The MASSIVE Survey - VII. The Relationship of Angular Momentum, Stellar Mass and Environment of Early-Type Galaxies

TL;DR

This study investigates how the angular momentum of early-type galaxies relates to their stellar mass and environment, finding that mass primarily drives kinematic properties with minimal environmental influence.

Contribution

It demonstrates that galaxy spin is mainly correlated with stellar mass rather than environment, clarifying the origins of kinematic diversity in early-type galaxies.

Findings

Strong correlation between stellar mass and spin parameter.

Almost no environmental dependence of galaxy spin at fixed mass.

Higher fraction of slow rotators in dense environments, partly explained by mass.

Abstract

We analyse the environmental properties of 370 local early-type galaxies (ETGs) in the MASSIVE and ATLAS3D surveys, two complementary volume-limited integral-field spectroscopic (IFS) galaxy surveys spanning absolute $K$-band magnitude $-21.5 > M_K > -26.6$, or stellar mass $8 \times 10^{9} < M_* < 2 \times 10^{12} M_\odot$. We find these galaxies to reside in a diverse range of environments measured by four methods: group membership (whether a galaxy is a brightest group/cluster galaxy, satellite, or isolated), halo mass, large-scale mass density (measured over a few Mpc), and local mass density (measured within the $N$th neighbour). The spatially resolved IFS stellar kinematics provide robust measurements of the spin parameter $\lambda_e$ and enable us to examine the relationship among $\lambda_e$, $M_*$, and galaxy environment. We find a strong correlation between $\lambda_e$ and $M_*$, where the average $\lambda_e$ decreases from $\sim 0.4$ to below 0.1 with increasing mass, and the fraction of slow rotators $f_{\rm slow}$ increases from $\sim 10$% to 90%. We show for the first time that at fixed $M_*$, there are almost no trends between galaxy spin and environment; the apparent kinematic morphology-density relation for ETGs is therefore primarily driven by $M_*$ and is accounted for by the joint correlations between $M_*$ and spin, and between $M_*$ and environment. A possible exception is that the increased $f_{\rm slow}$ at high local density is slightly more than expected based only on these joint correlations. Our results suggest that the physical processes responsible for building up the present-day stellar masses of massive galaxies are also very efficient at reducing their spin, in any environment.

Tables

Table 1. Table 1: Properties of MASSIVE survey galaxies

Galaxy	R.A.	Dec.	$D$	$M_K$	${\log }_{10}{M}_{\ast }$	$\epsilon$	${\lambda }_e$	Rot.	env	${\log }_{10}{M}_{\mathrm{halo}}$	Cluster	$\left(1+{\delta }_g\right)$	${\nu }_{10}$
	[deg]	[deg]	[Mpc]	[mag]	[$M_{\odot }$]					[$M_{\odot }$]			[$\overline{\nu }$]
(1)	(2)	(3)	(4)	(5)	(6)	(7)	(8)	(9)	(10)	(11)	(12)	(13)	(14)
NGC 0057	$3.8787$	$17.3284$	$76.3$	$-25.75$	$11.79$	$0.17$	$0.02$	S	I			$2.3$	$4.9$
NGC 0080	$5.2952$	$22.3572$	$81.9$	$-25.66$	$11.75$	$0.09$	$0.04$	S	B	$14.1$		$3.0$	$6600$
NGC 0128	$7.3128$	$2.8641$	$59.3$	$-25.35$	$11.61$	$0.59$			I			$1.4$	$7.8$
NGC 0227	$10.6534$	$-1.5288$	$75.9$	$-25.32$	$11.60$	$0.25$			B	$13.5$		$4.0$	$4.6$
NGC 0315	$14.4538$	$30.3524$	$70.3$	$-26.30$	$12.03$	$0.28$	$0.06$	S	B	$13.5$		$6.0$	$280$
NGC 0383	$16.8540$	$32.4126$	$71.3$	$-25.81$	$11.82$	$0.14$	$0.25$	F	S	$14.4$		$7.2$	$4400$
NGC 0393	$17.1540$	$39.6443$	$85.7$	$-25.44$	$11.65$	${0.18}^{\ast }$			I			$1.5$	$1.4$
NGC 0410	$17.7453$	$33.1520$	$71.3$	$-25.90$	$11.86$	$0.25$	$0.03$	S	B	$14.4$		$7.4$	$3200$
NGC 0467	$19.7922$	$3.3008$	$75.8$	$-25.40$	$11.64$	$0.05$			I			$3.9$	$18$
PGC 004829	$20.1287$	$50.1445$	$99.0$	$-25.30$	$11.59$	${0.34}^{\ast }$			I			$2.6$	$10$
NGC 0499	$20.7978$	$33.4601$	$69.8$	$-25.50$	$11.68$	$0.35$	$0.06$	S	S	$14.4$		$7.2$	$36000$
NGC 0507	$20.9164$	$33.2561$	$69.8$	$-25.93$	$11.87$	$0.09$	$0.05$	S	B	$14.4$		$7.2$	$59000$
NGC 0533	$21.3808$	$1.7590$	$77.9$	$-26.05$	$11.92$	$0.26$	$0.03$	S	B	$13.5$		$4.3$	$13$
NGC 0545	$21.4963$	$-1.3402$	$74.0$	$-25.83$	$11.83$	$0.28$	$0.13$	S	B	$14.5$	A194	$5.9$	$14000$
NGC 0547	$21.5024$	$-1.3451$	$74.0$	$-25.83$	$11.83$	$0.14$	$0.06$	S	S	$14.5$	A194	$5.9$	$14000$
NGC 0665	$26.2338$	$10.4230$	$74.6$	$-25.51$	$11.68$	$0.24$	$0.40$	F	B	$13.7$		$3.0$	$58$
UGC 01332	$28.0755$	$48.0878$	$99.2$	$-25.57$	$11.71$	${0.30}^{\ast }$	$0.04$	S	B	$13.8$		$3.7$	$170$
NGC 0708	$28.1937$	$36.1518$	$69.0$	$-25.65$	$11.75$	${0.40}^{\ast }$	$0.04$	S	B	$14.5$	A262	$5.8$	$12000$
UGC 01389	$28.8778$	$47.9550$	$99.2$	$-25.41$	$11.64$	${0.34}^{\ast }$			S	$13.8$		$3.8$	$150$
NGC 0741	$29.0874$	$5.6289$	$73.9$	$-26.06$	$11.93$	$0.17$	$0.04$	S	B	$13.8$		$2.9$	$130$
NGC 0777	$30.0622$	$31.4294$	$72.2$	$-25.94$	$11.87$	$0.17$	$0.05$	S	B	$13.5$		$5.0$	$78$
NGC 0890	$35.5042$	$33.2661$	$55.6$	$-25.50$	$11.68$	${0.38}^{\ast }$	$0.10$	S	I			$4.7$	$1.4$
NGC 0910	$36.3616$	$41.8243$	$79.8$	$-25.33$	$11.61$	${0.16}^{\ast }$			S	$14.8$	A347	$6.2$	$12000$
NGC 0997	$39.3103$	$7.3056$	$90.4$	$-25.40$	$11.64$	$0.13$			B	$13.0$		$3.0$	$26$
NGC 1016	$39.5815$	$2.1193$	$95.2$	$-26.33$	$12.05$	$0.06$	$0.03$	S	B	$13.9$		$4.8$	$56$
NGC 1060	$40.8127$	$32.4250$	$67.4$	$-26.00$	$11.90$	$0.24$	$0.02$	S	B	$14.0$		$3.9$	$2100$
NGC 1066	$40.9579$	$32.4749$	$67.4$	$-25.31$	$11.60$	$0.16$			S	$14.0$		$3.9$	$2200$
NGC 1132	$43.2159$	$-1.2747$	$97.6$	$-25.70$	$11.77$	$0.37$	$0.06$	S	B	$13.6$		$3.4$	$8.3$
NGC 1129	$43.6141$	$41.5796$	$73.9$	$-26.14$	$11.96$	${0.15}^{†}$	$0.12$	S	B	$14.8$		$10.7$	$16000$
NGC 1167	$45.4265$	$35.2056$	$70.2$	$-25.64$	$11.74$	$0.17$	$0.43$	F	B	$13.1$		$5.0$	$15$
NGC 1226	$47.7723$	$35.3868$	$85.7$	$-25.51$	$11.68$	${0.18}^{\ast }$	$0.03$	S	B	$13.2$		$3.5$	$3.1$
IC0 310	$49.1792$	$41.3248$	$77.5$	$-25.35$	$11.61$	$0.06$			S	$14.8$	Perseus	$13.2$	$15000$
NGC 1272	$49.8387$	$41.4906$	$77.5$	$-25.80$	$11.81$	$0.07$	$0.02$	S	S	$14.8$	Perseus	$13.5$	$400000$
UGC 02783	$53.5766$	$39.3568$	$85.8$	$-25.44$	$11.65$	$0.11$			B	$12.6$		$6.3$	$17$
NGC 1453	$56.6136$	$-3.9688$	$56.4$	$-25.67$	$11.75$	${0.14}^{\ast }$	$0.20$	F	B	$13.9$		$2.3$	$89$
NGC 1497	$60.5283$	$23.1329$	$87.8$	$-25.31$	$11.60$	${0.40}^{\ast }$			I			$2.7$	$89$
NGC 1600	$67.9161$	$-5.0861$	$63.8$	$-25.99$	$11.90$	${0.26}^{\ast }$	$0.03$	S	B	$14.2$		$6.0$	$1200$
NGC 1573	$68.7666$	$73.2624$	$65.0$	$-25.55$	$11.70$	${0.34}^{\ast }$	$0.04$	S	B	$14.1$		$4.1$	$590$
NGC 1684	$73.1298$	$-3.1061$	$63.5$	$-25.34$	$11.61$	${0.24}^{\ast }$			B	$13.7$		$6.2$	$1600$
NGC 1700	$74.2347$	$-4.8658$	$54.4$	$-25.60$	$11.72$	${0.28}^{\ast }$	$0.20$	F	B	$12.7$		$3.5$	$23$
NGC 2208	$95.6444$	$51.9095$	$84.1$	$-25.63$	$11.74$	${0.32}^{\ast }$	$0.06$	S	I			$2.8$	$7.2$
NGC 2256	$101.8082$	$74.2365$	$79.4$	$-25.87$	$11.84$	${0.20}^{\ast }$	$0.02$	S	B	$13.7$		$2.7$	$21$
NGC 2274	$101.8224$	$33.5672$	$73.8$	$-25.69$	$11.76$	${0.10}^{\ast }$	$0.07$	S	B	$13.3$		$3.1$	$110$
NGC 2258	$101.9425$	$74.4818$	$59.0$	$-25.66$	$11.75$	${0.24}^{\ast }$	$0.04$	S	B	$12.2$		$3.8$	$9.8$
NGC 2320	$106.4251$	$50.5811$	$89.4$	$-25.93$	$11.87$	${0.30}^{\ast }$	$0.23$	F	B	$14.2$		$7.9$	$660$
UGC 03683	$107.0582$	$46.1159$	$85.1$	$-25.52$	$11.69$	${0.26}^{\ast }$	$0.09$	S	B	$13.6$		$5.8$	$27$
NGC 2332	$107.3924$	$50.1823$	$89.4$	$-25.39$	$11.63$	${0.34}^{\ast }$			S	$14.2$		$7.8$	$1500$
NGC 2340	$107.7950$	$50.1747$	$89.4$	$-25.90$	$11.86$	${0.44}^{\ast }$	$0.03$	S	S	$14.2$		$7.8$	$1300$
UGC 03894	$113.2695$	$65.0791$	$97.2$	$-25.58$	$11.72$	$0.10$	$0.12$	F	B	$13.7$		$1.5$	$1.5$
NGC 2418	$114.1563$	$17.8839$	$74.1$	$-25.42$	$11.64$	$0.20$			I			$2.2$	$1.4$
NGC 2456	$118.5444$	$55.4953$	$107.3$	$-25.33$	$11.61$	${0.24}^{\ast }$			I			$2.4$	$3.7$
NGC 2492	$119.8738$	$27.0264$	$97.8$	$-25.36$	$11.62$	$0.19$			B	$13.0$		$1.1$	$0.8$
NGC 2513	$120.6028$	$9.4136$	$70.8$	$-25.52$	$11.69$	$0.20$	$0.10$	S	B	$13.6$		$2.3$	$5.2$
NGC 2672	$132.3412$	$19.0750$	$61.5$	$-25.60$	$11.72$	$0.14$	$0.10$	S	B	$13.0$		$1.3$	$1.3$
NGC 2693	$134.2469$	$51.3474$	$74.4$	$-25.76$	$11.79$	$0.25$	$0.29$	F	I			$1.7$	$6.9$
NGC 2783	$138.4145$	$29.9929$	$101.4$	$-25.72$	$11.78$	$0.39$	$0.04$	S	B	$12.8$		$3.2$	$4.7$
NGC 2832	$139.9453$	$33.7498$	$105.2$	$-26.42$	$12.08$	$0.31$	$0.07$	S	B	$13.7$	A779	$4.0$	$7.9$
NGC 2892	$143.2205$	$67.6174$	$101.1$	$-25.70$	$11.77$	$0.06$	$0.05$	S	I			$2.2$	$2.3$
NGC 2918	$143.9334$	$31.7054$	$102.3$	$-25.49$	$11.68$	$0.17$			I			$3.0$	$2.5$
NGC 3158	$153.4605$	$38.7649$	$103.4$	$-26.28$	$12.02$	$0.18$	$0.26$	F	B	$13.3$		$2.7$	$9.8$
NGC 3209	$155.1601$	$25.5050$	$94.6$	$-25.55$	$11.70$	$0.27$	$0.04$	S	B	$11.8$		$2.4$	$2.8$
NGC 3332	$160.1182$	$9.1825$	$89.1$	$-25.38$	$11.63$	$0.16$			I			$1.0$	$0.6$

