meeting 2025 07 24 n57 - JacobPilawa/TriaxSchwarzschild_wiki

Context

Following up on diagnostics/sanity checks/fixing a few of the plots from the last meeting.

Tracking down additional information on the galaxies/background information going into the paper

One item on my agenda was to find any potential rb measurements for the galaxies in our table:
- NGC 708 = 2.29 arcsec/0.76 kpc
Halo masses
- N57 and H15 use M200 (N57 I computed; H15 from Habas+18)
- NGC 4486 = 10^14.7 Msun from Massive VII
- NGC 4649 = 10^14.7 Msun from Massive VII
- NGC 4889 = 10^15.2 Msun from Massive VII
- All other halo masses are from Crook 2007 HDC projected mass estimator columns
Group HDC Counts
- Galaxy group counts are also from Crook 2007's HDC catalog; N57 does not appear so we use N=1 for this galaxy
- H15's group galaxy counts are from Agulli+17 (assuming r-K colors of 2.8 and using the M_K = -23.0 magnitude cut to be consistent with HDC catalog)
Currently (I was wrong in the last meeting), I am not "correcting" any of the black hole masses for cosmologicaly effects (e.g., using the redshifts). I'm not sure if we want to do this or simply leave the black hole masses as they are measured and reported in the original references. I am somewhat inclined to leave the masses as they are quoted since (a) the corrections will be very small, (b) we're not doing anything super quantitative with the black hole masses, and (c) avoid errors in doing this correction.

Verifying Radii

We wanted to ensure that the radii were being computed properly, so here are some diagnostics on that front.
For some additional context:
- We have a routine called R_from_ap_and_bin() which takes in the aperture and bin files as they appear in TriOS, and outputs either the bin center (x,y) positions, or the bin radii. I've first verified that these give equivalent answers. In my radial plots, the radius I am plotting is from the R_from_ap_and_bin() (x,y) centers, converted to radii, and taking the absolute value (since we have switched to folded radial plots instead of the unfolded versions).

Here are the numbers from R_from_ap_and_bin() for the radii, (x,y) centers, and then manually computing the radii from the (x,y) centers

