$ crosses these high-energy sub-bands at Q/Q’, sharp kinks are expected to appear in the transconductance of gated FL-MoS$_2$[@BrummePRB2016] (see | 0 | non_member_973 |
Fig.\[figure:bands\]e). These are reminiscent of a similar behavior in liquid-gated FL graphene, where their appearance was linked to the opening of interband scattering channels upon the crossing of high-energy sub-bands[@YePNAS2011; @Gonnelli2dMater2017; @PiattiAppSS2017]. Even in the absence of energy-dependent scattering, Ref. showed that $\sigma_{2d}$ can be expressed as: $$\sigma_{2d}=e^{2}\tau\langle v_{\parallel}^{2}\rangle N\left(E_{F}\right)\propto e^{2}\langle v_{\parallel}^{2}\rangle\label{eq:sigma_velocity}$$ where $\tau\propto N(E_F)^{-1}$ is the average scattering time, and $N\left(E_{F}\right)$ is the density of states (DOS) at $E_F$. This implies that $\sigma_{2d}$ is proportional to the average of the squared in-plane velocity $\text{\ensuremath{\langle}}\ensuremath{v_{\parallel}^{2}}\text{\ensuremath{\rangle}}$ over the FS[@BrummePRB2016]. Since $\text{\ensuremath{\langle}}\ensuremath{v_{\parallel}^{2}}\text{\ensuremath{\rangle}}$ linearly increases with $n_{2d}$ and drops sharply as soon as a new band starts to get doped[@BrummePRB2016], the kinks in $\sigma_{2d}$ (or, equivalently, $R_s$) at $T\lesssim15$K can be used to determine the onset of doping of the sub-bands in the Q/Q’ valleys. At T=0, the kink is a sharp drop in $\sigma_{2d}$, emerging for the doping value at which $E_F$ crosses the bottom of the next sub-band. This correspondence is lost due to thermal broadening for T$>$0, leading to a smoother variation in $\sigma_{2d}$. If T is sufficiently large the broadening smears out any signature of the kinks, Fig.\[figure:transport\]. Ref. calculated that, at finite T, the conductivity kinks define a | 0 | non_member_973 |
*doping range* where the sub-band crossing occurs (between $R_s$ minimum and maximum, i.e. the *lower* and *upper* bounds of each kink sets the resolution of this approach). Each sub-band crossing starts after the $R_s$ minimum at lower doping, then develops in correspondence of the inflection point, and is complete once the $R_s$ maximum is reached.
We show evidence for this behavior in Fig.\[figure:kinks\], where we plot $T_c$ (panel d) and $R_s^0$ (panel e) as a function of $n_{2d}$. The electric field is applied both in liquid-top-gate (filled dots and dashed line) and dual-gate (solid red line) configurations. For comparable values of $n_{2d}$, the liquid-gate geometry features larger $R_s^0$ than back-gated. This difference is due to increased disorder introduced when $n_{2d}$ is modulated via ionic gating[@Gonnelli2dMater2017; @PiattiAppSS2017; @PiattiAPL2017; @GallagherNatCommun2015; @OvchinnikovNatCommun2016]. Two kinks appear in the $n_{2d}$ dependence of $R_s$: a low-doping one for $1.5\cdot 10^{13} \lesssim n_{2d} \lesssim 2\cdot 10^{13}$cm$^{-2}$, and a high-doping one for $7\cdot 10^{13} \lesssim n_{2d} \lesssim 9\cdot 10^{13}$cm$^{-2}$. The plot of the SC dome of gated MoS$_2$ on the same $n_{2d}$ scale shows that the low-doping kink appears well before the SC onset, while the second appears immediately after, before the maximum $T_c$ is reached.
These results | 0 | non_member_973 |
can be interpreted as follows. When $n_{2d} \lesssim 1\cdot10^{13}$cm$^{-2}$, only the spin-orbit split sub-bands at K/K’ are populated, and the FS is composed only by two pockets, Fig.\[figure:kinks\]a. For $n_{2d}$ between$\sim1.5$ and $2\cdot 10^{13}$cm$^{-2}$, $E_F$ crosses the bottom of the Q$_1$ sub-band and two extra pockets appear in the FS at Q/Q’[@GePRB2013; @BrummePRB2015], Fig.\[figure:kinks\]b. The emergence of these pockets induces a Lifshitz transition, i.e. an abrupt change in the topology of the FS[@LifshitzJETP1960]. Once Q$_1$ is populated and $E_F$ is large enough ($n_{2d}\sim6\cdot 10^{13}$cm$^{-2}$), the system becomes superconducting[@LuScience2015; @YeScience2012]. For slightly larger $E_F$ ($7\cdot 10^{13} \lesssim n_{2d} \lesssim 9\cdot 10^{13}$cm$^{-2}$), $E_F$ crosses the bottom of Q$_2$ resulting in a second Lifshitz transition, and other two pockets emerge in the FS at Q/Q’[@BrummePRB2015], Fig.\[figure:kinks\]c.
We note that the experimentally observed kinks are at different $n_{2d}$ with respect to the theoretical ones for 3L-MoS$_2$[@BrummePRB2016]. Ref. predicted that for a $1.28\%$ in-plane tensile strain, Q$_1$ and Q$_2$ should be crossed for $n_{2d}\sim5\cdot 10^{13}$ and $\sim1\cdot 10^{14}$. Since the positions of the sub-band crossings are strongly dependent on strain[@BrummePRB2016], we estimate the strain in our devices by monitoring the frequency of the E$_{2g}^1$ mode via Raman spectroscopy.
![a) Raman spectra of the 4L-MoS$_2$ device | 0 | non_member_973 |
in Fig.\[figure:device\]c from 4 to 292K. b) Shift in the position of the E$_{2g}^1$ mode as a function of T for as-prepared bulk flake (black circles), a 4L-MoS$_2$ flake (blue circles), and a 4L-MoS$_2$ device with Au contacts (red circles).[]{data-label="fig:strain"}](Figure_strain_2.eps){width="\columnwidth"}
Strain can arise due to a mismatch in the thermal expansion coefficients (TECs) of MoS$_2$[@GanPRB2016], SiO$_2$ substrate[@NIST_standard1991] and Au electrodes[@NixPR1941]. Upon cooling, MoS$_2$, SiO$_2$ and Au would normally undergo a contraction. However the flake is also subject to a tensile strain due to TEC mismatch[@YoonNL2011]. The strain, $\epsilon_{MoS_2}$, due to the MoS$_2$-SiO$_2$ TEC mismatch is: $$\epsilon_{MoS_2}=\int_{T}^{292K} [\alpha_{MoS_2}(T)-\alpha_{SiO_2}(T)] dT$$ whereas the strain, $\epsilon_{Au}$, due to the Au contacts is: $$\epsilon_{Au}=\int_{T}^{292K} [\alpha_{Au}(T)-\alpha_{SiO_2}(T)] dT$$ $\epsilon_{MoS_2}$ and $\epsilon_{Au}$ are$\sim$0.1% and$\sim$0.3% at $\sim$4K, respectively[@YoonNL2011].
Any FL-MoS$_2$ on SiO$_2$ will be subject to $\epsilon_{MoS_2}$ at low T. When the flake is contacted, an additional contribution is present due to $\epsilon_{Au}$. This can be more reliably estimated performing T-dependent Raman scattering and comparing the spectra for contacted and un-contacted flakes[@YoonNL2011; @MohiPRB2009]. Figs.\[fig:strain\]a,b show how a T decrease results in the E$_{2g}^1$ mode shifting to higher frequencies for both as-prepared and contacted 4L-MoS$_2$, due to anharmonicity[@KlemPR1966]. However, in the as-prepared 4L-MoS$_2$, the up-shift is$\sim$1cm$^{-1}$ larger with respect to | 0 | non_member_973 |
the contacted one. This difference points to a further tensile strain. Refs. suggested that uniaxial tensile strain on 1L-MoS$_2$ induces a E$_{2g}^1$ softening and a splitting in two components: E$_{2g}^{1+}$ and E$_{2g}^{1-}$[@LeeNC2017; @ConlNL2013]. The shift rates for E$_{2g}^{1+}$ and E$_{2g}^{1-}$ are from -0.9 to -1.0cm$^{-1}$/% and from -4.0 to -4.5cm$^{-1}$/%, respectively[@LeeNC2017; @ConlNL2013]. We do not observe splitting, pointing towards a biaxial strain. As for Ref., we calculate a shift rate of E$_{2g}^1$ for biaxial strain from -7.2 to -8.2cm$^{-1}$/%. The amount of tensile strain on the 4L-MoS$_2$ device can thus be estimated. The E$_{2g}^1$ up-shift difference between contacted and as-prepared 4L-MoS$_2$, $\Delta$Pos(E$_{2g}^1$), at 4K is$\sim$-1.0cm$^{-1}$, corresponding to an additional $\sim$0.13% biaxial tensile strain. Thus, assuming a 0.1% strain for the as-prepared 4L-MoS$_2$ due to TEC mismatch with SiO$_2$, we estimate the total strain in the contacted 4L-MoS$_2$ to be$\sim$0.23% at$\sim$4K.