Table 2. Table 2: continued

Galaxy	R.A.	Dec.	$D$	$M_K$	${\log }_{10}{M}_{\ast }$	$\epsilon$	${\lambda }_e$	Rot.	env	${\log }_{10}{M}_{\mathrm{halo}}$	Cluster	$\left(1+{\delta }_g\right)$	${\nu }_{10}$
	[deg]	[deg]	[Mpc]	[mag]	[$M_{\odot }$]					[$M_{\odot }$]			[$\overline{\nu }$]
(1)	(2)	(3)	(4)	(5)	(6)	(7)	(8)	(9)	(10)	(11)	(12)	(13)	(14)
NGC 3343	$161.5432$	$73.3531$	$93.8$	$-25.33$	$11.61$	${0.32}^{\ast }$			I			$2.0$	$16$
NGC 3462	$163.8378$	$7.6967$	$99.2$	$-25.62$	$11.73$	$0.26$	$0.09$	S	I			$2.2$	$2.5$
NGC 3562	$168.2445$	$72.8793$	$101.0$	$-25.65$	$11.75$	${0.16}^{\ast }$	$0.04$	S	B	$13.5$		$2.2$	$8.5$
NGC 3615	$169.5277$	$23.3973$	$101.2$	$-25.58$	$11.72$	$0.38$	$0.40$	F	B	$13.6$		$3.1$	$5.2$
NGC 3805	$175.1736$	$20.3430$	$99.4$	$-25.69$	$11.76$	$0.36$	$0.50$	F	S	$14.8$	A1367	$5.6$	$440$
NGC 3816	$175.4502$	$20.1036$	$99.4$	$-25.40$	$11.64$	$0.31$			S	$14.8$	A1367	$5.8$	$1900$
NGC 3842	$176.0090$	$19.9498$	$99.4$	$-25.91$	$11.86$	$0.22$	$0.04$	S	B	$14.8$	A1367	$5.9$	$19000$
NGC 3862	$176.2708$	$19.6063$	$99.4$	$-25.50$	$11.68$	$0.06$	$0.06$	S	S	$14.8$	A1367	$5.9$	$18000$
NGC 3937	$178.1776$	$20.6313$	$101.2$	$-25.62$	$11.73$	$0.20$	$0.07$	S	B	$14.2$		$5.9$	$71$
NGC 4055	$181.0059$	$20.2323$	$107.2$	$-25.40$	$11.64$	$0.18$			S	$14.3$		$7.1$	$2300$
NGC 4065	$181.0257$	$20.2351$	$107.2$	$-25.47$	$11.67$	$0.22$			B	$14.3$		$7.1$	$2500$
NGC 4066	$181.0392$	$20.3479$	$107.2$	$-25.35$	$11.61$	$0.06$			S	$14.3$		$7.1$	$4200$
NGC 4059	$181.0471$	$20.4098$	$107.2$	$-25.41$	$11.64$	$0.06$			S	$14.3$		$7.1$	$4900$
NGC 4073	$181.1128$	$1.8960$	$91.5$	$-26.33$	$12.05$	$0.32$	$0.02$	S	B	$13.9$		$4.4$	$89$
NGC 4213	$183.9064$	$23.9819$	$101.6$	$-25.44$	$11.65$	$0.17$			B	$13.4$		$4.7$	$16$
NGC 4472	$187.4450$	$8.0004$	$16.7$	$-25.72$	$11.78$	${0.17}^{†}$	$0.20$	F	B	$14.7$	Virgo	$8.9$	$1900$
NGC 4486	$187.7059$	$12.3911$	$16.7$	$-25.31$	$11.60$	$0.10$			S	$14.7$	Virgo	$9.1$	$14000$
NGC 4555	$188.9216$	$26.5230$	$103.6$	$-25.92$	$11.86$	$0.20$	$0.12$	S	I			$5.9$	$6.3$
NGC 4649	$190.9167$	$11.5526$	$16.5$	$-25.36$	$11.62$	$0.20$			S	$14.7$	Virgo	$9.1$	$2600$
NGC 4816	$194.0506$	$27.7455$	$102.0$	$-25.33$	$11.61$	$0.20$			S	$15.3$	Coma	$13.2$	$1900$
NGC 4839	$194.3515$	$27.4977$	$102.0$	$-25.85$	$11.83$	$0.35$	$0.05$	S	S	$15.3$	Coma	$13.2$	$2600$
NGC 4874	$194.8988$	$27.9594$	$102.0$	$-26.18$	$11.98$	$0.09$	$0.07$	S	S	$15.3$	Coma	$13.2$	$24000$
NGC 4889	$195.0338$	$27.9770$	$102.0$	$-26.64$	$12.18$	$0.36$	$0.03$	S	B	$15.3$	Coma	$13.2$	$19000$
NGC 4914	$195.1789$	$37.3153$	$74.5$	$-25.72$	$11.78$	$0.39$	$0.05$	S	I			$1.1$	$1.2$
NGC 5129	$201.0417$	$13.9765$	$107.5$	$-25.92$	$11.86$	$0.37$	$0.40$	F	I			$4.3$	$4.9$
NGC 5208	$203.1163$	$7.3166$	$105.0$	$-25.61$	$11.73$	$0.63$	$0.61$	F	B	$13.0$		$5.0$	$16$
PGC 047776	$203.4770$	$3.2836$	$103.8$	$-25.36$	$11.62$	$0.18$			B	$14.1$		$4.0$	$17$
NGC 5252	$204.5661$	$4.5426$	$103.8$	$-25.32$	$11.60$	$0.52$			S	$14.1$		$4.9$	$52$
NGC 5322	$207.3133$	$60.1904$	$34.2$	$-25.51$	$11.68$	$0.33$	$0.05$	S	B	$13.7$		$2.5$	$21$
NGC 5353	$208.3613$	$40.2831$	$41.1$	$-25.45$	$11.66$	$0.56$			B	$13.6$		$2.6$	$63$
NGC 5490	$212.4888$	$17.5455$	$78.6$	$-25.57$	$11.71$	$0.20$	$0.14$	S	I			$2.1$	$9.8$
NGC 5557	$214.6071$	$36.4936$	$51.0$	$-25.46$	$11.66$	$0.17$			B	$13.3$		$2.6$	$8.5$
IC1 143	$232.7345$	$82.4558$	$97.3$	$-25.45$	$11.66$	${0.14}^{\ast }$			B	$13.0$		$2.0$	$13$
UGC 10097	$238.9303$	$47.8673$	$91.5$	$-25.43$	$11.65$	$0.23$			B	$12.7$		$1.5$	$5.0$
NGC 6223	$250.7679$	$61.5789$	$86.7$	$-25.59$	$11.72$	${0.20}^{\ast }$	$0.32$	F	B	$13.5$		$1.5$	$6.2$
NGC 6364	$261.1139$	$29.3902$	$105.3$	$-25.38$	$11.63$	$0.15$			I			$0.8$	$0.5$
NGC 6375	$262.3411$	$16.2067$	$95.8$	$-25.53$	$11.69$	${0.10}^{\ast }$	$0.24$	F	I			$1.2$	$1.5$
UGC 10918	$264.3892$	$11.1217$	$100.2$	$-25.75$	$11.79$	${0.14}^{\ast }$	$0.03$	S	I			$1.8$	$4.8$
NGC 6442	$266.7139$	$20.7611$	$98.0$	$-25.40$	$11.64$	${0.12}^{\ast }$			I			$1.1$	$3.0$
NGC 6482	$267.9534$	$23.0719$	$61.4$	$-25.60$	$11.72$	${0.36}^{\ast }$	$0.14$	S	B	$13.1$		$1.6$	$1.1$
NGC 6575	$272.7395$	$31.1162$	$106.0$	$-25.58$	$11.72$	${0.28}^{\ast }$	$0.12$	S	I			$2.1$	$5.0$
NGC 7052	$319.6377$	$26.4469$	$69.3$	$-25.67$	$11.75$	${0.50}^{\ast }$	$0.15$	S	I			$1.3$	$0.8$
NGC 7242	$333.9146$	$37.2987$	$84.4$	$-26.34$	$12.05$	${0.28}^{\ast }$	$0.04$	S	B	$14.0$		$6.3$	$2800$
NGC 7265	$335.6145$	$36.2098$	$82.8$	$-25.93$	$11.87$	${0.22}^{\ast }$	$0.04$	S	B	$14.7$		$6.9$	$5200$
NGC 7274	$336.0462$	$36.1259$	$82.8$	$-25.39$	$11.63$	${0.06}^{\ast }$			S	$14.7$		$6.9$	$3200$
NGC 7386	$342.5089$	$11.6987$	$99.1$	$-25.58$	$11.72$	$0.28$	$0.07$	S	B	$13.9$		$2.6$	$3.2$
NGC 7426	$344.0119$	$36.3614$	$80.0$	$-25.74$	$11.79$	${0.34}^{\ast }$	$0.56$	F	B	$13.8$		$3.8$	$8.5$
NGC 7436	$344.4897$	$26.1500$	$106.6$	$-26.16$	$11.97$	$0.12$	$0.09$	S	B	$14.4$		$4.1$	$100$
NGC 7550	$348.8170$	$18.9614$	$72.7$	$-25.43$	$11.65$	$0.07$			B	$11.9$		$0.9$	$1.0$
NGC 7556	$348.9353$	$-2.3815$	$103.0$	$-25.83$	$11.83$	$0.25$	$0.05$	S	B	$14.0$		$2.0$	$17$
NGC 7618	$349.9468$	$42.8526$	$76.3$	$-25.44$	$11.65$	${0.28}^{\ast }$			B	$13.7$		$3.2$	$250$
NGC 7619	$350.0605$	$8.2063$	$54.0$	$-25.65$	$11.75$	$0.23$	$0.12$	S	B	$14.0$		$1.5$	$22$
NGC 7626	$350.1772$	$8.2170$	$54.0$	$-25.65$	$11.75$	$0.14$	$0.03$	S	S	$14.0$		$1.5$	$21$