#	(x,y)	r	(x,y)-->r
1	[2.1553 1.1367]	2.4366	2.4366
2	[1.5189 2.8107]	3.1948	3.1948
3	[-0.2053 0.1033]	0.2298	0.2298
4	[0.7184 0.4133]	0.8288	0.8288
5	[-0.1026 0.4133]	0.4259	0.4259
6	[1.4955 2.1848]	2.6476	2.6476
7	[ 1.2572 -2.0925]	2.4411	2.4411
8	[ 0.6842 -0.8956]	1.127	1.127
9	[-3.0789 -0.9743]	3.2294	3.2294
10	[-3.3575 2.7162]	4.3186	4.3186
11	[-0.4105 -0.1033]	0.4233	0.4233
12	[0.8895 1.3089]	1.5825	1.5825
13	[-0.5132 -1.0333]	1.1537	1.1537
14	[ 0.6158 -1.705 ]	1.8128	1.8128
15	[-2.1758 -1.4053]	2.5902	2.5902
16	[ 0.1368 -1.3778]	1.3846	1.3846
17	[ 1.7447 -0.7233]	1.8887	1.8887
18	[-2.3263 0.0689]	2.3273	2.3273
19	[0.4105 0.7233]	0.8317	0.8317
20	[-3.6827 0.4498]	3.71	3.71
21	[0.2053 0.1033]	0.2298	0.2298
22	[-3.6332 1.7567]	4.0356	4.0356
23	[-0.1026 -1.0333]	1.0384	1.0384
24	[1.3 1.1022]	1.7044	1.7044
25	[ 1.0947 -1.3089]	1.7064	1.7064
26	[-1.3342 1.55 ]	2.0451	2.0451
27	[0.4105 0.1033]	0.4233	0.4233
28	[1.0263 0.5167]	1.149	1.149
29	[ 0. -0.7233]	0.7233	0.7233
30	[ 1.5737 -0.3444]	1.6109	1.6109
31	[-3.0379 0.5373]	3.085	3.085
32	[-0.2053 0.7233]	0.7519	0.7519
33	[-2.2836 2.2217]	3.186	3.186
34	[-3.0789 -0.4133]	3.1066	3.1066
35	[-0.9032 1.8187]	2.0306	2.0306
36	[-0.8211 -2.6571]	2.7811	2.7811
37	[ 1.7789 -0.0689]	1.7803	1.7803
38	[-1.7789 1.5156]	2.337	2.337
39	[-2.0184 -1.8944]	2.7682	2.7682
40	[-2.3811 2.8107]	3.6836	3.6836
41	[-1.3 -1.1022]	1.7044	1.7044
42	[-0.1026 1.9633]	1.966	1.966
43	[-0.2053 -0.31 ]	0.3718	0.3718
44	[ 0.7184 -0.62 ]	0.949	0.949
45	[-2.1211 -0.1378]	2.1255	2.1255
46	[ 2.1553 -0.7233]	2.2734	2.2734
47	[ 0.7245 -2.711 ]	2.8061	2.8061
48	[-0.8895 1.1022]	1.4164	1.4164
49	[-0.8211 0.7233]	1.0942	1.0942
50	[0.3079 1.0333]	1.0782	1.0782
51	[2.3459 2.7162]	3.589	3.589
52	[2.3605 0.31 ]	2.3808	2.3808
53	[-1.7789 -0.5511]	1.8624	1.8624
54	[0.6158 2.8675]	2.9329	2.9329
55	[-2.6171 0.2067]	2.6253	2.6253
56	[ 0.9237 -1.0333]	1.386	1.386
57	[-0.6158 0.7233]	0.95	0.95
58	[-2.5145 0.62 ]	2.5898	2.5898
59	[0.6158 0.7233]	0.95	0.95
60	[-0.7389 -1.8187]	1.9631	1.9631
61	[ 0.9237 -1.86 ]	2.0767	2.0767
62	[ 0.7184 -0.2067]	0.7476	0.7476
63	[0.3849 2.2475]	2.2802	2.2802
64	[2.1406 1.7124]	2.7412	2.7412
65	[ 1.7447 -1.1367]	2.0823	2.0823
66	[3.0533 0.8525]	3.1701	3.1701
67	[-2.7095 0.9507]	2.8714	2.8714
68	[3.4895 1.55 ]	3.8182	3.8182
69	[-0.8397 2.5552]	2.6896	2.6896
70	[ 0.2053 -0.31 ]	0.3718	0.3718
71	[1.7961 1.0333]	2.0721	2.0721
72	[ 0.7184 -0.4133]	0.8288	0.8288
73	[-2.5658 -0.93 ]	2.7291	2.7291