![a) Surface carrier densities required to cross the Q$_1$ sub-band in FL-MoS$_2$ as a function of tensile strain. Theoretical values for 1L (black dots and line) and 3L (red triangle and line) from Ref.; values for 4L (green diamonds and line) are by linear extrapolation. Blue diamond is the present experiment. b) EPC enhancement due to the crossing of the | 0 | non_member_973 |
Q$_2$ sub-band, $\Delta \lambda$, as a function of $n_{2d}$, assuming $\omega_{ln}=230\pm30$K and $\mu^*=0.13$[@GePRB2013]. Filled blue circles are our experiments. Black and magenta open circles from Refs.. The blue dashed line is a guide to the eye.[]{data-label="figure:coupling"}](Figure_EPC_4.eps){width="\columnwidth"}
Fig.\[figure:coupling\]a shows that, for $0.23$% tensile strain, the experimentally observed positions of the kinks agree well with a linear extrapolation of the data of Ref. to 4L-MoS$_2$ (representative of our experiments) and for in-plane strain between $0\%$ (bulk) and $1.28\%$ (fully relaxed). These findings indicate that, while the mechanism proposed in Ref. qualitatively describes the general behavior of gated FL-MoS$_2$, quantitative differences arise due to the spin-orbit split of the Q$_1$ and Q$_2$ sub-bands. The main reason for the EPC (and, hence, $T_c$) increase is the same, i.e. the increase in the number of phonon branches involved in the coupling when the high-energy valleys are populated[@GePRB2013]. However, the finite spin-orbit-split between the sub-bands significantly alters the FS connectivity upon increasing doping[@BrummePRB2015]. If we consider the relevant phonon wave vectors ($q$=$\Gamma$,K,M,$\Gamma$K$/2$) for 1L- and FL-MoS$_2$[@MolinaPRB2011; @AtacaJPCC2011], and only the K/K’ valleys populated, then only phonons near $\Gamma$ and K can contribute to EPC[@GePRB2013]. The former strongly couple e$^-$ within the same valley[@GePRB2013], but cannot contribute | 0 | non_member_973 |
significantly due to the limited size of the Fermi sheets[@GePRB2013]. The latter couple e$^-$ across different valleys[@GePRB2013], and provide a larger contribution[@GePRB2013], insufficient to induce SC. MoS$_2$ flakes are metallic but not superconducting before the crossing of Q$_1$. When this crossing happens, the total EPC increases due to the contribution of longitudinal phonon modes near K[@GePRB2013] (coupling states near two different Q or Q’), near $\Gamma$K$/2$[@GePRB2013] (coupling states near Q to states near Q’), and near M[@GePRB2013] (coupling states near Q or Q’ to states near K or K’). However, this first EPC increase associated with Q$_1$ is not sufficient to induce SC, as the SC transition is not observed until immediately before the crossing of the spin-orbit-split sub-band Q$_2$ and the second doping-induced Lifshitz transition. Additionally, the SC dome shows a maximum in the increase of $T_c$ with doping ($dT_c/dn_{2d}$) across the Q$_2$ crossing, i.e. when a new FS emerges. Consistently, the subsequent reduction of $T_c$ for $n_{2d}\geq 13\cdot 10^{13}$cm$^{-2}$ can be associated with the FS shrinkage and disappearance at K/K’[@GePRB2013; @BrummePRB2015], and might also be promoted by the formation of an incipient Charge Density Wave[@RosnerPRB2014; @Piattiarxiv2018] (characterized by periodic modulations of the carrier density coupled to a distortion | 0 | non_member_973 |
of the lattice structure[@GrunerBook2009]).
Since the evolution of the bandstructure with doping is similar in several semiconducting TMDs[@BrummePRB2015; @DasPRB2015; @BrummePRB2016; @WuNatCommun2016; @KangNanoLett2017], this mechanism is likely not restricted to gated MoS$_2$. The $T_c$ increase in correspondence to a Lifshitz transition is reminiscent of a similar behavior observed in CaFe$_2$As$_2$ under pressure[@GonnelliSciRep2016], suggesting this may be a general feature across different classes of materials.
We note that the maximum $T_c\sim11$K is reached at $n_{2d}\simeq 12\cdot 10^{13}$cm$^{-2}$, as reported in Ref.. This is a doping level larger than any doping level which can be associated with the kink. Thus, the Q$_2$ sub-band must be highly populated when the maximum $T_c$ is observed. We address this quantitatively with the Allen-Dynes formula[@AllenDynes], which describes the dependence of $T_c$ by a numerical approximation of the Eliashberg theory accurate for materials with a total $\lambda\lesssim1.5$[@AllenDynes]: $$\label{EqAllenDynes}
T_c(n_{2d})=\frac{\omega_{ln}}{1.2}\mathrm{exp}\left\lbrace\frac{-1.04\left[1+\lambda(n_{2d})\right]}{\lambda(n_{2d})-\mu^*\left[1+0.62\lambda(n_{2d})\right]}\right\rbrace$$ where $\lambda(n_{2d})$ is the total EPC as a function of doping, $\omega_{ln}$ is the representative phonon frequency and $\mu^*$ is the Coulomb pseudo-potential. It is important to evaluate the increase in EPC between the non-superconducting region ($n_{2d}\lesssim6\times10^{13}$cm$^{-2}$) and the superconducting one, i.e. the enhancement in $\lambda$ due to the crossing of the sub-band at Q$_2$. $\Delta\lambda=\lambda(T_c)-\lambda(T_c=0)$ indicates the EPC | 0 | non_member_973 |
increase due to the appearance of e$^-$ pockets at Q$_2$. By setting $\omega_{ln}=230\pm30$K and $\mu^*=0.13$ (as for Ref.), and using Eq.\[EqAllenDynes\], we find that the limit of $\lambda(T_c)$ for $T_c\to0$ is $\sim0.25$. The corresponding $\Delta\lambda$ vs. $n_{2d}$ dependence is shown in Fig.\[figure:coupling\]b. The crossing at Q$_2$ results in a maximum $\Delta\lambda=0.63\pm0.1$, with a maximum EPC enhancement of $350\pm40\%$ with respect to the non-superconducting region. This indicates that the largest contribution to the total EPC, hence to the maximum $T_c\sim11$K, is associated with the population of the Q$_2$ sub-band. This is consistent with the reports of a reduced $T_c\sim2$K in 1L-MoS$_2$[@CostanzoNatNano2016; @FuQuantMater2017], shown to be superconducting for smaller $n_{2d}\sim5.5\cdot10^{13}$cm$^{-2}$[@FuQuantMater2017], hence likely to populate Q$_1$ only. $n_{2d}\sim5\cdot10^{13}$cm$^{-2}$ is also the doping expected for the crossing of Q$_1$ in 1L-MoS$_2$ in presence of a low-T strain similar to that in our 4L-MoS$_2$ devices (see Fig.\[figure:coupling\]a).
In summary, we exploited the large carrier density modulation provided by ionic gating to explore sub-band population and multivalley transport in MoS$_2$ layers. We detected two kinks in the conductivity, associated with the doping-induced crossing of the two sub-bands at Q/Q’. By comparing the emergence of these kinks with the doping dependence of $T_c$, we showed how superconductivity | 0 | non_member_973 |
---
abstract: 'It is well known that reverberation mapping of active galactic nuclei (AGN) reveals a relationship between AGN luminosity and the size of the broad-line region, and that use of this relationship, combined with the Doppler width of the broad emission line, enables an estimate of the mass of the black hole at the center of the active nucleus based on a single spectrum. This has been discussed in numerous papers over the last two decades. An unresolved key issue is the choice of parameter used to characterize the line width; generally, most researchers use FWHM in favor of line dispersion (the square root of the second moment of the line profile) because the former is easier to measure, less sensitive to blending with other features, and usually can be measured with greater precision. However, use of FWHM introduces a bias, stretching the mass scale such that high masses are overestimated and low masses are underestimated. Here we investigate estimation of black hole masses in AGNs based on individual or “single epoch” observations, with a particular emphasis in comparing mass estimates based on line dispersion and FWHM. We confirm the recent findings that, in addition to luminosity and line | 0 | non_member_974 |
width, a third parameter is required to obtain accurate masses and that parameter seems to be Eddington ratio. We present simplified empirical formulae for estimating black hole masses from the $\lambda4861$ and $\lambda1549$ emission lines.'
author:
- Elena Dalla Bontà
- 'Bradley M. Peterson'
- 'Misty C. Bentz'
- 'W. N. Brandt'
- 'S. Ciroi'
- Gisella De Rosa
- Gloria Fonseca Alvarez
- 'Catherine J. Grier'
- 'P. B. Hall'
- 'Juan V. Hernández Santisteban'
- 'Luis C. Ho'
- 'Y. Homayouni'
- Keith Horne
- 'C. S. Kochanek'
- 'Jennifer I-Hsiu Li'
- 'L. Morelli'
- 'A. Pizzella'
- 'R. W. Pogge'
- 'D. P. Schneider'
- Yue Shen
- 'J. R. Trump'
- Marianne Vestergaard
title: 'THE SLOAN DIGITAL SKY SURVEY REVERBERATION MAPPING PROJECT: ESTIMATING MASSES OF BLACK HOLES IN QUASARS WITH SINGLE-EPOCH SPECTROSCOPY'
---
Introduction {#section:intro}
============
Reverberation-Based Black Hole Masses
-------------------------------------
The presence of emission lines with Doppler widths of thousands of kilometers per second is one of the defining characteristics of active galactic nuclei [@Burbidge67; @Weedman76]. It was long suspected that the large line widths were due to motions in a deep gravitational potential and this implied very large central masses [e.g., @Woltjer59], as | 0 | non_member_974 |
did the Eddington limit [@Tarter73]. Under a few assumptions, the central mass is $M
\propto V^2 R$, where $V$ is the Doppler width of the line and $R$ is the size of the broad-line region (BLR). It is the latter quantity that is difficult to determine. An early attempt to estimate $R$ by [@Dibai80] was based on the assumption of constant emissivity per unit volume, but led to an incorrect dependence on luminosity as in this case, luminosity is proportional to volume, so $R \propto L^{1/3}$. [@Wandel85] inferred the BLR size from the luminosity. Other attempts were based on photoionization physics [see @Ferland85; @Osterbrock85]. [@Davidson72] found that the relative strength of emission lines in ionized gas could be characterized by an ionization parameter $$U= \frac{Q({\rm H})}{4\pi R^2 c n_{\rm H}},
\label{eq:Udef}$$ where $Q({\rm H})$ is the rate at which H-ionizing photons are emitted by the central source and $n_{\rm H}$ is the particle density of the gas. The ionization parameter $U$ is proportional to the ratio of ionization rate to recombination rate in the BLR clouds. The similarity of emission-line flux ratios in AGN spectra over orders of magnitude in luminosity suggested that $U$ is constant, and the presence of | 0 | non_member_974 |
C[iii]{}\]$\lambda1909$ set an upper limit on the density $n_{\rm H} \la 10^{9.5}$ [@Davidson79]. Since $L \propto Q({\rm H})$, this naturally led to the prediction that the BLR radius would scale with luminosity as $R \propto L^{1/2}$. Unfortunately, best-estimate values for $Q({\rm
H})$ and $n_{\rm H}$ led to a significant overestimate of the BLR radius [@Peterson85] as a consequence of the simple but erroneous assumption that all the broad lines arise cospatially (i.e., models employed a single representative BLR cloud).