Table 3. Table 3: p 𝑝 p -values from KS test on slow and fast rotators. Small p 𝑝 p -values indicate slow and fast rotators are likely drawn from different distributions in the given quantity. For the three environment measures we run the KS test separately on the two broad M K subscript 𝑀 𝐾 M_{K} bins of ATLAS 3D ; M K subscript 𝑀 𝐾 M_{K} for each bin is given in magnitudes below.

	$M_K$	$M_{\mathrm{halo}}$	${\delta }_g$	${\nu }_{10}$
$\mathrm{MASSIVE}$	0.314	$0.305$	$0.710$	$0.414$
${\mathrm{ATLAS}}^{3D}$$\left(-23.7\mathrm{to}-25.5\right)$	0.007	$0.835$	$0.156$	$0.112$
${\mathrm{ATLAS}}^{3D}$$\left(>-23.7\right)$		$0.765$	$0.572$	$0.316$
combined	${10}^{-8}$	-	-	-

Equations

$$\lambda (<R)\equiv \frac{⟨R|V|⟩}{⟨R\sqrt{V^2+{\sigma }^2}⟩},{\lambda }_e\equiv \lambda (<R_e),$$

$${\nu }_{10}=\frac{{\Sigma }_{i=0}^{10}10^{-0.4(M_{i,K}-M_{\odot ,K})}}{\frac{4}{3}\pi r_{10}^3}$$

$$P(x)\propto x^{\alpha -1}(1-x{)}^{\beta -1}$$

Full text

††pubyear: 2017††pagerange: The MASSIVE Survey - VII. The Relationship of Angular Momentum, Stellar Mass and Environment of Early-Type Galaxies–C

1 Introduction

As a group, elliptical galaxies obey the fundamental plane and have predominantly old stellar populations (e.g. Dressler et al. 1987; Djorgovski & Davis 1987; Baldry et al. 2004; Thomas et al. 2005). The properties of elliptical galaxies vary considerably with mass, however, and can be grouped into two general families. Lower-mass elliptical galaxies have flattened isophotes, power-law central light profiles, and some net rotation that aligns with their shortest axis. More massive elliptical galaxies, in contrast, show boxy isophotes, cored light profiles, and small net rotation (e.g., Illingworth 1977; Davies et al. 1987; Bender et al. 1989; Kormendy & Bender 1996; Kormendy et al. 2009). The standard interpretation of these differences is that massive elliptical galaxies experience a large number of relatively gas-free mergers that effectively erase any record of their spin, while at lower mass gas accretion and gas-rich mergers tend to preserve a net spin to the galaxies (e.g. Hoffman et al., 2010; Bois et al., 2011; Moody et al., 2014; Khochfar et al., 2011; Martizzi et al., 2014; Naab et al., 2014; Choi & Yi, 2017; Penoyre et al., 2017). Stellar mass strongly determines whether a galaxy is a fast or slow rotator (e.g. Emsellem et al., 2011; Cappellari, 2013; Veale et al., 2017). However, given that the merger and accretion history of a galaxy is partially determined by its surrounding large-scale environment, it is also important to assess the impact of galaxy environment on galaxy rotation.

Motivated by the classic morphology-density relation of Dressler (1980), a number of papers have investigated an analagous kinematic morphology-density relation using integral field spectrograph (IFS) data, comparing how late-type galaxies (LTGs), fast rotating early-type galaxies (ETGs), and slow rotating ETGs populate different density environments. Cappellari et al. (2011b) find that substituting kinematic morphology (i.e. fast or slow rotator status) for Hubble type (lenticular versus elliptical) yields a cleaner relationship than the traditional morphology-density relation. While a significant fraction of elliptical galaxies populate low-density environments, they find that nearly all of them are fast rotators more similar to inclined lenticular galaxies than to genuine spheroidal ellipticals. They also find that the fraction of slow rotators within the ETG population (excluding LTGs) increased at the highest local densities.

A handful of subsequent studies based on individual clusters also report an increased fraction of slow rotators in dense cluster centres (D’Eugenio et al., 2013; Houghton et al., 2013; Scott et al., 2014; Fogarty et al., 2014). However, although the fraction of slow rotators was higher in each cluster centre than in the outskirts, the overall fraction of slow rotators in each cluster did not depend on the size of the cluster or on the large-scale density of the cluster’s environment. Jimmy et al. (2013) searched for signs of recent merging in several brightest cluster galaxies and companions, and find no particular connection between merging signatures and galaxy rotation. Recently, Oliva-Altamirano et al. (2017) find only a tentative increase in the fraction of slow rotators with cluster mass for central galaxies.

Most existing studies to date have been limited to a small number of rich environments, or a small volume probing the field. It is difficult, using these data sets, to decouple the correlated impact of stellar mass and environment on the demographics of slow and fast rotators. In order to control for stellar mass when investigating the importance of environment, it is necessary to span a full range of environments at the highest masses.

We designed the volume-limited MASSIVE survey to investigate systematically the most massive galaxies in the northern sky within a distance of 108 Mpc (Ma et al., 2014), targeting 116 galaxies with $M_{K}\la-25.3$ mag ($M_{*}\ga 10^{11.5}\;M_{\odot}$). This volume is large enough to sample a wide range of environments including several galaxy clusters, many galaxy groups, and galaxies in the field. The sample is thus complementary to the ATLAS3D survey, which includes about twice as many galaxies but from a volume about ten times smaller; it is dominated by the Virgo cluster, and contains only six galaxies more massive than $10^{11.5}\;M_{\odot}$. Details of the kinematic analysis of our IFS data were presented in Veale et al. (2017), which focused on the brightest 41 galaxies ($M_{K}<-25.7$ mag) in the MASSIVE survey. We have since completed observations and analysis of the larger sample of 75 galaxies with a limiting magnitude of $M_{K}=-25.5$ mag.

In this paper we perform a detailed analysis of the environments of the entire MASSIVE sample and present measurements of the spin of the 75 MASSIVE galaxies with Mitchell/VIRUS-P IFS data. Together with the ATLAS3D sample of ETGs at lower masses, we investigate the influence of galaxy mass and environment on the spin of ETGs over a wide range of stellar mass ($8\times 10^{9}\la M_{*}\la 2\times 10^{12}M_{\odot}$) and environment. In particular, the combined MASSIVE and ATLAS3D sample from the two volume-limited IFS surveys has well-defined stellar mass selection and is large enough for us to conduct the first analysis of the relationship between spin and environment at fixed $M_{*}$ for present-day ETGs, and assess how much of the kinematic morphology-density relation is driven by stellar mass.

Since different methods for quantifying environment probe different physical scales, we compare four approaches in this paper: (1) group membership, i.e., if a galaxy is the brightest galaxy or a satellite in a group/cluster, or is relatively isolated; (2) halo mass; (3) smoothed large-scale galaxy density field; and (4) local galaxy density based on the $N$th nearest neighbour. The fact that these measures cover differing length scale implies that they may correlate differently with quantities such as galaxy merger rates, assembly histories, and masses.

Section 2 of this paper summarizes the selection and properties of MASSIVE survey galaxies and ATLAS3D galaxies. Section 3 presents results for the individual measurements and the statistics of the four environmental quantities. Technical details of our local density calculation are in Appendix A. Section 4 summarizes the kinematic analysis of our IFS data and presents results for galaxy spin versus stellar mass. Section 5 connects spin to environment, and explores how to decouple those trends from the influence of stellar mass. Section 6 discusses implications and conclusions. Details of the error bars we use to determine whether any trends are significant are presented in Appendix B. An application of our analysis to the original ATLAS3D sample densities is presented in Appendix C. We assume $h=0.7$ throughout the paper.

2 Galaxy Samples

The MASSIVE survey consists of a volume-limited sample of early-type galaxies, targeting all 116 galaxies111The actual count is 115 galaxies after we remove NGC 7681, as discussed in Veale et al. (2017), which our data showed to be a close pair of bulges. We likewise exclude NGC 7681 from this paper. with $K$-band magnitudes $M_{K}$ brighter than $-25.3$ mag (i.e. stellar masses $M^{*}\gtrsim 10^{11.5}M_{\odot}$) and distances within $D<108$ Mpc, in the northern hemisphere and away from the galactic plane (Ma et al., 2014). The galaxies were selected from the Extended Source Catalogue (XSC; Jarrett et al., 2000) of the Two Micron All Sky Survey (2MASS; Skrutskie et al., 2006). Distances are taken from the surface-brightness fluctuation method (Blakeslee et al., 2009, 2010; Blakeslee, 2013) for galaxies in Virgo and Coma, from the High Density Contrast (HDC) group catalogue (Crook et al., 2007, see also Section 3.1) for other galaxies if available, and from the flow model of Mould et al. (2000) otherwise. We use $M_{K}$ as a proxy for the stellar mass, estimated using equation 2 of Ma et al. (2014). That equation was taken from equation 2 of Cappellari (2013), a fit to stellar masses derived from dynamical modeling.