74	[1.0263 0.1033]	1.0315	1.0315
75	[-0.7184 -0.62 ]	0.949	0.949
76	[ 1.3 -0.2756]	1.3289	1.3289
77	[-1.0947 -0.7578]	1.3314	1.3314
78	[-0.5132 1.0333]	1.1537	1.1537
79	[ 0.2053 -0.93 ]	0.9524	0.9524
80	[-0.9579 1.3778]	1.678	1.678
81	[-2.6428 -1.8342]	3.2169	3.2169
82	[1.0263 0.93 ]	1.385	1.385
83	[-1.0263 -0.1033]	1.0315	1.0315
84	[0.7184 1.0333]	1.2585	1.2585
85	[-0.4105 -0.7233]	0.8317	0.8317
86	[-0.1026 1.0333]	1.0384	1.0384
87	[-2.1553 -0.5167]	2.2163	2.2163
88	[1.95 0.31]	1.9745	1.9745
89	[ 1.0263 -0.31 ]	1.0721	1.0721
90	[-0.4105 0.7233]	0.8317	0.8317
91	[ 3.4895 -0.8956]	3.6026	3.6026
92	[-2.1553 1.1367]	2.4366	2.4366
93	[ 3.4602 -0.2214]	3.4672	3.4672
94	[ 2.6684 -0.1033]	2.6704	2.6704
95	[1.3342 1.7567]	2.2059	2.2059
96	[0.78 1.7773]	1.941	1.941
97	[0.7184 0. ]	0.7184	0.7184
98	[-1.1289 -1.86 ]	2.1758	2.1758
99	[ 1.3684 -0.9644]	1.6741	1.6741
100	[-1.3342 0.4133]	1.3968	1.3968
101	[0.2737 1.3089]	1.3372	1.3372
102	[-3.6827 -0.6565]	3.7407	3.7407
103	[-0.1368 1.3089]	1.316	1.316
104	[-1.3684 1.1711]	1.8011	1.8011
105	[-1.6421 -0.31 ]	1.6711	1.6711
106	[-0.325 2.9278]	2.9458	2.9458
107	[-2.6684 -1.3433]	2.9875	2.9875
108	[-0.2053 -0.7233]	0.7519	0.7519
109	[1.3 0.6889]	1.4712	1.4712
110	[ 0. -0.31]	0.31	0.31
111	[-1.7105 2.2389]	2.8175	2.8175
112	[-0.6158 0.31 ]	0.6894	0.6894
113	[ 2.9103 -2.8048]	4.0419	4.0419
114	[-1.6421 1.8083]	2.4427	2.4427
115	[-2.1211 0.3444]	2.1488	2.1488
116	[1.5395 1.3433]	2.0432	2.0432
117	[ 2.6684 -0.5167]	2.718	2.718
118	[-1.3342 0. ]	1.3342	1.3342
119	[ 0.2822 -2.1183]	2.1371	2.1371
120	[-2.6684 1.3433]	2.9875	2.9875
121	[-1.3342 -0.4133]	1.3968	1.3968
122	[1.3 0.2756]	1.3289	1.3289
123	[-0.7184 0. ]	0.7184	0.7184
124	[0.8211 0.7233]	1.0942	1.0942
125	[-3.3581 -2.5131]	4.1943	4.1943
126	[ 0.7526 -1.3778]	1.5699	1.5699
127	[ 2.0526 -0.3617]	2.0843	2.0843
128	[-1.7789 -0.8956]	1.9917	1.9917
129	[ 0.3079 -0.62 ]	0.6922	0.6922
130	[3.6407 0.5547]	3.6827	3.6827
131	[-3.2842 1.0953]	3.4621	3.4621
132	[-1.7105 0.5511]	1.7971	1.7971
133	[2.6311 1.9164]	3.255	3.255
134	[ 2.2836 -1.6275]	2.8042	2.8042
135	[2.5658 0.7233]	2.6658	2.6658
136	[ 2.8737 -1.4467]	3.2173	3.2173
137	[-1.7105 1.1711]	2.073	2.073
138	[-3.0789 0.1033]	3.0807	3.0807
139	[ 3.6415 -2.0054]	4.1572	4.1572
140	[0. 0.1033]	0.1033	0.1033
141	[2.0937 0.7027]	2.2085	2.2085
142	[-1.7105 -1.1711]	2.073	2.073
143	[-2.5316 -0.5511]	2.5909	2.5909
144	[0.2053 0.7233]	0.7519	0.7519
145	[-0.9237 -0.4133]	1.0119	1.0119