With the advent of reverberation mapping [hereafter RM; @Blandford82; @Peterson93], direct measurements of $R$ enabled improved black hole mass determinations. Attempts to estimate black hole masses based on early RM results and the $R \propto L^{1/2}$ prediction included those of [@Padovani88], [@Koratkar91], and [@Laor98]. The first multiwavelength RM campaigns demonstrated ionization stratification of the BLR [@Clavel91; @Krolik91; @Peterson91] and this eventually led to identification of the virial relationship, $R \propto V^{-2}$ [@PetersonWandel99; @PetersonWandel00; @Onken02; @Kollatschny03; @Bentz10], that gave reverberation-based mass measurements higher levels of credibility. Of course, the virial relationship demonstrates only that the central force has a $R^{-2}$ dependence, which is also characteristic of radiation pressure; whether or not radiation pressure from the continuum source is important has not been | 0 | non_member_974 |
clearly established [@Marconi08; @Marconi09; @Netzer10]. If radiation pressure in the BLR turns out to be important, then the black hole masses, as we discuss them here, are underestimated.
Masses of AGN black holes are computed as $$\label{eq:masseqn}
M_{\rm BH} = f \left(\frac{V^2 R}{G}\right),$$ where $V$ is the line width, $R$ is the size of the BLR from the reverberation lag, and $G$ is the gravitational constant. The quantity in parentheses is often referred to as the virial product $\mu$; it incorporates the two observables in RM, line width and time delay $\tau = R/c$, and is in units of mass. The scaling factor $f$ is a dimensionless quantity of order unity that depends on the geometry, kinematics, and inclination of the AGN. Throughout most of this work, we ignore $f$ (i.e., set it to unity) and work strictly with the virial product.
While reverberation mapping has emerged as the most effective technique for measuring the black hole masses in AGNs [@Peterson14], it is resource intensive, requiring many observations over an extended period of time at fairly high cadence. Fortunately, observational confirmation of the $R$–$L$ relationship [@Kaspi00; @Kaspi05; @Bentz06a; @Bentz09a; @Bentz13] enables “single-epoch” (SE) mass estimates because, in principle, a single | 0 | non_member_974 |
spectrum could yield $V$ and also $R$, through measurement of $L$ [e.g., @Wandel99; @McLure02; @Vestergaard02; @Corbett03; @Vestergaard04; @Kollmeier06; @Vestergaard06; @Fine08; @Shen08a; @Shen08b; @Vestergaard08]. Of the three strong emission lines generally used to estimate central black hole masses, the $R$–$L$ relationship is only well-established for $\lambda4861$ [@Bentz13 and references therein, but see the discussion in §\[section:hbpredictor\]]. Empirically establishing the $R$–$L$ relationship for $\lambda2798$ [@Homayouni20] and $\lambda1549$ [@Peterson05; @Kaspi07; @Trevese14; @Lira18; @Grier19; @Hoormann19] has been difficult.
Masses based on the $\lambda1549$ emission line, in particular, have been somewhat controversial. Some studies claim that there is good agreement between masses based on and those measured from other lines [@Vestergaard06; @Greene10b; @Assef11]. On the other hand, there are several claims that there is inadequate agreement with masses based on other emission lines [@Baskin05; @Netzer07; @Sulentic07; @Shen08b; @Shen12; @Trak12]. [@Denney09a] and [@Denney13], however, note that there are a number of biases that can adversely affect single-epoch mass estimates, with low $S/N$ “survey quality” data being an important problem with some of the studies for which poor agreement between and other lines is found. It has also been argued, however, that some fitting methodologies are more affected by this than others [@Shen19]. There have been more | 0 | non_member_974 |
recent papers that attempt to correct mass determinations to better agree with those based on other lines [e.g., @Runnoe13; @Coatman17; @Mejia18; @Marziani19].
Characterizing Line Widths
--------------------------
It is our suspicion that the apparent difficulties with -based masses trace back not only to the $S/N$ issue, but also to how the line widths are characterized. It has been customary in AGN studies to characterize line widths by one of two parameters, either FWHM or the line dispersion , which is defined by $${\mbox{$\sigma_{\rm line}$}}= \left[ \frac{ \int (\lambda - \lambda_0)^2 P(\lambda)\,d\lambda}{\int P(\lambda)\,d\lambda} \right]^{1/2},
\label{eq:Defsigl}$$ where $P(\lambda)$ is the emission-line profile as a function of wavelength and $\lambda_0$ is the line centroid, $$\lambda_0 = \frac{\int \lambda P(\lambda)\,d\lambda}{\int P(\lambda)\,d\lambda}.
\label{eq:Deflambda0}$$ While both FWHM and have been used in the virial equation to estimate AGN black hole masses, they are not interchangeable. It is well known that AGN line profiles depend on the line width [@Joly85]: broader lines have lower kurtosis, i.e., they are “boxier” rather than “peakier.” Indeed, for AGNs, the ratio ${\rm FWHM}/{\mbox{$\sigma_{\rm line}$}}$ has been found to be a simple but useful characterization of the line profile [@Collin06; @Kollatschny13].
Each line-width measure has practical strengths and weaknesses [@Peterson04; @Wang20]. The line | 0 | non_member_974 |
dispersion is more physically intuitive, but it is sensitive to the line wings, which are often badly blended with other features. All three of the strong lines usually used to estimate masses — $\lambda4861$, $\lambda2798$, and $\lambda1549$ — are blended with other features: the $\lambda4570$ and $\lambda\lambda$5190, 5320 complexes [@Phillips78] and $\lambda4686$ in the case of , the UV complexes in the case of , and $\,\lambda1640$ in the case of . The FWHM can usually be measured more precisely than (although @Peterson04 note that the opposite is true for the rms spectra, described below, which are sometimes quite noisy), but it is not clear that FWHM yields more [*accurate*]{} mass measurements. In practice, FWHM is used more often than because it is relatively simple to measure and can be measured more precisely while often requires deblending or modeling the emission features, which does not necessarily yield unambiguous results.
There are, however, a number of reasons to prefer to FWHM as the line-width measure for estimating AGN black hole masses. [@Fromerth00] point out that better characterizes an arbitrary or irregular line profile. [@Peterson04] note that produces a tighter virial relationship than FWHM, and [@Denney13] find better agreement between -based and | 0 | non_member_974 |
-based mass estimates by using rather than FWHM (these latter two are essentially the same argument). In the case of NGC 5548, for which there are multiple reverberation-based mass measures, a possible correlation with luminosity is stronger for FWHM-based masses than for -based masses, suggesting that the former are biased as the same mass should be recovered regardless of the luminosity state of the AGN [@Collin06; @ShenKelly12]. A possibly more compelling argument for using instead of FWHM is bias in the mass scale that is introduced by using FWHM as the line width. [@Steinhardt10] used single-epoch masses for more than 60,000 SDSS quasars [@Shen08b] with masses computed using FWHM. They found that, in any redshift bin, if one plots the distribution of mass versus luminosity, the higher mass objects lie increasingly below the Eddington limit; they refer to this as the “sub-Eddington boundary.” There is no physical basis for this. [@Rafiee11] point out, however, that if the quasar masses are computed using instead of FWHM, the sub-Eddington boundary disappears: the distribution of quasar black hole masses approaches the Eddington limit at all masses. Referring to Figure 1 of [@Rafiee11], the distribution of quasars in the mass vs. luminosity diagram is | 0 | non_member_974 |
an enlongated cloud of points whose axis is roughly parallel to the Eddington ratio when is used to characterize the line width. However, when FWHM is used, the axis of the distribution rotates as the higher masses are underestimated and the lower masses are overestimated. However, the apparent rotation of the mass distribution is in the same sense that is expected from the Malmquist bias and a bottom heavy quasar mass function [@Shen13]. Unfortunately, these arguments are not statistically compelling. Examination of the –$\sigma{*}$ relation using FWHM-based and -based masses is equally unrevealing [@Wang19].
In reverberation mapping, a further distinction among line-width measures must be drawn since either FWHM or can be measured in the mean spectrum, $$\overline{F}(\lambda) =\frac{1}{N} \sum_{1}^{N} F_i(\lambda),
\label{eq:meanspec}$$ where $F_i(\lambda)$ is the flux in the $i^{th}$ spectrum of the time series at wavelength $\lambda$ and $N$ is the number of spectra, or they can be measured in the rms residual spectrum (hereafter simply “rms spectrum”), which is defined as $$\sigma_{\rm rms}(\lambda) = \left\{ \frac{1}{N-1}
\sum_{1}^{N}\left[ F_i(\lambda) -
\overline{F}(\lambda)\right]^2 \right\}^{1/2}.
\label{eq:rmsspec}$$ In this paper, we will specifically refer to the measurements of in the mean spectrum as and in the rms spectrum as . Similarly, refers to | 0 | non_member_974 |
FWHM of a line in the mean spectrum or a single-epoch spectrum and is the FWHM in the rms spectrum. It is common to use as the line-width measure for determining black hole masses from reverberation data — it is intuitatively a good choice as it isolates the gas in the BLR that is actually responding to the continuum variations. As noted previously, the strong and strongly variable broad emission lines can be hard to isolate as they are blended with other features. In the rms spectra, however, the contaminating features are much less of a problem because they are generally constant or vary either slowly or weakly and thus nearly disappear in the rms spectra.
Since the goal is to measure a black hole mass from a single (or a few) spectra, we must use a proxy for . Here we will attempt to determine if either or in a single or mean spectrum can serve as a suitable proxy for ; we know [*a priori*]{} that there are good, but non-linear, correlations between and both and . It therefore seems likely that either or could be used as a proxy for .