We have completed observations of the “priority sample” of the MASSIVE survey, which consists of the 75 galaxies with $M_{K}<-25.5$ mag ($M_{*}\ga 10^{11.7}\;M_{\odot}$). The observations were performed using the Mitchell/VIRUS-P IFS at the McDonald Observatory (Hill et al., 2008), which has a large 107″$\times$107″ field of view and consists of 246 evenly-spaced 4″-diameter fibres with a one-third filling factor. We observed each galaxy with three dither positions of equal exposure time to obtain contiguous coverage of the field of view. The spectral range spanned 3650 Å to 5850 Å, covering the Ca H+K region, the G-band region, H$\beta$, the Mgb region, and many Fe absorption features, with $\sim 2000$ pixels (log-spaced in wavelength) and an instrumental resolution of $\sim 4.5$ Å full width at half-maximum.

We spatially bin our IFS spectra to obtain a mean signal-to-noise ratio (S/N) of at least 20 (per pixel) for each spectrum, folding across the major axis to combine symmetrical bins and obtain our minimum S/N with the smallest possible bin size. To obtain the stellar line-of-sight velocity distribution (LOSVD) for each spectrum, we use the penalized pixel-fitting (pPXF) method of Cappellari & Emsellem (2004) and parametrize the LOSVD as a Gauss-Hermite series up to order 6. For each spectrum we thus obtain velocity $V$, dispersion $\sigma$, and higher order moments $h_{3},h_{4},h_{5}$ and $h_{6}$. See Veale et al. (2017) for a more detailed description of the analysis.

There are two main sources of uncertainty on $M_{K}$. First, the $K$-band magnitudes of 2MASS are likely underestimated somewhat due to the shallowness of the survey (Lauer et al., 2007; Schombert & Smith, 2012). Second, the choice of distance estimate and extinction (see Ma et al. 2014 for details) can also impact $M_{K}$ and $M_{*}$; based on the galaxies in common between MASSIVE and ATLAS3D, typical differences in $M_{K}$ due to different estimates is around 0.1 mag, and up to nearly 0.5 mag for extreme cases. These are both larger than the uncertainty due to formal errors in $K$-band magnitude from 2MASS, which are generally less than $0.03$ mag. Together, combined with the 0.14 dex scatter in the $M_{*}$-$M_{K}$ relation (Cappellari, 2013), these correspond to an underestimation of $M_{*}$ of up to $\sim 0.3$ dex, and an uncertainty of $\sim 0.2$ dex.

Where available, additional photometric data is taken from the NASA-Sloan Atlas (NSA, http://www.nsatlas.org) based on the SDSS DR8 catalogue (York et al., 2000; Aihara et al., 2011); otherwise the 2MASS values are used. The effective radius $R_{e}$ from these sources, like $M_{K}$, may be underestimated due to the shallowness of the surveys. We have scaled the 2MASS $R_{e}$ to be comparable to the NSA values (see Ma et al., 2014, for details) to partly account for the 2MASS underestimation. Comparing ellipticity $\varepsilon$ and position angle (PA) of galaxies with both NSA and 2MASS data gives an estimate of uncertainties in those quantities of $\sim 0.05$ and $\sim 15$ degrees, respectively. Very round galaxies may have much higher uncertainties in PA; we discuss in Section 4.1 how this impacts our kinematic analysis, and why we do not expect these uncertainties (or the underestimation of $R_{e}$) to impact our results. Results from our deep CFHT photometry will be reported in upcoming MASSIVE papers.

As a comparison sample to MASSIVE, we examine the 260 nearby galaxies in the ATLAS3D survey (also selected from the 2MASS XSC), which targets all early-type galaxies with $M_{K}<-21.5$ mag ($M_{*}\ga 10^{9.9}\;M_{\odot}$) and located within $D<42$ Mpc, also in the northern hemisphere and excluding the galactic plane (Cappellari et al., 2011a). Due to the larger volume and brighter $M_{K}$ cutoff of the MASSIVE survey, only 6 ATLAS3D galaxies overlap with MASSIVE. The two surveys therefore target complementary parameter space in stellar mass and volume. Of the 6 common galaxies, NGC 4472 and NGC 5322 are in the priority sample ($M_{K}<-25.5$ mag) presented in this paper, while the remaining 4 (NGC 4486, 4649, 5353, 5557) are fainter than this limit. Our kinematic measurements of $V$, $\sigma$, $h_{3}$ and $h_{4}$ agree well with ATLAS3D for the inner $\sim 25$″ region of each galaxy covered by ATLAS3D, but our data extend to at least $\sim 60$″ in radius (see Figure B1 of Veale et al. 2017, where we compare kinematics from both surveys for NGC 4472, 5322, and 5557). We use our wide-field kinematic results for NGC 4472 and NGC 5322 below and remove these two galaxies from the ATLAS3D sample in our plots, to avoid double-counting and create a clean separation in $M_{K}$ between the two samples.

We note that $\sim 20\%$ of galaxies in the ATLAS3D sample are in the Virgo cluster. These galaxies are powerful probes of the intra-cluster environments within Virgo, but they probe only a single cluster. By contrast, no MASSIVE galaxy resides in a galaxy group or cluster (as defined in Section 3.1) containing more than three other MASSIVE galaxies. The MASSIVE sample therefore tends to probe distinct group/cluster environments. When appropriate, we will denote Virgo galaxies in the ATLAS3D sample with distinct symbols below so that the rest of ATLAS3D sample can be compared with the MASSIVE sample more fairly.

3 Galaxy Environments

In this paper we use four different measures to quantify galaxy environments and to investigate the connection between galaxy environments and stellar kinematics for the galaxies in the MASSIVE and ATLAS3D surveys: (1) group membership from the group catalogues of Crook et al. (2007) constructed from the 2MASS Redshift Survey (2MRS; Huchra et al., 2012); (2) halo mass from the same group catalogues, available for galaxies in a group with 3 or more members; (3) a smoothed large-scale density field from Carrick et al. (2015) based on the 2M++ Redshift Catalogue of Lavaux & Hudson (2011); and (4) a local galaxy luminosity density calculated within the volume to the 10th neighbour, similar to $\nu_{10}$ in Cappellari et al. (2011b). We discuss the differences and caveats of the four methods in the subsections below. See also, e.g., Muldrew et al. (2012) for a comprehensive study of different definitions of galaxy environments, and Carollo et al. (2013) for a discussion of how a similar set of environment measures connects to other galaxy properties such as size, color, and star formation rate. Each measure of environment is tabulated in Table 1. An overview of the MASSIVE volume is shown in Figure 1.

3.1 Group Membership

Crook et al. (2007, 2008) published redshift-limited catalogues of galaxy groups based on a 2MRS sample complete to an apparent magnitude (corrected for extinction) of $K<11.25$ mag (Huchra et al., 2005a, b). This limiting magnitude corresponds to an absolute magnitude of approximately $M_{K}<-23.9$ mag at our maximum distance of 108 Mpc, and approximately $M_{K}<-21.9$ mag at the maximum ATLAS3D distance of 42 Mpc. The group catalogues thus cover both MASSIVE ($M_{K}<-25.3$ mag) and ATLAS3D galaxies ($M_{K}<-21.5$ mag), with only two ATLAS3D galaxies (PGC 029321 and UGC 05408) falling outside the magnitude cut.

Crook et al. (2007) applied the Friends-of-Friends (FOF) algorithm with two sets of linking parameters to create two group catalogues of differing density contrasts. The High Density Contrast (HDC) catalogue used a linking length of 350 km s*-1* along the line of sight and 0.89 Mpc in the transverse direction, corresponding to a density contrast of \delta\rho/\rho\;\hbox to0.0pt{\lower 2.5pt\hbox{\sim}\hss}\raise 1.5pt\hbox{>}\;80. The Low Density Contrast (LDC) catalogue used larger linking lengths of 399 km s*-1* and 1.63 Mpc for a density contrast of \delta\rho/\rho\;\hbox to0.0pt{\lower 2.5pt\hbox{\sim}\hss}\raise 1.5pt\hbox{>}\;12. In Ma et al. (2014) we compared and discussed the agreement between the group assignments of the HDC and the 2M++ redshift catalogue of Lavaux & Hudson (2011).

We classify MASSIVE and ATLAS3D galaxies into three types according to their group membership in the HDC catalogue: (1) “brightest group galaxy” (BGG) that belongs to a group and is the most luminous galaxy in the group; (2) “satellite” that belongs to a group but is not the BGG; and (3) “isolated” galaxy that does not belong to a group of at least 3 members in the catalogue. We make no attempt to determine whether the BGG of a group is also the central galaxy of the group, which it may not be (e.g. Skibba et al., 2011); see Oliva-Altamirano et al. (2017) for a discussion of rotation in central galaxies.

Figure 2 shows that among the 116 MASSIVE galaxies, 56% are BGGs, 21% are satellites in groups, and 23% are “isolated”, whereas in ATLAS3D, most of the galaxies are either satellites (51%) or isolated (39%), with only 10% being BGGs. The much lower percentage of BGGs in ATLAS3D than MASSIVE is largely a result of the smaller survey volume and lower mass limit of the ATLAS3D survey. It may also be further suppressed by the relative incompleteness of the HDC catalogue near the edges of the ATLAS3D volume. While only two ATLAS3D galaxies are outside the magnitude cut of Crook et al. (2007) entirely, group membership status also depends on whether nearby galaxies are inside or outside this cut. For example, the same galaxy might be classified as BGG of a group with 4 members at a close distance, but classified as isolated at a farther distance due to all 3 satellite galaxies falling outside the magnitude cut. Thus, ATLAS3D galaxies near the edge of the sample and near the completeness limit of the Crook et al. (2007) catalogues may be biased towards classification as isolated.

In principle all MASSIVE galaxies beyond the ATLAS3D volume are also subject to this relative bias towards being classified as isolated, but this will only occur if the (unidentified) rank 2 galaxy is below the $K=11.25$ magnitude cut ($M_{K}=-23.9$ mag at $D=108$ Mpc, compared to $M_{K}<-25.3$ for MASSIVE galaxies). Most groups have a smaller gap between BGG and rank 2 galaxies than this, so we proceed with the classifications as they are. We are consistent with high-redshift studies showing progenitors of our very massive galaxies in isolated environments (Vulcani et al., 2016).