146	[ 1.0263 -0.7233]	1.2556	1.2556
147	[-1.3684 -0.6889]	1.532	1.532
148	[1.3342 0. ]	1.3342	1.3342
149	[-0.1026 -2.79 ]	2.7919	2.7919
150	[ 0.4105 -0.31 ]	0.5144	0.5144
151	[0.9237 2.2733]	2.4538	2.4538
152	[ 2.6684 -0.93 ]	2.8258	2.8258
153	[-1.7447 -1.55 ]	2.3338	2.3338
154	[0.7184 0.2067]	0.7476	0.7476
155	[-0.9237 0.2067]	0.9465	0.9465
156	[ 2.799 -2.0855]	3.4905	3.4905
157	[0.0684 1.5844]	1.5859	1.5859
158	[-0.9237 0.4133]	1.0119	1.0119
159	[ 1.3 -0.6889]	1.4712	1.4712
160	[-1.3684 -1.3778]	1.9419	1.9419
161	[-0.7184 -0.8267]	1.0952	1.0952
162	[-2.5658 -2.8704]	3.85	3.85
163	[-2.1211 -2.3652]	3.1769	3.1769
164	[-2.1992 1.6533]	2.7514	2.7514
165	[-1.3342 -0.2067]	1.3501	1.3501
166	[0.3421 1.7911]	1.8235	1.8235
167	[-0.5132 -0.4133]	0.6589	0.6589
168	[0.5474 1.3778]	1.4825	1.4825
169	[-1.8474 -0.1033]	1.8503	1.8503
170	[0.3079 0.4133]	0.5154	0.5154
171	[ 1.7242 -1.612 ]	2.3604	2.3604
172	[ 0.4105 -0.93 ]	1.0166	1.0166
173	[-1.0263 0.7233]	1.2556	1.2556
174	[-0.4105 0.31 ]	0.5144	0.5144
175	[2.8737 0.31 ]	2.8904	2.8904
176	[-1.5053 -2.7211]	3.1097	3.1097
177	[ 1.6216 -2.7797]	3.2181	3.2181
178	[-0.9579 -1.3778]	1.678	1.678
179	[-2.6684 -0.2583]	2.6809	2.6809
180	[1.5737 0.3444]	1.6109	1.6109
181	[-0.1368 -1.3089]	1.316	1.316
182	[1.7789 1.7567]	2.5001	2.5001
183	[-0.5337 1.9013]	1.9748	1.9748
184	[-1.3342 0.62 ]	1.4712	1.4712
185	[-0.3421 1.5844]	1.621	1.621
186	[-1.8474 0.31 ]	1.8732	1.8732
187	[1.5737 0.7578]	1.7466	1.7466
188	[-1.5189 -1.9013]	2.4336	2.4336
189	[ 1.3547 -1.488 ]	2.0123	2.0123
190	[-0.3519 -2.0667]	2.0964	2.0964
191	[-2.8326 1.86 ]	3.3887	3.3887
192	[-1.3342 0.2067]	1.3501	1.3501
193	[-1.2316 2.0667]	2.4058	2.4058
194	[-2.1553 0.7233]	2.2734	2.2734
195	[1.0947 1.5156]	1.8696	1.8696
196	[-0.3421 -1.5844]	1.621	1.621
197	[ 2.2168 -1.1573]	2.5008	2.5008
198	[-1.6421 0.1033]	1.6454	1.6454
199	[-0.5474 -1.3089]	1.4187	1.4187
200	[ 0.1026 -1.7567]	1.7597	1.7597
201	[ 1.9911 -2.1287]	2.9147	2.9147
202	[2.6428 1.2658]	2.9303	2.9303
203	[-1.3 0.8956]	1.5786	1.5786
204	[3.4024 2.5677]	4.2625	4.2625
205	[ 0.4789 -1.3089]	1.3938	1.3938
206	[-1.7789 0.8956]	1.9917	1.9917
207	[ 2.1895 -0.0689]	2.1906	2.1906
208	[-3.5361 -1.6815]	3.9156	3.9156
209	[-0.5474 1.3089]	1.4187	1.4187
210	[-0.7184 -0.2067]	0.7476	0.7476
211	[-0.2053 2.3767]	2.3855	2.3855
212	[-0.8895 -1.1022]	1.4164	1.4164
213	[-1.4882 2.8288]	3.1963	3.1963
214	[-2.1553 -0.93 ]	2.3474	2.3474
215	[0. 0.7233]	0.7233	0.7233