Investigation of the relationship among the | 0 | non_member_974 |
line-width measures motivated a broader effort to produce easy-to-use prescriptions for computing [*accurate*]{} black hole masses using and emission lines and nearby continuum fluxes measurements for each line. We note that we do not discuss RM results in this contribution as the present situation has been addressed rather thoroughly by [@Bahk19] and new SDSS-RM results will be published soon [@Homayouni20]. In §[2]{}, the data used in this investigation are described. In §[3]{}, the relationship between the reverberation lag and different measures of the AGN luminosity are considered, and we identify the physical parameters to lead to accurate black-hole mass determinations. In §[4]{}, we will similarly discuss masses based on . In §[5]{}, we present simple empirical formulae for estimating black hole masses from or . The results of this investigation and our future plans to improve this method are outlined in §[6]{}. Our results are briefly summarized in §[7]{}. Throughout this work, we assume $H_0 = 72$Mpc$^{-1}$, $\Omega_{\rm matter} =0.3$ and $\Omega_\Lambda = 0.7$.
Observational Database and Methodology
======================================
Data {#section:Data}
----
We use two high-quality databases for this investigation:
1. Spectra and measurements for previously reverberation-mapped AGNs, for (Table A1) and for (Table A2). These are mostly taken from | 0 | non_member_974 |
the literature (see also @Bentz15 for a compilation[^1]). Sources without estimates of host-galaxy contamination to the optical luminosity $L(5100\,{\rm \AA})$ have been excluded. This database provides the fundamental $R$–$L$ calibration for the single-epoch mass scale. In this contribution, we will refer to these collectively as the “reverberation-mapping database (RMDB)”.
2. Spectral measurements from the Sloan Digital Sky Survey Reverberation Mapping Program [@Shen15 hereafter “SDSS-RM” or more compactly simply as “SDSS”]. We use both (Table A3) and (Table A4) data from the 2014–2018 SDSS-RM campaign [@Grier17b; @Shen19; @Grier19]. Each spectrum is comprised of the average of the individual spectra obtained for each of the 849 quasars in the SDSS-RM field.
In addition, because RM measurements remain rather scarce, we augmented the sample with measurements from [@Vestergaard06] (hereafter VP06), who combined single-epoch luminosity and line-width measurements from archival UV spectra with -based mass measurements of the objects in Table A1. The UV parameters are given in Table A5; we note, however, that we have excluded 3C273 and 3C390.3 because they both have uncertainities in their virial product larger than 0.5dex; the former was a particular problem because there were far more measurements of UV parameters for this source than for any other | 0 | non_member_974 |
and the combination of a large number of measurements and a poorly constrained virial product conspired to disguise real correlations.
All SDSS-RM spectra have been reduced and processed as described by [@Shen15] and [@Shen16b], including post-processing with [PrepSpec]{} (Horne, in preparation). We note that only lags ($\tau$), line dispersion in the rms spectrum (${\mbox{$\sigma_{\rm R}$}}$), and virial products (${\mbox{$\mu_{\rm RM}$}}= {\mbox{$\sigma_{\rm R}$}}^2 c \tau/G$) are taken from [@Grier17b] and [@Grier19]; all luminosities and other line-width measures are from [@Shen19] (Tables A3 and A4 are included here for the sake of clarity).
For each SDSS AGN, there are two determinations of both and ; one is the best-fit (BF) to the mean spectrum, and the other is the mean of multiple Monte Carlo (MC) realizations. For each MC realization, $N$ independent random selections of the $N$ spectra are combined and the line width is measured for both and . After a large number of realizations, the mean $\langle V
\rangle$ and rms $\Delta V$, for $V = {\mbox{${\rm FWHM}_{\rm M}$}}$ and $V = {\mbox{$\sigma_{\rm M}$}}$ are computed, and the rms values are adopted as the uncertainties in each line-width measure.
For the purpose of mass estimation, we need to establish relationships | 0 | non_member_974 |
based on the most reliable data. Many of the SDSS average spectra are still quite noisy, so we imposed quality cuts. Even though we are for the most part restricting our attention to the SDSS-RM quasars for which there are measured lags for (44 quasars) or (48 quasars), we impose these cuts on the entire sample for the sake of later discussion. The first quality condition is that $$\label{eq:minwidth}
V \geq 1000\,{\mbox{\rm km~s$^{-1}$}}$$ for both $V = {\mbox{${\rm FWHM}_{\rm M}$}}$ and $V ={\mbox{$\sigma_{\rm M}$}}$, since AGNs with lines narrower than 1000 are probably Type 2 AGNs; there are some Type 1 AGNs with line widths narrower than this, including several in Table A1, but these are low-luminosity AGNs [e.g., @Greene07], not SDSS quasars. The second quality condition is that the best fit value $V({\rm BF})$ must lie in the range $$\label{eq:consistency}
\langle V \rangle - \Delta V \leq V({\rm BF}) \leq \langle V \rangle + \Delta V$$ for both FWHM and . A third quality condition is a “signal-to-noise” ($S/N$) requirement that the line width must be significantly larger than its uncertainty. Some experimentation showed that $$\label{eq:s2n}
\frac{V}{\Delta V} \geq 10$$ is a good criterion for both $V = {\mbox{${\rm | 0 | non_member_974 |
FWHM}_{\rm M}$}}$ and $V = {\mbox{$\sigma_{\rm M}$}}$ to remove the worst outliers from the line-width comparisons discussed in §[\[section:hblinewidths\]]{} and §[\[section:civfundamental\]]{}.
Finally, we removed quasars that were flagged by [@Shen19] as having broad absorption lines (BALs), mini-BALs, or suspected BALs in .
The effect of each quality cut on the size of the database available for each emission line is shown in Table 1. Of the 44 SDSS-RM quasars with measured lags, 12 failed to meet at least one of the quality criteria, usually the $S/N$ requirement, thus reducing the SDSS-RN sample to 32 quasars. Three quasars with reverberation measurements (RMID 362, 408, and 722) were rejected for significant BALs, thus reducing the SDSS-RM reverberation sample to 45 quasars. As we will show in §[\[section:massformulae\]]{}, another effect of imposing quality cuts is, not surprisingly, that it removes some of the lower luminosity sources from the sample.
[lcc]{} Original sample & 221 & 540\
(a) Minimum Line Width (eq. \[eq:minwidth\]) & 199 & 520\
(b) Consistency (eq. \[eq:consistency\]) & 194 & 368\
(c) $S/N$ (eq. \[eq:s2n\]) & 121 & 462\
(a) + (b) & 174 & 352\
(a) + (c) & 108 & 450\
(b) + (c) & 107 & 309\
| 0 | non_member_974 |
(a) + (b) + (c) & 96 & 299\
All + BAL removal & 96 & 248 \[table:qcuts\]
Fitting Procedure
-----------------
Throughout this work, we use the fitting algorithm described by [@Cappellari13] that combines the Least Trimmed Squares technique of [@Rousseeuw06] and a least-squares fitting algorithm which allows errors in all variables and includes intrinsic scatter, as implemented by [@DallaBonta18]. Briefly, the fits we perform here are of the general form $$\label{eq:powerlaw}
y = a + b\left(x - x_0 \right),$$ where $x_0$ is the median value of the observed parameter $x$. The fit is done iteratively with $5 \sigma$ rejection (unless stated otherwise) and the best fit minimizes the quantity $$\label{eq:chi2line}
\chi^2=\sum_{i=1}^N \frac{[a+b (x_i-x_0) - y_i]^2}
{(b \Delta x_i)^2 + (\Delta y_i)^2 + \varepsilon_y^2},$$ where $\Delta x_i$ and $\Delta y_i$ are the errors on the variables $x_i$ and $y_i$, and $\varepsilon_y$ is the sigma of the Gaussian describing the distribution of intrinsic scatter in the $y$ coordinate; $\varepsilon_y$ is iteratively adjusted so that the $\chi^2$ per degree of freedom $\nu=N-2$ has the value of unity expected for a good fit. The observed scatter is $$\label{eq:Deltadef}
\Delta = \left\{ \frac{1}{N-2} \sum_{i=1}^N
\left[y_i - a - b\left(x_i - x_0\right) \right]^2 \right\}^{1/2}.$$ The | 0 | non_member_974 |
value of $\varepsilon_y$ is added in quadrature when $y$ is used as a proxy for $x$.
The bivariate fits are intended to establish the physical relationships among the various parameters and also to fit residuals. The actual mass estimation equations that we use will be based on multivariate fits of the general form $$\label{eq:twoparameters}
z = a + b\left( x - x_0 \right) +c\left(y - y_0\right),$$ where the parameters are as described above, plus an additional observed parameter $y$ that has median value $y_0$. Similarly to linear fits, the plane fitting minimizes the quantity $$\label{eq:chi2plane}
\chi^2=\sum_{i=1}^N \frac{[a + b (x_i-x_0) + c (y_i-y_0) - z_i]^2}
{(b \Delta x_i)^2 + (c \Delta y_i)^2 + (\Delta z_i)^2 + \varepsilon_z^2},$$ with $\Delta x_i$, $\Delta y_i$ and $\Delta z_i$ as the errors on the variables $(x_i,y_i,z_i)$, and $\varepsilon_z$ as the sigma of the Gaussian describing the distribution of intrinsic scatter in the $z$ coordinate; $\varepsilon_z$ is iteratively adjusted so that the $\chi^2$ per degrees of freedom $\nu=N-3$ has the value of unity expected for a good fit. The observed scatter is $$\label{eq:twoparmDeltadef}
\Delta = \left\{ \frac{1}{N-3} \sum_{i=1}^N
\left[y_i - a - b\left(x_i - x_0\right)
-c\left(y_i - y_0\right)\right]^2 \right\}^{1/2}.$$
Masses Based on
================
The $R$–$L$ | 0 | non_member_974 |
Relationships
-------------------------
In this section, we examine the calibration of the fundamental $R$–$L$ relationship using various luminosity measures. The analysis in this section is based only on the RMDB sample in Table A1 because all these sources have been corrected for host-galaxy starlight. To obtain accurate masses from , contaminating starlight from the host galaxy must be accounted for in the luminosity measurement, or the mass will be overestimated. For reverberation-mapped sources, this has been done by modeling unsaturated images of the AGNs obtained with the [*Hubble Space Telescope*]{} [@Bentz06a; @Bentz09a; @Bentz13]. The AGN contribution was removed from each image by modeling the images as an extended host galaxy plus a central point source representing the AGN. The starlight contribution to the reverberation-mapping spectra is determined by using simulated aperture photometry of the AGN-free image. In the left panel of Figure \[Figure:HbRL\], we show the lag as a function of the AGN continuum with the host contribution removed in each case. This essentially reproduces the result of [@Bentz13] as small differences are due solely to improvements in the quality and quantity of the RM database \[cf. Table A1\]; we give the best-fit values to equation (\[eq:powerlaw\]) in the first row | 0 | non_member_974 |
of Table 2.