We will use both HDC and LDC catalogues to provide group distances in Section 3.4 and Appendix A, instead of the improved distances calculated in Ma et al. (2014), for purposes of calculating local density $\nu_{10}$; see those sections for details.

3.2 Halo Mass

The group catalogues of Crook et al. (2007) also include dynamical estimates of dark matter halo masses for groups with at least 3 members. Two measurements of each halo are listed, one based on the standard virial estimator and the other based on the projected mass estimator (Heisler et al., 1985). We use the latter since the former is sensitive to close pairs and can be noisy for groups not uniformly sampled spatially (e.g. Bahcall & Tremaine, 1981). Uncertainties on the projected mass estimator can be up to 0.5 dex for groups with only a few members, though they become smaller for groups with many members (Heisler et al., 1985).

For the MASSIVE galaxies, 89 are in groups with 3 or more members and have available $M_{\rm halo}$ from the HDC catalogue. For the 258 ATLAS3D galaxies not observed with MASSIVE, 158 are in groups with $M_{\rm halo}$ measurements in the same catalogue. Additional halo mass measurements based on more detailed analyses are available for the three well-studied clusters of Virgo, Coma, and Perseus. For Virgo, we use $M_{\rm halo}=5.5\times 10^{14}M_{\odot}$ (same as in Durrell et al., 2014), which is a combination of the Virgo A and B subcluster masses (Ferrarese et al., 2012) and the M86 subcluster mass (Schindler et al., 1999). For Coma, we use $M_{\rm halo}=1.8\times 10^{15}M_{\odot}$, an average between $2.7\times 10^{15}M_{\odot}$ (Kubo et al., 2007) from weak gravitational lensing measurements and $9.2\times 10^{14}M_{\odot}$ (Falco et al. 2014, see also Rines et al. 2003) from galaxy dynamics. For Perseus, we use $M_{\rm halo}=6.7\times 10^{14}M_{\odot}$ from spatially-resolved Suzaku X-ray observations (Simionescu et al., 2011).

The left panel of Figure 3 compares the distribution of $M_{\rm halo}$ in the two surveys. Nearly 40% of MASSIVE galaxies are in haloes above $10^{14}M_{\odot}$, whereas only $\sim 5$% of ATLAS3D galaxies outside of the Virgo region are in such massive haloes. The leftmost bin shows "isolated" galaxies with fewer than 3 group members in the HDC catalogue and hence with no available $M_{\rm halo}$ measurements; a higher fraction of ATLAS3D galaxies belong to this category than MASSIVE galaxies ($\sim 40$% versus 23%; see also Section 3.1 and Figure 2).

3.3 Large-scale density

The group membership and group halo mass that we have investigated thus far provide information about galaxy environment on scales of a few hundred kpc to $\sim 1$ Mpc. Another useful measure of galaxy environment is the large-scale density field surrounding a galaxy on the scale of several Mpc. To this end, we use the density field of Carrick et al. (2015) constructed from the 2M++ redshift catalogue of Lavaux & Hudson (2011). The 2M++ catalogue contains 69,160 galaxy redshifts from 2MRS, the Sloan Digital Sky Survey Data Release 7 (SDSS-DR7; Abazajian et al., 2009), and the 6dF galaxy redshift survey Data Release 3 (6dFGRS-DR3; Jones et al., 2009). It covers nearly the full sky and reaches a depth of $K=12.5$ mag, deeper than $K=11.75$ mag for the 44,599 galaxies in 2MRS alone.

Carrick et al. (2015) presents a luminosity-weighted galaxy density contrast, $\delta_{g}\equiv(\rho_{g}-\overline{\rho_{g}})/\rho_{g}$, smoothed with a 5.7 Mpc Gaussian kernel. This density field is computed with weights assigned to each galaxy’s luminosity to account for the magnitude limit of the survey and incompleteness. It is also rescaled to account for the impact of luminosity dependent galaxy-matter bias on the density field calculated at different redshifts. The result is a smoothed density field complete out to a distance of 178 Mpc (and partial coverage to a further distance of 286 Mpc). Our survey (out to 108 Mpc) is well within this radius.

The grid spacing of the published density field is approximately 2.2 Mpc. We compute the density at the location of each galaxy using a simple trilinear interpolation, which results in interpolation errors of approximately 0.1 dex. For galaxy distance, we use the LDC group-corrected distance where available, and redshift distances from Huchra et al. (2012) otherwise (assuming $h=0.73$ as in Crook et al. 2007). We use these distances here instead of the distances from Ma et al. (2014) because they are more comparable to the reconstruction procedure in Carrick et al. (2015), but uncertainties in distance estimates will result in uncertainties in the density.

The middle panel of Figure 3 shows the distributions of $\delta_{g}$ for the entire MASSIVE versus ATLAS3D sample. The values of $\delta_{g}$ for the 75 MASSIVE galaxies with stellar kinematics are listed in Table 1. While all MASSIVE galaxies are in regions above or near the cosmic mean density, about 20% of ATLAS3D galaxies are in underdense or mean-density regions. The Coma cluster and Perseus cluster are the two highest-density regions sampled by the MASSIVE survey, both with $\delta_{g}\approx 12$. In comparison, the Virgo cluster is the highest-density region sampled by the ATLAS3Dsurvey with $\delta_{g}\approx 8$ (dotted line in Figure 3). Because $\delta_{g}$ is smoothed over a scale larger than the size of a galaxy cluster, all galaxies in the same cluster have the same $\delta_{g}$.

Figure 1 is a sky map of $\delta_{g}$ contours over the MASSIVE volume. The MASSIVE galaxies (black circles) are located in regions with $\delta_{g}\ga 0$ (yellow and light green), whereas many ATLAS3D galaxies (grey squares) are in the Virgo cluster or lower density regions (dark green). As expected, the parent sample of $\sim 15,000$ early-type galaxies with $M_{K}<-23.0$ mag from 2MRS (white dots) traces the $\delta_{g}$ contours quite well.

3.4 Local density

Finally, we calculate a local galaxy density by finding the distance to the $N$th nearest neighbour of a galaxy and estimating the luminosity (or mass) enclosed in this region. Several versions of local densities were tabulated in Cappellari et al. (2011b) for the ATLAS3D sample. We will focus on $\nu_{10}$, the luminosity density of galaxies in a sphere enclosing the $10^{\rm th}$ nearest neighbour (where the galaxy itself is counted as the 0th neighbour).

The galaxy sample used to estimate $\nu_{10}$ in Cappellari et al. (2011b) included all 2MRS galaxies (including spirals) with $M_{K}<-21.5$ mag in the ATLAS3D volume. This cut matches the completeness limit $K=11.75$ mag of 2MRS, which corresponds to $M_{K}\approx-21.5$ mag at the edge of the ATLAS3D volume (42 Mpc). At the edge of the much larger volume probed by MASSIVE (108 Mpc), however, 2MRS is complete only to $M_{K}\approx-23.4$ mag. Using the same parent sample as in ATLAS3D to calculate $\nu_{10}$ for MASSIVE galaxies would thus suffer substantially from incompleteness. Instead, We choose a magnitude cut of $M_{K}=-23.0$ mag for defining the parent 2MRS sample and compute $\nu_{10}$ from this sample of approximately $10^{4}$ galaxies for both MASSIVE and ATLAS3D galaxies, for a fair comparison between the two surveys. For simplicity and uniformity, we use HDC group distances (where available; LDC group distances otherwise, and redshift distances as a last resort) for all galaxies in this calculation. This includes the survey galaxies, even if they have more accurate distances tabulated in Ma et al. (2014) or Cappellari et al. (2011a). Using group distances results in typical uncertainties of about 0.5 dex on $\nu_{10}$, due to individual galaxy distances being flattened to the group distace. Rarely, $\nu_{10}$ may be inflated by more than this because the distance flattening also re-orders the closest neighbors, but this impacts only a handful of galaxies. Additional details are discussed in Appendix A.

The distribution of the resulting $\nu_{10}$ for each survey is shown in the right panel of Figure 3. The values of $\nu_{10}$ for the MASSIVE priority sample are listed in Table 1. We express $\nu_{10}$ in units of the mean $K$-band luminosity density $\overline{\nu}\sim 2.8\times 10^{8}\;{\rm L}_{\odot}\;{\rm Mpc}^{-3}$ for magnitude ranges of $-21>M_{K}>-25$ mag from Table 2 of Lavaux & Hudson (2011). As their Table 2 shows, enlarging the range to $-17>M_{K}>-25$ mag would raise $\overline{\nu}$ by only 5%, so the accuracy of the magnitude range is not a concern.

Figure 3 and Table 1 show that MASSIVE galaxies span about five orders of magnitude in $\nu_{10}$, reaching $\nu_{10}$ above $10^{4}$ for galaxies at the centres of the Coma cluster, Perseus cluster, Virgo cluster, Abell 194, Abell 262, Abell 347, and Abell 1367.

An alternate measure of local density is $\Sigma_{3}$, defined as the number density of galaxies in a cylinder of 600 km s*-1* in height, enclosing the 3 nearest neighbours, centred on the galaxy. This avoids the requirement of good redshift-independent distances by replacing it with a flat cutoff requiring neighbours to have heliocentric velocities within 300 km s*-1* of the central galaxy. It is more sensitive to issues of survey completeness, and the overall results using $\Sigma_{3}$ are not much different than those using $\nu_{10}$, so we do not discuss it in the body of the paper. For completeness, we do present results in Appendix C for the original $\Sigma_{3}$ and $\nu_{10}$ densities calculated in Cappellari et al. (2011a).

3.5 Relationships among different measures of environment

Here we examine how the environmental measures discussed above – group membership, $M_{\rm halo}$, $\delta_{g}$, and $\nu_{10}$ – correspond to one another. The distributions in $\delta_{g}$ for isolated galaxies in the two surveys are shown at the bottom of Figure 4. The top panel of Figure 4 shows $M_{\rm halo}$ vs $\delta_{g}$ for BGGs (magenta) and satellites (yellow), where higher-mass haloes generally reside in higher-density regions. This is expected since clusters of mass above $\sim 10^{14}M_{\odot}$ dominate the overdensity within 5.7 Mpc, the smoothing scale of $\delta_{g}$. If a galaxy halo with $M_{\rm halo}$ were the only mass within the smoothing volume $V$ (of radius 5.7 Mpc), the overdensity would be $M_{\rm halo}=(1+\delta)\overline{\rho}\,V$, as indicated by the dotted line in Figure 4. The galaxies with $M_{\rm halo}\ga 10^{14}M_{\odot}$ indeed lie near this line. Many lower-mass halos, on the other hand, lie to the right of this line since $\delta_{g}$ measures the large-scale density field rather than the halo mass itself in this regime; $\delta_{g}$ therefore spans a wide range of values, from near the cosmic mean to regions with $\delta_{g}$ near 10.