The (x,y) centers are plotted here, with the origin as the red star:

(x,y) centers with origin

I've also pulled out a few of the specific radii from the table above that are duplicated, and drew concentric circles at these radii. These lines intersect multiple points (or in some cases, are just very, very slightly off from the centers but virtually indistinguishable by eye):
- Note that the specific radii I've plotted are: [0.2298, 0.3718, 0.4233, 0.5144, 0.7184, 0.7233, 0.7476, 0.7519, 0.8288, 0.8317, 0.949 , 0.95 , 1.0119, 1.0315, 1.0384, 1.0942, 1.1537, 1.2556, 1.316 , 1.3289, 1.3342, 1.3501, 1.3968, 1.4164, 1.4187, 1.4712, 1.6109, 1.621 , 1.678 , 1.7044, 1.9917, 2.073 , 2.2734, 2.4366, 2.9875]

(x,y) centers with circles

We can also spot check a few of the bins to see why they have identical radii.
- I think that these plots are a bit more clear -- I essentially went through and found which bins have identical radii (out to 4 decimal places), and plotted those with identical radii with a black outline.
- In these plots/in the table of numbers in the dropdown above, you can see that the symmetry comes from the bins having the same (x,y) centers up to a factor of -1, meaning that there is some symmetric voronoi strucutre sprinkled in with the asymmetric binning scheme.
There is no difference in the columns here, I just needed to create a 7x5 table for the repeated radii. The numbers are simply column indices:

1	2	3	4	5

Kinematic Maps

We also wanted to see both the PAS and raw set of maps, which I've plotted here:

Raw	PAS	Model

Radial Plot Uniquely Colored

Unique Colors Radial Plot

Updated Spectra Plot

I also added the mask over the first 9 pixels which had been absent from the previous iteration of the spectra plot. Here's an updated version:

Spectra

V(R, Theta)

Still running these and uploading the results soon!
We wanted to ensure that the V(R, Theta) results are robust to changes of bin size/number of bins included in each fit, so I have a few different set of results here exploring that sensitivity.

Changing number of bins

	3 bins	5 bins	7 bins	9 bins
Raw Data
Sym Data
Model

3 bins case: Tables of Fit Information

Unsym data:

Bin Extent	V1 (km/s)	Phase (PA + V)	V0 (km/s)
R = ( 0.0" - 1.6")	6.8 +/- 0.9	113.9 +/- 7.1	1.3 +/- 0.6
R = ( 1.4" - 3.3")	7.1 +/- 0.7	97.6 +/- 5.6	-0.8 +/- 0.5
R = ( 2.6" - 5.0")	4.6 +/- 1.1	31.1 +/- 19.5	-3.6 +/- 0.9

Sym data:

Bin Extent	V1 (km/s)	Phase (PA + V)	V0 (km/s)
R = ( 0.0" - 1.6")	5.5 +/- 0.9	108.1 +/- 8.9	0.0 +/- 0.6
R = ( 1.4" - 3.3")	6.6 +/- 0.7	93.3 +/- 6.1	0.0 +/- 0.5
R = ( 2.6" - 5.0")	4.8 +/- 1.1	61.7 +/- 18.3	-0.4 +/- 0.9

Model:

Bin Extent	V1 (km/s)	Phase (PA + V)	V0 (km/s)
R = ( 0.0" - 1.6")	6.5 +/- 0.9	112.0 +/- 7.4	-0.1 +/- 0.6
R = ( 1.4" - 3.3")	7.1 +/- 0.7	96.5 +/- 5.6	0.1 +/- 0.5
R = ( 2.6" - 5.0")	4.6 +/- 1.0	45.1 +/- 19.7	-0.4 +/- 0.9