![Left: The rest-frame lag in days is shown as a function of the AGN luminosity $L_{\rm AGN}(5100\,{\rm \AA})$ in . The host-galaxy starlight contribution has been removed by using unsaturated [[*HST*]{}]{} images [see @Bentz13]. Right: The lag in days is shown as a function of the broad luminosity $L({\mbox{\rm H$\beta$}}_{\rm broad})$ in . The narrow component of has been removed in each case where it was sufficiently strong (i.e., easily identifiable) to isolate. In both panels, the solid line shows the best-fit to the data using equation (\[eq:powerlaw\]) with coefficients given in Table 2. The short dashed lines show the $\pm1\,\sigma$ uncertainty (equivalent to enclosing 68% of the values for a Gaussian distribution) and the long dashed lines show the $2.6\sigma$ uncertainties (equivalent to enclosing 99% of the values for a Gaussian distribution). []{data-label="Figure:HbRL"}](Figure1.eps)
[llccccc]{} $\log L_{\scriptsize\rm AGN}(5100\,{\rm \AA})$ & $\log \tau({\mbox{\rm H$\beta$}})$ & $1.228 \pm 0.025$ & $0.482 \pm 0.029$ & $43.444$ & $0.213 \pm 0.021$ & $0.241$\
$\log L({\mbox{\rm H$\beta$}}_{\scriptsize\rm broad})$ & $\log \tau({\mbox{\rm H$\beta$}})$ & $1.200\pm 0.025$ & $0.492 \pm 0.030$ & $41.746$ & $0.218 \pm 0.022$ & $0.244$\
$\log L(1350\,{\rm \AA})$ & $\log \tau({\mbox{\rm C\,{\sc iv}}})$ & $1.915 \pm 0.047$ & $0.517 \pm | 0 | non_member_974 |
0.036$ & $45.351$ & $0.336 \pm 0.041$ & $0.361$\
$\log L_{\scriptsize\rm AGN}(5100\,{\rm \AA})$ & $\log L({\mbox{\rm H$\beta$}}_{\rm broad})$ & $41.797 \pm 0.017$ & $0.960 \pm 0.020$ & $43.444$ & $0.158 \pm 0.014$ &$0.171$\
$\log L({\mbox{\rm H$\beta$}}_{\rm broad})$ &$\log L_{\scriptsize\rm AGN}(5100\,{\rm \AA})$ & $43.396 \pm 0.018$ & $1.003 \pm 0.022$ & $41.746$ & $0.161 \pm 0.015$ & $0.174$ \[table:RL\]
Accounting for the host-galaxy contribution in the same way for large number of AGNs, such as those in SDSS-RM (not to mention the entire SDSS catalog), is simply not feasible. It is well-known, however, that there is a tight correlation between the AGN continuum luminosity and the luminosity of [e.g., @Yee80; @Ilic17], and it has indeed been argued that the emission-line luminosity can be used as a proxy for the AGN continuum luminosity for reverberation studies [@Vestergaard06; @Greene10a]. However, in some of the reverberation-mapped sources, narrow-line contributes significantly to the total flux; NGC 4151 is an extreme example [e.g., @Antonucci83; @Bentz06a; @Fausnaugh17]. Whenever the narrow-line component can be isolated, it has been subtracted from the total flux. In Figure \[Figure:LAGNLHb\], we show the tight relationship between $L_{\rm AGN}(5100\,{\rm \AA})$ and $L({\mbox{\rm H$\beta$}}_{\rm broad})$; the best-fit coefficients for this relationship are given in | 0 | non_member_974 |
Table 2.
In the right panel of Figure \[Figure:HbRL\], we show the lag as a function of the luminosity of the broad component of , with the narrow component removed whenever possible. We give the best-fit values to the equation (\[eq:powerlaw\]) in the second row of Table 2, which shows that the slope of this relationship is nearly identical to the slope of the $R$–$L$ relationship using the AGN continuum. The luminosity of the broad component is thus an excellent proxy for the AGN luminosity and requires only removal of the narrow component (at least when it is significant) which is much easier than estimating the starlight contribution to the continuum luminosity at 5100Å. Moreover, by using the line flux instead of the continuum flux, we can include core-dominated radio sources where the continuum may be enhanced by the jet component [@Greene05]. This is therefore the $R$–$L$ relationship we prefer for the purpose of estimating single-epoch masses and we will focus on this relationship through the remainder of this contribution.
![The relationship between the broad emission line luminosity and the starlight-corrected AGN luminosity for the sources in Table A1. The black solid line is the regression of $L({\mbox{\rm H$\beta$}}_{\rm broad})$ | 0 | non_member_974 |
on $L_{\rm AGN}(5100\,{\rm \AA})$; the red dotted line is the regression of $L_{\rm AGN}(5100\,{\rm \AA})$ on $L({\mbox{\rm H$\beta$}}_{\rm broad})$, which we use in equation (\[eq:LHbLAGN\]). The coefficients for both fits are given in Table 2.[]{data-label="Figure:LAGNLHb"}](Figure2.eps)
Line-Width Relationships {#section:hblinewidths}
------------------------
We now consider the use of and as proxies for [cf. @Collin06; @Wang19]. The left panel of Figure \[Figure:windowhbwidths\] shows the relationship between ${\mbox{$\sigma_{\rm R}$}}({\mbox{\rm H$\beta$}})$, the line dispersion in the rms spectrum, and ${\mbox{$\sigma_{\rm M}$}}({\mbox{\rm H$\beta$}}),$ the line dispersion in the mean spectrum. The relationship is nearly linear (slope $ = 1.085\pm0.045$) and the intrinsic scatter is small ($0.079$dex). The fit coefficients are given in the first row of Table 3.
![The relationship between line dispersion in the rms ${\mbox{$\sigma_{\rm R}$}}({\mbox{\rm H$\beta$}})$ and mean ${\mbox{$\sigma_{\rm M}$}}({\mbox{\rm H$\beta$}})$ spectra is shown on the left. The relationship between line dispersion in the rms spectrum () and FWHM in the mean spectrum () is shown on the right. Blue filled circles are for the RMDB sample (Table A1) and open green triangles are for the SDSS sample (Table A3). The solid lines are best fits to equation (\[eq:powerlaw\]) with coefficients in Table 3. The short dashed and long dashed lines indicate the $\pm | 0 | non_member_974 |
1 \sigma$ and $\pm 2.6 \sigma$ envelopes, respectively, and the red dotted lines indicate where the two line-width measures are equal. Crosses are points that were rejected at the 2.6$\sigma$ (99%) level and are colored-coded like the circles. The relationship on the left is nearly linear (slope $= 1.085 \pm 0.045$) and the scatter $\varepsilon_y$ is low (0.079dex). It is clear in the right panel that () and () are well-correlated, but the relationship is significantly non-linear (slope $= 0.535 \pm 0.042$), the scatter $\varepsilon_y$ is slightly larger (0.106dex), and there are several significant outliers.[]{data-label="Figure:windowhbwidths"}](Figure3.eps)
[llccccc]{} $\log {\mbox{$\sigma_{\rm M}$}}({\mbox{\rm H$\beta$}})$ & $\log {\mbox{$\sigma_{\rm R}$}}({\mbox{\rm H$\beta$}})$ & $3.260 \pm 0.008$ & $1.085 \pm 0.045$ & 3.297 & $0.079 \pm 0.006$ & 0.087\
$\log {\mbox{${\rm FWHM}_{\rm M}$}}({\mbox{\rm H$\beta$}})$ & $\log {\mbox{$\sigma_{\rm R}$}}({\mbox{\rm H$\beta$}}) $ & $3.205 \pm 0.011$ & $0.535 \pm 0.042$ & 3.559 & $0.106 \pm 0.001$ & 0.114\
$\log {\mbox{$\sigma_{\rm M}$}}({\mbox{\rm C\,{\sc iv}}})$ & $\log {\mbox{$\sigma_{\rm R}$}}({\mbox{\rm C\,{\sc iv}}})$ & $3.436 \pm 0.009$ &$0.822 \pm 0.059$ & 3.394 & $0.064 \pm 0.008$ & 0.067\
$\log {\mbox{${\rm FWHM}_{\rm M}$}}({\mbox{\rm C\,{\sc iv}}})$ & $\log {\mbox{$\sigma_{\rm R}$}}({\mbox{\rm C\,{\sc iv}}})$ & $3.447 \pm 0.016$ & $0.445 \pm 0.101$ & 3.580 & $0.121 \pm 0.014$ | 0 | non_member_974 |
& 0.121 \[table:LW\]
We also show in the right panel of Figure \[Figure:windowhbwidths\] the relationship between () and the FWHM of in the mean spectrum, (). The fit coefficients are given in the second row of Table 3. The relationship is far from linear (slope $= 0.535 \pm 0.042$), and the scatter $\varepsilon_y$ is larger than it is for the ()–() relationship, even after removal of the notable outliers. While it is clear that () is an excellent proxy for (), the value of () is less clear. Nevertheless we will fit both versions in order to understand the relative merits of each.
Single-Epoch Predictors of the Virial Product {#section:hbpredictor}
---------------------------------------------
In the previous subsections, we have re-established the correlations between $\tau({\mbox{\rm H$\beta$}})$ and $L({\mbox{\rm H$\beta$}}_{\rm broad})$ and between () and both () and (). As a first approximation for a formula to estimate single-epoch masses, we fit the following equations: $$\log {\mbox{$\mu_{\rm RM}$}}({\mbox{\rm H$\beta$}}) = a + b\left[\log L({\mbox{\rm H$\beta$}}_{\rm broad}) - x_0\right]
+c\left[\log{\mbox{$\sigma_{\rm M}$}}({\mbox{\rm H$\beta$}}) - y_0\right],
\label{eq:Fit_139}$$ and $$\log {\mbox{$\mu_{\rm RM}$}}({\mbox{\rm H$\beta$}}) = a + b\left[ \log L({\mbox{\rm H$\beta$}}_{\rm broad}) - x_0\right]
+c\left[ \log {\mbox{${\rm FWHM}_{\rm M}$}}({\mbox{\rm H$\beta$}}) - y_0\right].