Satellite galaxies are sometimes visible as “tails” to the left of the BGG galaxy in the same halo in Figure 4, when the halo is large enough for the outskirts to show noticeably lower density on the smoothing scale of $\delta_{g}$. Many more satellite galaxies, however, are hidden behind their BGG galaxies on this figure; in particular, all galaxies defined as Virgo galaxies by ATLAS3D are within one symbol-width of the Virgo BGG. The other satellites defined as part of the same halo by the HDC catalogue were not designated as Virgo galaxies in ATLAS3D.

Figure 5 shows the relationship between local density $\nu_{10}$ and large-scale density $\delta_{g}$. At low densities, where both the distance to the $10^{\rm th}$ neighbour and the smoothing scale of $\delta_{g}$ are at scales beyond the size of the host halo, $\nu_{10}$ follows $\delta_{g}$ almost exactly. In dense regions, however, $\nu_{10}$ and $\delta_{g}$ can deviate significantly, where $\nu_{10}$ is determined by the innermost 10 galaxies well within a single host halo, while $\delta_{g}$ continues to measure the overdensity on several Mpc scale surrounding the galaxy. The Virgo galaxies (open symbols) illustrate this difference, showing a large spike in $\nu_{10}$ relative to $\delta_{g}$. Perseus shows the highest $\nu_{10}$, and other groups/clusters (including Coma) in dense environments are similarly far above the smoothed density $\delta_{g}$.

4 Galaxy spin versus stellar mass

In this section we present measurements of rotation in MASSIVE galaxies and investigate the dependence of galaxy spin on $M_{*}$ and $M_{K}$ in the ATLAS3D and MASSIVE surveys. Here, and throughout the rest of the paper, we consider the 75 galaxies of the priority sample ($M_{K}<-25.5$ mag, $M_{*}\ga 10^{11.7}\;M_{\odot}$) of the MASSIVE survey, for which we have completed the Mitchell IFS observations. This sample nearly doubles the 41 galaxies brighter than $M_{K}=-25.7$ mag reported in Veale et al. (2017).

4.1 Galaxy spin $\lambda_{e}$

As discussed in Section 2, our kinematic data provide measurements of the stellar velocity ($V$), dispersion ($\sigma$), and higher velocity moments ($h_{N}$) for each spatial bin folded across the major axis. We use the dimensionless parameter $\lambda$ to quantify the relative importance of rotation in a galaxy:

[TABLE]

where the angle brackets represent luminosity-weighted averages over all bins within $R$ (Emsellem et al., 2007). The luminosity-weighting and cumulative nature of $\lambda(<R)$ prevents it from varying rapidly past $R_{e}$, unlike the local $\lambda(R)$ sometimes used to investigate radial gradients in rotation structure. The fact that our $\lambda(<R)$ is largely flat by $R_{e}$ is important to minimize any bias in $\lambda_{e}$ due to possibly underestimated $R_{e}$ measurements (Veale et al., 2017).

The values of $\lambda_{e}$ for the 75 MASSIVE galaxies are plotted against ellipticity in Figure 6 (circles) and are listed in Table 1. The ATLAS3D sample is also shown (squares) for comparison. We note that our measurements of $\lambda_{e}$ use circular bins (Veale et al., 2017), whereas ATLAS3D uses elliptical apertures. Since most MASSIVE galaxies are quite round ($\varepsilon<0.4$), the difference between circular and elliptical apertures is likely to be insignificant. Emsellem et al. (2011) classified galaxies as fast or slow rotators according to $\lambda_{e}=0.31\sqrt{\varepsilon}$, where all galaxies with $\lambda_{e}$ below this cutoff are defined as slow rotators. A flat cutoff at $\lambda_{e}=0.2$ had also been suggested (Lauer, 2012), and could be appropriate for our galaxies for several reasons as discussed in Veale et al. (2017). We have tested that the conclusions of this paper are not changed if we use this flat cutoff, which by extension ensures that our conclusions are not impacted by uncertain measurements of $\varepsilon$. For simplicity, we use only the ATLAS3D cutoff hereafter.

We note that the IFS observations of many ATLAS3D galaxies do not reach $R_{e}$, so their tabulated $\lambda_{e}$ is calculated within a smaller radius than in the MASSIVE survey. About half of ATLAS3D galaxies have data extending to between $0.5R_{e}$ and $R_{e}$, and about 8% have observations extending to less than $0.5R_{e}$ (Emsellem et al., 2011). Because many galaxies show a rising $\lambda(<R)$ profile within $R_{e}$, this may result in under-estimated $\lambda_{e}$ for ATLAS3D galaxies, and perhaps in fast rotators being misclassified as slow rotators. For a galaxy with $\lambda(<R)$ measured to $0.5R_{e}$, the appropriate slow/fast cutoff would be reduced by a factor of $\sim 1.2$ (Emsellem et al., 2011). Based on this rough scaling, only 4 ATLAS3D galaxies may be misclassified as slow rotators. One of these is NGC 4472 (discussed in detail in Veale et al. 2017), which has $\lambda_{e}=0.077$ according to Emsellem et al. (2011), measured with data going to only $0.26R_{e}$, and has $\lambda_{e}=0.2$ in our sample. Of the remaining overlap galaxies between MASSIVE and ATLAS3D, only NGC 5322 and NGC 5557 (below the $M_{K}<-25.5$ cut of this paper) have been observed, and they both agree to within about $\Delta\lambda_{e}\sim 0.02$ with the results of ATLAS3D.

Another source of uncertainty is the measured PA of the galaxies; because we fold our data over the photometric PA, any misalignment between the rotation axis and photometric minor axis would wash out the rotation. NGC 1129 and NGC 4874 were misaligned in this way, so we manually adjusted the folding PA (see Veale et al., 2017). Based on those manual adjustments we estimate minor misalignments ($\sim 10$ degrees) to reduce $\lambda_{e}$ by $\sim 0.01$, although the impact may be greater for faster rotators. Formal errors on $\lambda_{e}$, calculated from the uncertainties on $V$ and $\sigma$ in each bin according to the formulae in Houghton et al. (2013), are shown in Figure 6 for MASSIVE galaxies. They are generally smaller than the potential systematic uncertainties discussed here, with over half of MASSIVE galaxies having formal errors less than $0.01$. Finally, the $V$ and $\sigma$ used to calculate $\lambda_{e}$ may differ by up to 10% between Gaussian-only fits (as used by ATLAS3D) and Gauss-Hermite fits including higher moments (as we use) (van der Marel & Franx, 1993).

4.2 $\lambda_{e}$ versus $M_{*}$

We find a strong anti-correlation between $\lambda_{e}$ and $M_{*}$, similar to our earlier finding from the smaller sample of 41 MASSIVE galaxies in (Veale et al., 2017). The top panel of Figure 7 shows $\lambda_{e}$ versus $M_{*}$ (and $M_{K}$) for each galaxy in MASSIVE and ATLAS3D. Even though the range of $\lambda_{e}$ at a given $M_{*}$ is similar for almost the entire range of $M_{*}$, the average $\lambda_{e}$ over 9 $M_{*}$ bins (colour symbols) drops from $\sim 0.4$ for galaxies below $M_{*}\sim 10^{10.5}M_{\odot}$, to below 0.1 for galaxies above $M_{*}\sim 10^{12}M_{\odot}$. The bottom panel of Figure 7 plots the corresponding steep rise in the fraction of slow rotators $f_{\rm slow}$, from $\sim 10$% to $\sim 90$% with increasing $M_{*}$. The average behaviour of $\lambda_{e}$ versus $M_{*}$ in the top panel does not substantially change if we normalize $\lambda_{e}$ by the slow/fast cutoff ($0.31\sqrt{\varepsilon}$). Some individual round galaxies have very high normalized $\lambda_{e}$, but the average behaviour is qualitatively very similar.

There is a slight decrease in average $\lambda_{e}$ and slight increase in $f_{\rm slow}$ at the low mass end of the ATLAS3D sample. This is not due to any incompleteness of the sample (Cappellari et al., 2011a), and also occurs if we use a flat cutoff to define the slow rotators, so it is not due to changes in $\epsilon$ influencing the classification. Perhaps coincidentally, the peak in the ETG mass function at $M_{K}\sim-22.5$ mag is near the peak of $\langle\lambda_{e}\rangle$ and the minimum of $f_{\rm slow}$. It is also near the inferred peak in star formation efficiency at approximately $10^{10.5}M_{\odot}$ (e.g. Behroozi et al., 2013). Since we are focusing on the highest mass galaxies, we will not discuss these trends further.

5 Galaxy spin versus environment

In this section we examine the relation between galaxy spin and the various galaxy environmental measures defined in Section 3 for the combined MASSIVE and ATLAS3D sample. Section 5.1 and Section 5.2 present simple tests of the correlation between slow rotator fraction $f_{\rm slow}$ and $M_{\rm halo}$, $\delta_{g}$, and $\nu_{10}$, and Section 5.3 presents a more detailed test of how the joint correlation of $M_{*}$ vs. $\lambda_{e}$, and $M_{*}$ vs. environment, impacts our results.

5.1 $\lambda_{e}$ versus halo mass and density

Figure 8 shows the distribution of $\lambda_{e}$ (upper panels) and $f_{\rm slow}$ (lower panels) versus halo mass $M_{\rm halo}$ (left), large-scale density contrast $\delta_{g}$ (middle), and local density $\nu_{10}$ (right). Measurements of $\lambda_{e}$ for individual MASSIVE (black circles) and ATLAS3D (grey squares) galaxies are shown in the upper panels. The colored lines show the average $\lambda_{e}$ and $f_{\rm slow}$ for three $M_{*}$ bins: $M_{*}>10^{11.7}M_{\odot}$ (red) contains all MASSIVE galaxies in this study; and the other two bins contain ATLAS3D galaxies with $10^{10.9}M_{\odot}<M_{*}<10^{11.7}M_{\odot}$ (orange) and $10^{9.9}M_{\odot}<M_{*}<10^{11.7}M_{\odot}$ (olive), respectively.

The average $\lambda_{e}$ in Figure 8 is seen to decrease with increasing $M_{*}$ bin, and $f_{\rm slow}$ is seen to increase correspondingly, as discussed in the previous section. Within each $M_{*}$ bin, however, there is at most a weak correlation with environment. The lack of correlation applies regardless of the exact quantity we used to measure spin (i.e., individual $\lambda_{e}$, $\langle\lambda_{e}\rangle$, $f_{\rm slow}$) or environment (i.e., $M_{\rm halo}$, $\delta_{g}$, $\nu_{10}$). The MASSIVE galaxies occupy nearly the same range of $\lambda_{e}$, but have many more galaxies near $\lambda_{e}=0$. Those slow and non-rotating galaxies occupy the same range of environments as our overall sample, resulting in a low $\lambda_{e}$ and high $f_{\rm slow}$ in all environments for our high-mass galaxies.

Some subtle trends with environment may be seen, although none are obviously significant given the number statistics of our samples, with one or two galaxies being the margin of difference in many cases. For MASSIVE galaxies, larger halo mass correlates with a slightly lower average $\lambda_{e}$ and slightly higher $f_{\rm slow}$ (red lines in the left panels of Figure 8). Similar correlations also apply to $\nu_{10}$ for MASSIVE galaxies (red lines in the right panels of Figure 8), and to $\delta_{g}$ and $\nu_{10}$ for the more massive half of ATLAS3D galaxies (orange lines in the middle and right panels of Figure 8).

To quantify the significance of these trends, we run two-sample Kolmogorov-Smirnov (KS) tests to compare the distribution of slow rotators to the distribution of fast rotators in $M_{\rm halo}$, $\delta_{g}$, and $\nu_{10}$.222Because the KS test is not suitable for discrete parameters, the large number of ATLAS3D galaxies at the same $M_{\rm halo}$ in Virgo (and to a lesser degree the duplicated $M_{\rm halo}$ values of other haloes for both surveys) causes a problem. To solve this, we add a small random variable between $\pm 0.1$ to $\log_{10}M_{\rm halo}$ before computing the KS test, and run 1000 trials of this procedure to find the average $p$-value. The resulting $p$-values for MASSIVE and ATLAS3D are listed in Table 3. Many values are greater than 0.5, indicating that it is more likely than not that the slow and fast rotators are drawn from identical distributions in environment. The smallest $p$-values for environment are $\sim 0.1$ for the distributions in $\delta_{g}$ and $\nu_{10}$ of the more massive half of the ATLAS3D galaxies. These align with the qualitative trends we noted above, but are still not considered significant. In comparison, a KS-test for the distribution of slow and fast rotators with $M_{K}$ gives $p=0.007$ for the ATLAS3D sample, $p=0.31$ for the MASSIVE sample, and approximately $p\sim 10^{-8}$ for a combined sample.333For the combined sample, we copy each ATLAS3D galaxy 10 times before finding the KS statistic to account for the fact that the MASSIVE volume is approximately 10 times larger. This gives a reasonable overall distribution in $M_{K}$, with no kink in the cumulative distribution function to inflate differences between the slow and fast rotators. To convert the KS statistic into a $p$-value, we use the original sample sizes, so the $p$-value is not artificially small due to artificially large $N$.

5.2 $\lambda_{e}$ versus $M_{*}$, for two environmental bins

In the previous subsection we examined $\lambda_{e}$ as a function of environment for three $M_{*}$ bins. Here we investigate $\lambda_{e}$ as a function of $M_{*}$ for a low-density versus a high-density environmental bin.

Figure 9 shows the average $\lambda_{e}$ versus $M_{*}$, split into two samples (orange versus blue) by each of our three environment measures: $M_{\rm halo}$ (left), $\delta_{g}$ (middle), and $\nu_{10}$ (right). The trend of $\lambda_{e}$ with $M_{*}$ for each environmental group in Figure 9 follows closely the trend of the whole sample in Figure 7. Within a $M_{*}$ bin, however, there is again no statistically significant difference in the average $\lambda_{e}$ for the higher versus lower density sample for any of the three environmental variables. In the central $M_{*}$ bin, which spans a factor of $\sim 5$ in $M_{*}$ ($10^{11}M_{\odot}<M_{*}<10^{11.7}M_{\odot}$), the galaxies in the higher-density sample (orange) have a slightly higher average $M_{*}$ and a lower average $\lambda_{e}$ than the lower-density sample (blue). These points continue smoothly into the highest mass bins populated by the MASSIVE galaxies.

Like Figure 8, Figure 9 again illustrates that for local ETGs of similar $M_{*}$, the galaxy spin does not correlate noticeably with galaxy environment.

5.3 $\lambda_{e}$, $M_{*}$, and environment

We perform an additional test to gain further insight into the lack of correlation between galaxy spin and environment at fixed stellar mass. We create a randomized sample in which the galaxies’ $M_{*}$ and environment are preserved but the spins are shuffled. Specifically, we randomly assign each galaxy to be a fast or slow rotator, using the probability for being a slow rotator determined by the galaxy’s $M_{*}$ shown in Figure 7 (bottom panel). For example, a galaxy with $M_{*}=10^{10.6}M_{\odot}$ has a 6/68 = 8.8% chance of being a slow rotator, and a galaxy with $M_{*}=10^{12.1}M_{\odot}$ has a 8/9 = 89% chance of being a slow rotator, independent of their environment. We then count the fraction of slow rotators in each environment for each survey, and repeat the procedure 1000 times to obtain an estimate of the fraction and error bar 444The 68% confidence interval based on the cumulative distribution function of trials. that reflects expected errors due to small sample sizes. We estimate error bars on the observed $f_{\rm slow}$ with a Bayesian method described in Appendix B.

Figure 10 shows $f_{\rm slow}$ for this test sample (symbols with black error bars) for three $M_{*}$ bins for the three types of group memberships (isolated, satellite and BGG). The fact that the randomized test sample reproduces the observed $f_{\rm slow}$ supports our finding of the lack of an independent correlation between galaxy spin and environment as quantified by group membership. The much higher fraction of slow rotators in MASSIVE (red) than ATLAS3Dgalaxies (orange and olive) seen in Figure 10 arises from their higher $M_{*}$ and not galaxy environment. Likewise, the increasing fraction of slow rotators with group membership category for ATLAS3D galaxies can be accounted for entirely (within errors) by the joint correlations between $M_{*}$ and spin, and between $M_{*}$ and environment.

Figure 11 shows the same comparison as Figure 10 but for the other environmental measures. The observed trends of $f_{\rm slow}$ with environment for the three $M_{*}$ bins is again reproduced in the test sample despite the randomization of galaxy spin. One possible exception is the highest $\nu_{10}$ bin for the middle and high mass bins (orange and red in the right panel of Figure 11), where the test sample underpredicts the observed $f_{\rm slow}$ slightly. For completeness, we also apply this procedure to the local densities $\nu_{10}$ and $\Sigma_{3}$ originally tabulated in Cappellari et al. (2011b) (see Appendix C) and obtain similar results.

Figure 10 and Figure 11 show that the observed $f_{\rm slow}$ match closely the test samples in which galaxy spin and environment are randomized. We therefore conclude that there is likely little or no correlation between spin and environment for galaxies of the same mass, and the apparent kinematic morphology-density is largely driven by stellar mass.

6 Conclusions

In this paper we have analysed the detailed environmental properties of the 116 galaxies in the MASSIVE survey and the 260 galaxies in the ATLAS3D survey. The MASSIVE survey is designed to sample a much larger volume than ATLAS3D and to target exclusively galaxies above $M_{*}=10^{11.5}M_{\odot}$; only 6 ATLAS3D galaxies are above this mass limit (Section 2). These two complementary IFS surveys together span $-21.5\ga M_{K}\ga-26.6$ mag, or $8\times 10^{9}\la M_{*}\la 2\times 10^{12}M_{\odot}$, and provide the most comprehensive study to date of individual early-type galaxies in the local universe.

We examined different ways to quantify galaxy environment and presented results for group membership (BGG, satellite or isolated), halo mass, large-scale density $\delta_{g}$ measured over a few Mpc, and local density $\nu_{10}$ measured within the 10th nearest neighbour of each galaxy (Section 3). Despite their high stellar masses, MASSIVE galaxies reside in a diverse range of environments, and not all of them are the brightest galaxies in massive haloes at high densities, as is commonly assumed for massive ETGs. This key feature of massive galaxies was highlighted in Ma et al. (2014); here we have added environmental measures $\delta_{g}$ and $\nu_{10}$ to the analysis. About 20% of MASSIVE galaxies are “isolated”, having fewer than three group members in the 2MASS HDC catalogue (Figure 2); about 30% of MASSIVE galaxies are in regions of modest densities $\nu_{10}\la 10\bar{\nu}$, or $\delta_{g}\la 2$ (Figure 3). Compared to ATLAS3D galaxies, we found a higher fraction of MASSIVE galaxies to be BGGs ($\sim 60$% versus 10%) and to be located in more massive haloes and higher density regions (Figure 2 and Figure 3).

We then presented measurements of galaxy spin $\lambda_{e}$ for the 75 galaxies of the MASSIVE “priority sample” with IFS data (Section 4). We confirm the strong anti-correlation between spin and stellar mass that we reported earlier for a smaller sample of MASSIVE galaxies (Veale et al., 2017): the average $\lambda_{e}$ decreases from $\sim 0.4$ at $M_{*}\sim 10^{10}M_{\odot}$ to below 0.1 at $M_{*}\ga 10^{12}M_{\odot}$, and the fraction of slow rotators increases from $\sim 10$% to 90% (Figure 7).

The combined MASSIVE and ATLAS3D sample is sufficiently large for us to analyze for the first time the correlation between galaxy spin and environment – the so-called kinematic morphology-density relation – at fixed $M_{*}$. After controlling for $M_{*}$, we found almost no remaining correlations between galaxy environment and spin (Figure 8- Figure 11).

In particular, we find a high fraction of slow rotators ($\sim 80$%) in the MASSIVE sample in every environment, regardless of group membership, halo mass, or densities $\delta_{g}$ or $\nu_{10}$ (Figure 8, Figure 10 and Figure 11). Previous studies did not include high-mass galaxies at low environmental densities since they either sampled smaller volumes (Cappellari et al., 2011b) or investigated only individual clusters (D’Eugenio et al., 2013; Houghton et al., 2013; Fogarty et al., 2014). Strong correlations between slow rotator fraction and local density have been reported for these samples, with possible implications for slow rotator formation pathways (see, e.g. Cappellari 2016 for a review). However, without the high-mass low-density galaxies like those in the MASSIVE survey, it was difficult to assess whether the kinematic morphology-density relation applies at fixed mass, i.e., whether environmental overdensities play a role in the formation of slow rotators that is distinct from their role as a natural environment for massive ETGs in general. The results presented in this paper show that a dense environment is not required to create massive slow rotators.