\label{eq:Fit_141}$$
The results of these fits based | 0 | non_member_974 |
---
abstract: |
We use the very large Millennium Simulation of the concordance $\Lambda$CDM cosmogony to calibrate the bias and error distribution of Timing Argument estimators of the masses of the Local Group and of the Milky Way. From a large number of isolated spiral-spiral pairs similar to the Milky Way/Andromeda system, we find the interquartile range of the ratio of timing mass to true mass to be a factor of $1.8$, while the 5% and 95% points of the distribution of this ratio are separated by a factor of $5.7$. Here we define true mass as the sum of the “virial” masses $M_{200}$ of the two dominant galaxies. For current best values of the distance and approach velocity of Andromeda this leads to a median likelihood estimate of the true mass of the Local Group of $5.27\times 10^{12}{\rm \,M_{\odot}}$, or $\log M_{LG}/M_\odot = 12.72$, with an interquartile range of $[12.58, 12.83]$ and a 5% to 95% range of $[12.26, 13.01]$. Thus a 95% lower confidence limit on the true mass of the Local Group is $1.81\times 10^{12}{\rm \,M_{\odot}}$. A timing estimate of the Milky Way’s mass based on the large recession velocity observed for the distant satellite Leo I | 0 | non_member_975 |
works equally well, although with larger systematic uncertainties. It gives an estimated virial mass for the Milky Way of $2.43 \times
10^{12}{\rm \,M_{\odot}}$ with a 95% lower confidence limit of $0.80 \times
10^{12}{\rm \,M_{\odot}}$.
author:
- |
Yang-Shyang Li$^{1}$[^1] and Simon D. M. White$^{2}$[^2]\
$^{1}$Kapteyn Astronomical Institute, PO Box 800, 9700 AV, Groningen, The Netherlands\
$^{2}$Max–Planck–Institut für Astrophysik, Karl-Schwarzschild-Str. 1, D-85748 Garching, Germany
date: 'Accepted. Received ; in original form '
title: Masses for the Local Group and the Milky Way
---
\[firstpage\]
Galaxy: formation – galaxies: Local Group – galaxies: kinematics and dynamics – dark matter.
Introduction {#intro_section}
============
During the 1970’s it became generally accepted that most, perhaps all, galaxies are surrounded by extended distributions of dark matter, so-called dark halos [@eks74; @opy74]. These were soon understood to play an essential role in driving the formation and clustering of galaxies [@wr78]. With the introduction of the Cold Dark Matter (CDM) paradigm, these ideas took more concrete form, allowing quantitative predictions to be made both for the population properties [@blumenthal84] and for the large-scale clustering [@davis85] of galaxies.
Measurements of the fluctuation spectrum of the Cosmic Microwave Background [@smoot92; @spergel03] and of the apparent acceleration of the cosmic expansion | 0 | non_member_975 |
[@riess98; @perlmutter99] elevated the CDM model, in its variant with a cosmological constant ($\Lambda$CDM), to the status of a standard paradigm. At the same time improving numerical techniques and faster computers have enabled detailed simulation of the formation and evolution of the galaxy population within this paradigm throughout a significant fraction of the observable Universe [@mr_nature]. Nevertheless, direct observational evidence for halos as extended as the paradigm predicts around galaxies like our own has so far come only from statistical analyses of the dynamics of satellite galaxies [e.g. @zaritsky97; @prada03] and of the gravitational lensing of background galaxies [e.g. @seljak02; @mandelbaum06] based on large samples of field spirals.
The earliest observational indication that the effective mass of the Milky Way must be much larger than its stellar mass came from the Timing Argument (hereafter TA) of @kw59. These authors noted that the Local Group is dominated by the two big spirals, and that these are currently approaching each other at about $100{{\rm \,km\,s^{-1}}}$. (The next most luminous galaxy is M33 which is probably about a factor of $10$ less massive than M31 and the Galaxy.) This reversal of the overall cosmic expansion must have been generated by gravitational forces, and | 0 | non_member_975 |
since the distance to the nearest external bright galaxy is much greater than that between M31 and the Milky Way, these forces are presumably dominated by material associated with the two spirals themselves.
@kw59 set up a simple model to analyse this situation – two point masses on a radial orbit. These were at pericentre (i.e. at zero separation) at the Big Bang and must have passed through apocentre at least once in order to be approaching today. Clearly this requires an apocentric separation larger than the current separation and an orbital period less than twice the current age of the Universe. Together these requirements put a lower limit on on the total mass of the pair. A more precise estimate of the minimum possible mass is obtained from the parametric form of Kepler’s laws for a zero angular momentum orbit: $$r=a(1-\cos\chi)$$
$$t=\bigg(\frac{a^{3}}{GM}\bigg)^{1/2}(\chi-\sin\chi)$$
$$\frac{dr}{dt}=\sqrt{\frac{GM}{a}}\frac{\sin\chi}{1-\cos\chi}$$
where $r$ is the current separation, $dr/dt$ is the current relative velocity, $a$ is the semi-major axis, $\chi$ is the eccentric anomaly, $t$ is the time since the Big Bang (the age of the universe) and $M$ is total mass [@lb81]. Given observationally determined values for $r$, $dr/dt$ and $t$, these equations have an infinite set | 0 | non_member_975 |
of discrete solutions for $\chi$, $a$ and $M$ labelled by the number of apocentric passages since the Big Bang. The solution corresponding to a single apocentric passage gives the smallest (and only plausible) estimate for the mass, which is about $5\times 10^{12}{\rm \,M_{\odot}}$ for current estimates of $r$, $dr/dt$ and $t$. Note that this is still a lower limit on the total mass, even within the simple point-mass binary model, since any non-radial motions in the system would increase its present kinetic energy and so increase the mass required to reverse the initial expansion and bring the pair to their observed separation by the present day [see @el82].
As @kw59 realised, this timing estimate of the total mass of the Local group exceeds by more than an order of magnitude the mass within the visible regions of the galaxies, as estimated from their internal dynamics, in particular, from their rotation curves. Thus 90% of the mass must lie outside the visible galaxies and be associated with little or no detectable light. Modern structure formation theories like $\Lambda$CDM predict this mass to be in extended dark matter halos with $M(r)$ increasing very roughly as $r$ out to the point where the | 0 | non_member_975 |
halos of the two galaxies meet. Such structures have no well-defined edge, so any definition of their total mass is necessarily somewhat arbitrary. In addition, their dynamical evolution from the Big Bang until the present is substantially more complex than that of a point-mass binary. Thus the mass value returned by the Timing Argument cannot be interpreted without some calibration against consistent dynamical models with extended dark halos.
A first calibration of this type was carried out by @kc91 using simulations of an Einstein-de Sitter CDM cosmogony. Here we use the very much larger Millennium Simulation [@mr_nature] to obtain a more refined calibration based on a large ensemble of galaxy pairs with observable properties similar to those of the Local Group. We find that the standard timing estimate is, in fact, an almost unbiased estimate of the sum of the conventionally defined virial masses of the two large galaxies.
@zaritsky89 attempted to measure the halo mass of the Milky Way alone by measuring radial velocities for its dwarf satellites and assuming the population to be in dynamical equilibrium in the halo potential. They noted, however, that one of the most distant dwarfs, Leo I, has a very large recession velocity | 0 | non_member_975 |
and as a result provides a interesting lower limit on the Milky Way’s mass by a variant of the original Timing Argument. To reach its present position and radial velocity, Leo I must have passed pericentre at least once since the Big Bang and now be receding from the Galaxy for (at least) the second time.
Applying the point-mass radial orbit Equations (1) – (3) to this case gives a lower bound of about $1.6\times 10^{12}{\rm \,M_{\odot}}$. This seems likely to be a significant underestimate, since Leo I could not have passed through the centre of the Milky Way without being tidally destroyed so its orbit cannot be purely radial. Below we calibrate the Timing Argument for this case also, finding it to work well although with significantly more scatter than for the Local Group as a whole. This is because the $\Lambda$CDM paradigm predicts that the dynamics on the scale of Leo I’s orbit ($\sim 200{\rm \,kpc}$) is typically more complex than on the scale of the Local Group as a whole ($\sim 700{\rm \,kpc}$).
Our paper is organised as follows. In Section \[data\_section\], we briefly describe the Millennium Simulation and the selection criteria we use to define various | 0 | non_member_975 |
samples of ‘Local Group-like’ pairs and of ‘Milky Way-like’ halos. In Section \[result\_section\], we plot the ratio of true total mass to Timing Argument mass estimate for these samples, and we use its distribution to define an unbiased TA estimator of true mass with its associated confidence ranges. In Section \[app\_LG\_section\] this is then applied to the Local Group in order to obtain an estimate its true mass with realistic uncertainties. Section \[app\_MW\_section\] attempts to carry out a similar calibration for the TA-based estimate of the Galaxy’s halo mass from the orbit of Leo I. We conclude in Section \[conclusions\] with a summary and brief discussion of our results.
The Millennium Simulation {#data_section}
=========================
The *Millennium Simulation* is an extremely large cosmological simulation carried out by the Virgo Consortium [@mr_nature]. It followed the motion of $N=2160^{3}$ dark matter particles of mass $8.6 \times 10^{8}
~h^{-1}{\rm \,M_{\odot}}$ within a cubic box of comoving size $500~h^{-1}{\rm \,Mpc}$. Its comoving spatial resolution (set by the gravitational softening) is 5 $h^{-1}{\rm \,kpc}$. The simulation adopted the concordance $\Lambda$CDM model with parameters $\Omega_{m}=0.25, \Omega_{b}=0.045, h=0.73, \Omega_{\Lambda}=0.75,
n=1$ and $\sigma_{8}=0.9$, where, as usual, we define the Hubble constant by $H_{0}=100h{{\rm \,km\,s^{-1}}}{\rm \,Mpc}^{-1}$. The current age of | 0 | non_member_975 |
the universe is then $13.6
\times 10^{9}$ yr. The positions and velocities of all particles were stored at 63 epochs spaced approximately logarithmically in expansion factor at early times and at approximately 300 Myr intervals after $z=2$. For each such snapshot a friends-of-friends group-finder was used to locate all virialised structures, and their self-bound substructures (subhalos) were identified using [SUBFIND]{} [@swtk01]. Halos and subhalos in neighbouring outputs were then linked in order to build formation history trees for all the subhalos present at each time. These data are publicly available at the Millennium release site[^3]. A “Milky Way” halo at $z=0$ typically contains a few thousand particles and several resolved subhalos.