Our tests show that most observed correlations between galaxy environment and spin can be explained by the joint connections between $M_{*}$ and spin, and between $M_{*}$ and environment (Figure 9, Figure 11). This is consistent with a scenario in which mergers generally are responsible for both increasing the mass of a galaxy and decreasing the spin. A possible exception is the local density $\nu_{10}$, which shows evidence that the highest densities host a slightly larger fraction of slow rotators, even after controlling for $M_{*}$ (Figure 11). This residual dependence may indicate that certain types of assembly history – perhaps those including more minor mergers and non-merger interactions as suggested by simulations (e.g., Moody et al. 2014; Choi & Yi 2017) – are more likely to create slow rotators, even when controlling for the final mass of the galaxy, and that local density is a reasonable proxy for the type of assembly history.

The fast-slow kinematic transformation (e.g. this work, Cappellari, 2013) can be compared to the spiral-elliptical morphological transformation (e.g. Dressler, 1980). Galaxy kinematics and morphologies both transform with galaxy mass, so it is important to examine whether the transformation with environment applies at fixed mass. The morphology-density relation nearly disappears for galaxy samples at fixed stellar mass (e.g., Bamford et al. 2009; Tasca et al. 2009; Blanton & Moustakas 2009; Grützbauch et al. 2011; Muzzin et al. 2012; Alpaslan et al. 2015; Saracco et al. 2017). We have found the same to be true for the kinematic morphology-density relation, which, at fixed stellar mass, disappears completely for every environmental measure studied in this paper except perhaps $\nu_{10}$. A recent pre-print finds similar results for galaxies in eight clusters with halo mass above $10^{14.2}M_{\odot}$ from the SAMI survey, with no significant remaining relationship between the slow rotator fraction and local overdensity after controlling for the strong correlation with mass (Brough et al., 2017).

Increased statistics from ongoing and future surveys using IFS such as SAMI (Croom et al., 2012), CALIFA (Sánchez et al., 2012), MaNGA (Bundy et al., 2015), and HECTOR (Bryant et al., 2016) could provide more sensitive probes of the transition regime between fast and slow rotating ETGs. The MASSIVE survey is designed to explore new parameter space unprobed by ATLAS3D; the two samples therefore have little overlap. It is somewhat a coincidence that ETGs transition from being dominated by fast rotators to being dominated by slow rotators at $M_{K}\sim-25$ mag, the interface between the two surveys (Figure 7). A volume-limited survey targeting more galaxies brighter than $M_{K}\sim-24$ mag would be useful for gaining further insight into the kinematic transformation along the mass sequence of present-day ETGs.

Acknowledgements

We thank Mike Hudson for his assistance with the 2M++ catalogue. The MASSIVE survey is supported in part by NSF AST-1411945, NSF AST-1411642, HST-GO-14210, and HST-AR-1457.

Appendix A Calculation of $\nu_{10}$

The local luminosity density $\nu_{10}$ was described briefly in Section 3.4. Here we discuss some of the technical details of calculating $\nu_{10}$, which is defined as follows:

[TABLE]

where the solar $K$-band luminosity is $M_{\odot,K}=3.29$ mag (Blanton & Roweis, 2007). Index $i=0$ to 10 refers to the galaxy itself ($i=0$) and its ten nearest neighbours, so $r_{10}$ is the distance to the 10th neighbour and defines a sphere containing the 10 neighbours.

In Cappellari et al. (2011b), the ten nearest neighbours were chosen from the ATLAS3D parent sample, containing all galaxies (not just ETGs) in the ATLAS3D volume with $M_{K}<-21.5$ mag. This cut reflects the 2MRS survey limit of $K=11.75$, which is illustrated in Figure 12. Using the same $M_{K}$ cut to define a parent sample for MASSIVE would result in substantial incompletness; all galaxies between the red line and dashed grey line in Figure 12 would be missing. On the other hand, to guarantee zero impact from incompleteness to our $\nu_{10}$ calculation (moving the vertical dotted line all the way to the right of the figure) would require a cut of $M_{K}<-24.0$, which would cause most ATLAS3D galaxies to fall outside the cut entirely. We want to make a fair comparison to ATLAS3D galaxies, so we recompute $\nu_{10}$ instead of using the values in Cappellari et al. (2011b), and thus want to keep more than a few galaxies inside our cut. We choose to define the MASSIVE parent sample with $M_{K}<-23.0$ as a compromise between those two considerations. This cut allows us to keep about half of the ATLAS3D galaxies for comparison, while about half of the MASSIVE galaxies have $\nu_{10}$ possibly impacted by incompleteness of the parent sample.

To estimate the impact of incompleteness on $\nu_{10}$, we repeated our calculation for a parent sample cut at $M_{K}<-24.0$. Using only galaxies not impacted by incompleteness (i.e. those to the left of the dotted line in Figure 12), we found that expanding the parent sample from $M_{K}<-24.0$ to $M_{K}<-23.0$ results in a characteristic increase of $\Delta\log_{10}\nu_{10}\sim 0.6$. This represents a worst case scenario for the bias in $\nu_{10}$ of galaxies to the right of the dotted line, since only those at the very edge of the volume experience the maximum effect of incompleteness. Since $\nu_{10}$ covers six orders of magnitude, we judge this to be a minor impact.

Another difference between the MASSIVE and ATLAS3D surveys is the availability of accurate distance estimates (discussed in detail in Ma et al. 2014 and Cappellari et al. 2011a respectively). For most of the MASSIVE galaxies, we use group distances from the catalogues of Crook et al. (2007). This effectively flattens the galaxies in each group to the same distance, and would generally result in a higher $\nu_{10}$. Although more accurate distances would result in more accurate values of $\nu_{10}$, we wish to make a fair comparison to ATLAS3D galaxies, so we do not use the accurate distances tabulated by the survey papers even when they are available. Instead, we assign distances from the HDC catalogue first (if available), then from the LDC catalogue, and as a last resort use the raw 2MRS redshift distance.

Figure 13 compares our recalculated $\nu_{10}$ to the values from Cappellari et al. (2011b). Overall the agreement is reasonable, considering the two competing influences of our changes to the calculation. First, we have a more strict $M_{K}$ cut on the parent sample, which will reduce $\nu_{10}$. We see this at low densities, with reductions in $\nu_{10}$ up to an order of magnitude. This is roughly in line with our comparison between the $M_{K}<-24.0$ and $M_{K}<-23.0$ cuts discussed above. Second, we have ignored accurate distance estimates for nearby galaxies in favor of a more uniform assignment of group distances. Flattening the groups to a single distance reshuffles the order of which neighbouring galaxies are closest, which may have a small impact on the total luminosity, but the major impact on $\nu_{10}$ comes from reduced $r_{10}$. Even if the 10 neighbours are the same galaxies, $r_{10}$ is reduced to a 2-dimensional $R_{10}$ if all neighbours are in the same group. It can be reduced further if galaxies that are nearly coincident on-sky, but are at opposite sides of the group along the line of sight, are counted as neighbours when they would not be otherwise. A moderate change in $r_{10}$ has an impact of $r^{3}$ on the volume used to calculate $\nu_{10}$, and in a very few cases $\nu_{10}$ increases by up to 2 orders of magnitude.

The agreement between our new $\nu_{10}$ and the original values is good, considering the effects described above. We also stress that enabling a fair comparison between MASSIVE and ATLAS3D galaxies is more important than increased accuracy of $\nu_{10}$ for individual galaxies.

Appendix B Bayesian error estimates for fraction of slow rotators

In Section 5.3 we compare the observed fraction of slow rotators $f_{\rm slow}$ as a function of environment to what we predict using $M_{K}$. There is limited statistical power in certain bins, where the number of MASSIVE and/or ATLAS3D galaxies is small. Thus we require a reasonable estimate of the error on $f_{\rm slow}$ so that we can make the comparison fairly and not overstate any differences.

For simplicity, we will ignore error bars on $\lambda_{e}$ and treat the classification of each individual galaxy as slow or fast as a 100% certain measurement with no errors. Although this is not true, the statistical errors due to sample size are our main concern. The fraction of slow rotators must be between 0 and 1, and many simple estimates of the error are unsatisfactory. (For example, a simple bootstrapping method would yield zero error for a subsample of 5 galaxies containing 0 slow rotators, even though there should be significant uncertainty due to the small sample size.) Fortunately, our problem is equivalent to a well known example in Bayesian statistics, the problem of flipping a biased coin $N$ times and estimating the true probability of getting heads or tails.

For some fraction of coin flips (or slow rotators) $x$, the prior and posterior distributions can be conveniently defined by a Beta distribution:

[TABLE]

with a mean of $\mu=\alpha/(\alpha+\beta)$. The quantity $n=\alpha+\beta$ is often interpreted as the sample size, and the variance is $\mu(1-\mu)/(n+1)$. The parameters of the posterior distribution, given a prior distribution and the measured numbers of fast and slow rotators, are $\alpha_{\rm post}=\alpha_{\rm prior}+N_{\rm slow}$, $\beta_{\rm post}=\beta_{\rm prior}+N_{\rm total}-N_{\rm slow}$.

We choose a prior distribution based on $f_{\rm slow}$ for each survey: $\mu_{\rm prior}=0.78$ for MASSIVE and $\mu_{\rm prior}=0.13$ for ATLAS3D, with $n_{\rm prior}=5$ for both. Then to obtain the error on $f_{\rm slow}$ in each specific bin of environment, we find the 68% confidence interval of the posterior distribution. These choices of prior are somewhat arbitrary (i.e. there is nothing special about $n_{\rm prior}=5$), but qualitatively give the behaviour we expect. We have a weak prior assumption that any subsample of galaxies will have the same $f_{\rm slow}$ as the overall sample, so the errors will be slightly asymmetric towards that overall fraction, and the size of the error depends properly on the size of the subsample. The slow fraction for ATLAS3D BGG galaxies in the bottom panel of Figure 10 is a good illustration of these properties.

Appendix C Comparing to ATLAS3D densities

Figure 14 compares the observed slow rotatator fraction $f_{\rm slow}$ to the a test sample constructed by randomizing the slow/fast assignment of galaxies within bins of $M_{K}$. (See Section 5.3 for details.) This is similar to the results shown in Figure 11, but uses the local densities $\nu_{10}$ and $\Sigma_{3}$ tabulated in Cappellari et al. (2011b) for the ATLAS3D sample. The results in the top panel of Figure 14 for the entire ATLAS3D sample, with $\nu_{10}$ calculated using the best available distance estimates, are slightly different from the results in the top right panel of Figure 11, calculated for galaxies with $M_{K}<-23.0$ using our simplified distances.

Both measures of local density find an increase in $f_{\rm slow}$ in the highest density bin, but the test samples (grey points in Figure 14) show an important difference. The slow rotator fraction as a function of $\Sigma_{3}$ is well matched using the randomized test sample, but if the case of $\nu_{10}$ the test sample underpredicts $f_{\rm slow}$ at the highest bin. This is very similar to the results for MASSIVE galaxies in the top right panel of Figure 11.