Galaxy formation was simulated within these merging history trees by using semi-analytic models to follow the evolution of the baryonic components associated with each halo/subhalo. Processes included are radiative cooling of diffuse gas, star formation, the growth of supermassive black holes, feedback of energy and heavy elements from supernovae and AGN, stellar population evolution, galaxy merging and effects due to a reionising UV background. The $z=0$ galaxy catalogue we analyse here corresponds to the specific model of @croton06 and details of its assumptions and parameters can be found in | 0 | non_member_975 |
that paper. Data for the galaxy population at all redshifts are available at the Millennium web site for the updated model of @deluciab07, as well as for the independent galaxy formation model of @bower06. All these models are tuned to fit a wide variety of data on the nearby galaxy population, and in addition fit many (but not all!) available data at higher redshift [see, for example, @kw06]. The details of the galaxy formation modelling are not, however, important for the dynamical issues which are the focus of our own paper.
At $z=0$ there are $18.2 \times 10^{6}$ halos/subhalos identified in the simulation to its resolution limit of $20$ particles. The galaxy formation model populates these with $8,394,180$ galaxies brighter than an absolute magnitude limit of $M_{B} = -16.7$ above which the catalogue can be considered reasonably complete. These catalogues list a number of properties for the halos, subhalos and galaxies which will be important for us. Galaxies are categorised into three types according to the nature of their association with the dark matter. A Type 0 galaxy sits at the centre of the dominant or main subhalo and can be considered the central galaxy of the halo itself (formally, | 0 | non_member_975 |
of the FOF group). A Type 1 galaxy sits at the centre of one of the smaller non-dominant subhalos associated with a FOF group. Finally, a Type 2 galaxy is associated to a specific particle and no longer has an associated subhalo because the object within which it formed was tidally disrupted after accretion onto a larger halo. Such galaxies merge with the central galaxy of their new halo after waiting for a dynamical friction time.
Each galaxy in the catalogue has an associated “rotation velocity” $V_{max}$. This is the maximum of the circular velocity $V_c(r) =
(GM(r)/r)^{1/2}$ of its subhalo for Types 0 and 1; for Type 2 objects $V_{max}$ is frozen to its value at the latest time when the galaxy still occupied a subhalo. Type 0 and 1 galaxies also have an associated mass $M_{halo}$ which is the mass of the self-bound subhalo which surrounds them. Finally, halos of Type 0 galaxies have a conventional “virial mass” $M_{200}$, defined as the total mass within the largest sphere surrounding them with an enclosed mean density exceeding 200 times the critical value. Below we will consider both $M_{halo}$ and $M_{200}$ as possible definitions for the “true” masses of M31 | 0 | non_member_975 |
and the Galaxy.
We use the Millennium Simulation to construct samples of mock Milky Way/Andromeda galaxies and of mock Local Groups as follows. We begin by identifying all Type 0 or Type 1 galaxies with characteristic “rotation velocity” either in the narrow range $200 \le V_{max}< 250{{\rm \,km\,s^{-1}}}$ or in the wider range, $150 \le V_{max}< 300{{\rm \,km\,s^{-1}}}$. This produces samples of $166,090$ and $699,177$ galaxies respectively. The exclusion of Type 2 galaxies reduces the samples by about 5-6% in each case, but the excluded galaxies are in any case not plausible analogues for the Local Group giants since they are almost all members of large groups or clusters. We also consider subsamples in which the morphologies predicted by the semianalytic model are forced to approximate those of M31 and the Galaxy. Specifically, we require a bulge-to-total luminosity ratio in the range $1.2 \le M_{B,bulge}-M_{B,total}
< 2.5$ so that the disks are 2 to 9 times brighter than the bulges in the *B*-band. This morphology cut reduces the samples in the two $V_{max}$ ranges to $62,605$ and $271,857$ galaxies respectively.
We then identify Local Group analogues in each of these four samples by identifying isolated pairs with separations in the | 0 | non_member_975 |
range of $500-1,000{\rm \,kpc}$ and with negative relative radial velocities. (Note that this is the true relative velocity rather than the relative peculiar velocity, i.e. we have added the Hubble expansion to the relative peculiar velocity and have required the result to be negative.) We identify isolated pairs by keeping only those which have no “massive” companion, defined as a galaxy with $V_{max} \ge
150{{\rm \,km\,s^{-1}}}$, within a sphere of 1 Mpc radius centred on the mid-point of the binary, and no nearby cluster, defined as a halo with $M_{200}>3\times
10^{13}{\rm \,M_{\odot}}$ within 3 Mpc of the mid-point of the binary. These cuts ensure that the dynamics are dominated by mass associated with the two main systems, as appears to be the case for the Local Group. For galaxies selected in the narrower $V_{max}$ range we then find $178$ pairs when the morphology cut is applied and $1,128$ pairs when it is not. For the wider $V_{max}$ range the corresponding numbers are $2,815$ pairs and $16,479$ pairs respectively.
When calibrating the TA estimator it proves advantageous to use simulated pairs with dynamical state quite close to that of the real Local Group. As we will see below, this eliminates some | 0 | non_member_975 |
systems where the dominant motion is not in the radial direction and the TA therefore significantly underestimates the mass. We therefore make one final cut which requires the approach velocity of the two galaxies to lie between $0.5$ and $1.5$ times the value measured for the real system ($-130{{\rm \,km\,s^{-1}}}$). This results in our final sets of Local Group lookalikes. For the narrower $V_{max}$ range we end up with $117$ pairs when the morphology cut is applied and $758$ pairs when it is not, while for the wider $V_{max}$ range the corresponding numbers are $1,273$ pairs and $8,449$ pairs respectively.
When we study the application of the Timing Argument to the Milky Way–Leo I system, we consider individual galaxies from both our $V_{max}$ ranges. We require these to be isolated by insisting that there should be no bright/massive companion (with luminosity exceeding 10% of that of the host or $V_{max} >
150{{\rm \,km\,s^{-1}}}$) closer than 700 kpc and no massive group (defined as above) closer than 3 Mpc. This produces samples of $137,926$ and $266,229$ potential hosts in the cases with and without the morphology cut for the wider $V_{max}$ range, and $29,245$ and $57,816$ potential hosts for the narrower | 0 | non_member_975 |
range. We then search for Leo I analogues around these hosts by identifying companions in the separation range 200 to 300 kpc with $V_{max}({\rm comp}) \leq 80{{\rm \,km\,s^{-1}}}$, $M_{B} < -16.7$ and $V_{ra} \ge 0.7 V_{max}({\rm host})$ where $V_{ra}$ is the relative radial velocity of the two objects and the last condition reflects the fact that Leo I is useful for estimating the Milky Way’s mass only because its recession velocity is comparable to the Galactic rotation velocity ($V_{ra} \sim 0.8 V_{max}({\rm host})$ for the real Leo I–Milky Way system). Pairs sharing the same MW-like host are excluded in the final list.
With these cuts we find $344$ and $896$ satellite-host pairs in the samples with and without the morphology cut for the looser $V_{max}$ range, and $168$ and $374$ for the tighter range. These relatively small numbers reflect the fact that only about 10% of potential hosts actually have a faint companion in this distance range which is still bright enough to be resolved, and fewer than 5% of these satellites are predicted to have positive recession velocities comparable to that observed.
Results {#result_section}
=======
Calibration of the Timing Argument mass for the Local Group
-----------------------------------------------------------
For each simulated | 0 | non_member_975 |
Local Group analogue the separation and relative radial velocity of the two galaxies can be combined with the age of the Universe (taken to be 13.6 Gyr) to obtain a Timing Argument mass estimate $M_{TA}$ (Equations (1) to (3)). The true mass of the pair $M_{tr}$ is harder to define because of the extended and complex mass distributions predicted by the $\Lambda$CDM model. The mass of an individual dark halo is often taken to be $M_{200}$ the mass within a sphere of mean density 200 times the critical value, so a natural choice for $M_{tr}$ is the sum of $M_{200}$ for the two galaxies. The Millennium Simulation database only lists $M_{200}$ for Type 0 galaxies, those at the centre of the main subhalo of each friends-of-friends particle group. Many of our LG analogues lie within a single FOF group. One of the pair is then a Type 1 galaxy, the central object of a subdominant subhalo, and so has no listed value for $M_{200}$. In such cases we have gone back to the particle data for the simulation in order to measure $M_{200}$ directly also for these galaxies.
An alternative convention is to define $M_{tr}$ as the sum of the | 0 | non_member_975 |
values of $M_{halo}$, the maximal self-bound mass of each subhalo; this is included in the database for both Type 0 and Type 1 galaxies. In the following we use the notation $M_{tr,200}$ and $M_{tr,halo}$ to distinguish these two definitions. For either we can calculate the ratio of true mass to Timing Argument estimate, $$A_x = M_{tr,x}/M_{TA},$$ where the suffix $x$ is $200$ or $halo$ depending on the definition adopted for $M_{tr}$. If the Timing Argument is a good estimator of true mass, our samples of LG analogues should produce a narrow distribution of $A$ values. This distribution then allows the TA mass estimate for the real Local Group to be converted into a best estimate of its true mass, together with associated confidence intervals.
Our preferred sets of Local Group analogues contain simulated galaxy pairs which mimic the real system in terms of morphology, isolation, pair separation and pair approach velocity. In addition, they require the halos of the simulated galaxies to have $V_{max}$ values within about $\pm 10\% $ and $\pm
35\%$ of those estimated for M31 and the Galaxy for the tight and loose ranges of $V_{max}$, respectively. In order to understand the influence of these constraints we | 0 | non_member_975 |
give results below not only for our “best” samples but also for samples where the various constraints are relaxed. Thus, we consider samples in which 1) both morphology and isolation requirements are applied (our preferred case), 2) the isolation requirement is removed, 3) the morphology requirement is removed, and 4) both morphology and isolation requirements are removed. For each case, we compare results for the two allowed ranges of $V_{max}$ and we also examine the effect of loosening the radial velocity constraint to $V_{ra} < 0$.
Fig. \[hist\] gives histograms of the distribution of $A_{200}$ for a sample in the narrow $V_{max}$-range with our preferred isolation, morphology and radial velocity cuts, as well as for three samples with the same $V_{max}$ and $V_{ra}$ cuts but with reduced morphology and isolation requirements. Fig. \[narrow\_ratio\_cumu\_dist\] presents these same distributions in cumulative form and compares them with the corresponding distributions for samples with the loosened circular velocity requirement, $150{{\rm \,km\,s^{-1}}}\le V_{max} < 300{{\rm \,km\,s^{-1}}}$. In both plots black curves refer to class (1) samples for which both isolation and morphology cuts are imposed, while red, green and blue curves refer to samples in classes (2), (3) and (4) respectively. Results for the broader | 0 | non_member_975 |
$V_{max}$ selection are indicated by dashed curves in Fig. \[narrow\_ratio\_cumu\_dist\]. We give numerical results for various percentile points of these distributions in Table \[tb\_a200\_vra\], and repeat all these in Table \[tb\_a200\] for samples where the separation velocity requirement has been loosened to $V_{ra}
< 0$.
![Normalised histograms of $A_{200}$, the ratio of true mass to Timing Argument estimate, for samples of Local Group analogues with $200{{\rm \,km\,s^{-1}}}\le V_{max} < 250{{\rm \,km\,s^{-1}}}$ and $-195{{\rm \,km\,s^{-1}}}<V_{ra} < -65{{\rm \,km\,s^{-1}}}$. The black histogram refers to our preferred selection where both isolation and morphology requirements are imposed. For the red histogram the isolation requirement has been removed, for the green histogram the morphology requirement, and for the blue histogram both requirements.[]{data-label="hist"}](narrow_radial_a200_hist.ps){width="50.00000%"}
![Cumulative distributions of $A_{200}$ the ratio of true mass to Timing Argument estimate for Local Group analogues with $-195{{\rm \,km\,s^{-1}}}<
V_{ra} < -65{{\rm \,km\,s^{-1}}}$. The solid curves correspond to the four samples already plotted in Fig. \[hist\] while the dashed curves are for samples with $150{{\rm \,km\,s^{-1}}}\le V_{max} < 300{{\rm \,km\,s^{-1}}}$. The colour coding is the same as in Fig. \[hist\]; black indicates samples with our preferred isolation and morphology constraints.[]{data-label="narrow_ratio_cumu_dist"}](radial_a200_cumuhist.ps){width=".5\textwidth"}
The first and most important point to note from these these figures | 0 | non_member_975 |
and tables is that the median value of $A_{200}$ is very robust and only varies between $0.98$ and $1.34$ for our full range of sample selection criteria. With our preferred cuts the median values are $1.15$ and $0.99$ for the narrow and wide $V_{max}$ samples respectively. The best estimate of the true mass of the Local Group (for this definition) is thus very similar to its Timing Argument mass estimate, and depends very little on the calibrating sample of simulated pairs.
The second important point is that the width of the distribution of $A_{200}$ does depend on how the calibrating sample is defined. In particular, it is narrower for samples with the more restrictive $V_{max}$ range, and for given $V_{max}$ range it is smallest for samples with our preferred cuts, those which match the dynamical and morphological properties of the Local Group most closely. For the narrow $V_{max}$ sample the interquartile range is a factor of just $1.6$, and the upper and lower 5% points are separated by a factor of $3$. For the wider velocity range the interquartile range is a factor of $1.8$ and the 5% points are separated by a factor of $5.7$. This shows the Timing | 0 | non_member_975 |
Argument to be remarkably precise for systems similar to the Local Group.
The broadening of the distribution as the selection requirements are relaxed is easy to understand. Removing the isolation requirements allows third bodies to play a significant role in the dynamics. This can extend the upper tail of the $A_{200}$ distribution if mass from the third body falls inside $R_{200}$ for one of the pair galaxies or if the gravity of the third galaxy produces a tidal field which opposes the attraction between the pair members. It can extend the lower tail if the mass of the third body lies between the pair members but outside their $R_{200}$ spheres, thus enhancing their mutual attraction without adding to their mass. Removing the morphology constraint moves the whole distribution towards larger values and this effect is most pronounced in the large $A_{200}$ tail. This is because objects with more dominant bulges have more complex merger histories. They typically form in denser regions and their halos tend to be more massive and to have more complex structure.
Loosening the requirements on $V_{max}$ affects the distribution in a complex way. There is a tight relation between $V_{max}$ and $M_{200}$ (also $M_{halo}$) in the | 0 | non_member_975 |
$\Lambda$CDM structure formation model [e.g. @nfw97]. Thus if we place tight restrictions on the $V_{max}$ values of our galaxies, we will obtain a sample of Local Group analogues with a narrow range of $M_{tr}$ values. If, in addition, we force the parameters which enter in the Timing Argument (the pair separation and relative radial velocity) to lie in narrow ranges, then the TA mass estimate itself is tightly constrained. The distribution of $A_{200}$ is thus forced to be narrow as a consequence of our selection criteria.
A second effect is that most of the new pairs added by widening the requirement on $V_{max}$ have at least one galaxy with $150{{\rm \,km\,s^{-1}}}\le V_{max} < 200{{\rm \,km\,s^{-1}}}$, thus with relatively low $M_{tr}$. This simply reflects the strong dependence of halo abundance on $V_{max}$. Given that halo mass scales approximately as $V_{max}^3$ it is striking that the addition of so many pairs containing a “low mass” galaxy reduces the median value of $A_{200}$ by just 15%. The low tail of the distribution is more strongly affected, by almost a factor of $2$ at the lower 5% point, but the upper end of the distribution is barely affected at all. This demonstrates that the | 0 | non_member_975 |
main body of the distribution is weakly affected by restrictions on $V_{max}$, but that the lower tail (which is needed to place a lower limit on the true mass of the Milky Way) is suppressed if $V_{max}$ is not allowed to take small values.
5% 25% 50% 75% 95% \# of pairs
---------------------------------------------------------------- ------ ------ ------ ------ ------ -------------
$200{{\rm \,km\,s^{-1}}}\le V_{max} < 250{{\rm \,km\,s^{-1}}}$
morphology, isolation 0.67 0.93 1.15 1.47 2.05 117
morphology, no isolation 0.61 0.93 1.14 1.52 2.09 155
no morphology, isolation 0.67 0.97 1.20 1.50 2.32 758
no morphology, no isolation 0.63 0.96 1.22 1.55 2.54 1015
$150{{\rm \,km\,s^{-1}}}\le V_{max} < 300{{\rm \,km\,s^{-1}}}$
morphology, isolation 0.34 0.72 0.99 1.27 1.93 1273
morphology, no isolation 0.33 0.68 0.98 1.29 2.00 1650
no morphology, isolation 0.41 0.81 1.09 1.40 2.21 8449
no morphology, no isolation 0.35 0.77 1.08 1.43 2.41 11838
5% 25% 50% 75% 95% \# of pairs
---------------------------------------------------------------- ------ ------ ------ ------ ------ -------------
$200{{\rm \,km\,s^{-1}}}\le V_{max} < 250{{\rm \,km\,s^{-1}}}$
morphology, isolation 0.54 0.97 1.33 1.66 3.93 178
morphology, no isolation 0.45 0.94 1.26 1.66 3.93 241
no morphology, isolation 0.54 1.01 1.34 1.82 4.62 1128
no morphology, no isolation 0.42 0.96 1.34 1.93 5.11 1596
| 0 | non_member_975 |
$150{{\rm \,km\,s^{-1}}}\le V_{max} < 300{{\rm \,km\,s^{-1}}}$
morphology, isolation 0.28 0.85 1.19 1.64 3.30 2815
morphology, no isolation 0.22 0.77 1.16 1.64 3.38 3532
no morphology, isolation 0.31 0.89 1.23 1.76 4.06 16479
no morphology, no isolation 0.18 0.79 1.19 1.78 4.46 23429
In Table \[tb\_a200\] we show the effect of weakening the cut on relative radial velocity to require only that the two main galaxies be approaching. Again this has remarkably little effect on the median $A_{200}$ values. A comparison with Table \[tb\_a200\_vra\] shows them all to be increased by about 10%-15%. The effects on the tails of the distributions are more substantial. The 95% point is typically increased by about a factor of $2$. This is because the sample now includes a substantial number of pairs with small $V_{ra}$ (and thus smaller TA mass estimate) for which tangential motion is important for their current orbit. The 5% point of the distribution is significantly reduced, reflecting the fact that our restrictions on relative approach velocity exclude a non negligible number of systems with approach velocities [*larger*]{} than $195{{\rm \,km\,s^{-1}}}$, and thus with large TA mass estimates. Such systems must have more mass [*outside*]{} the conventional virial radii of the two | 0 | non_member_975 |
galaxies than do typical Local Group analogues in our samples.
In conclusion, we believe our most precise and robust estimate of the distribution of $A_{200}$ to be that obtained with our preferred morphology, isolation and radial velocity cuts for $150{{\rm \,km\,s^{-1}}}\le V_{max} <
300{{\rm \,km\,s^{-1}}}$, and we will use this distribution in the next section to estimate the true mass of the Local Group. Although the tails of the distribution are suppressed still further for a narrower range of $V_{max}$, this is at least in part due to the artificial effects mentioned above. In addition the number of Local Group analogues is too small in this case for the tails of the distribution to be reliably determined. From Table \[tb\_a200\_vra\] we see that the best estimate of the true mass of the Local Group (which we take to be that obtained using the median value of $A_{200}$) is almost identical to the direct TA estimate. The most probable range of true mass (given by the quartiles of $A_{200}$) extends to values about 30% above and below this, while the 95% confidence lower limit on the true mass (given by the 5% point of the $A_{200}$ distribution) is a factor of | 0 | non_member_975 |
Subsets and Splits