5 bins case: Tables of Fit Information

Unsym data:

Bin Extent	V1 (km/s)	Phase (PA + V)	V0 (km/s)
R = ( 0.0" - 0.9")	6.0 +/- 1.4	108.4 +/- 13.3	4.6 +/- 1.0
R = ( 0.8" - 1.9")	6.1 +/- 0.9	120.0 +/- 8.2	-0.7 +/- 0.6
R = ( 1.7" - 3.1")	8.3 +/- 0.9	94.0 +/- 6.3	-0.4 +/- 0.6
R = ( 2.4" - 3.7")	4.9 +/- 1.2	74.7 +/- 15.0	-3.3 +/- 0.9
R = ( 3.1" - 5.0")	4.2 +/- 1.4	43.8 +/- 31.8	-2.4 +/- 1.2

Sym data:

Bin Extent	V1 (km/s)	Phase (PA + V)	V0 (km/s)
R = ( 0.0" - 0.9")	5.6 +/- 1.3	95.4 +/- 14.3	-0.0 +/- 1.0
R = ( 0.8" - 1.9")	5.6 +/- 0.9	115.2 +/- 8.9	0.0 +/- 0.6
R = ( 1.7" - 3.1")	7.1 +/- 0.9	89.4 +/- 7.5	0.1 +/- 0.6
R = ( 2.4" - 3.7")	5.8 +/- 1.2	79.8 +/- 12.6	-0.4 +/- 0.9
R = ( 3.1" - 5.0")	4.0 +/- 1.5	52.6 +/- 33.3	0.0 +/- 1.2

Model:

Bin Extent	V1 (km/s)	Phase (PA + V)	V0 (km/s)
R = ( 0.0" - 0.9")	5.9 +/- 1.3	102.8 +/- 13.6	0.1 +/- 1.0
R = ( 0.8" - 1.9")	6.1 +/- 0.9	119.6 +/- 8.2	-0.0 +/- 0.6
R = ( 1.7" - 3.1")	8.3 +/- 0.9	91.6 +/- 6.4	0.0 +/- 0.6
R = ( 2.4" - 3.7")	5.5 +/- 1.2	80.9 +/- 13.3	-0.5 +/- 0.9
R = ( 3.1" - 5.0")	4.1 +/- 1.4	46.6 +/- 32.3	0.1 +/- 1.1

7 bins case: Tables of Fit Information

Unsym data:

Bin Extent	V1 (km/s)	Phase (PA + V)	V0 (km/s)
R = ( 0.0" - 0.6")	4.8 +/- 2.0	111.0 +/- 24.7	3.4 +/- 1.5
R = ( 0.6" - 1.3")	6.9 +/- 1.2	100.6 +/- 9.8	2.1 +/- 0.8
R = ( 1.2" - 2.0")	6.0 +/- 1.1	128.8 +/- 9.2	-1.3 +/- 0.7
R = ( 1.8" - 2.8")	8.3 +/- 1.0	88.7 +/- 7.2	-0.9 +/- 0.7
R = ( 2.4" - 3.6")	7.1 +/- 1.2	95.8 +/- 9.6	-2.2 +/- 0.8
R = ( 2.8" - 4.4")	5.4 +/- 1.8	6.0 +/- 21.3	-2.5 +/- 1.2
R = ( 3.4" - 5.0")	6.5 +/- 3.5	81.9 +/- 26.6	-0.3 +/- 2.2

Sym data:

Bin Extent	V1 (km/s)	Phase (PA + V)	V0 (km/s)
R = ( 0.0" - 0.6")	4.4 +/- 2.0	96.5 +/- 27.1	-0.2 +/- 1.5
R = ( 0.6" - 1.3")	6.1 +/- 1.2	98.9 +/- 11.0	-0.0 +/- 0.8
R = ( 1.2" - 2.0")	5.7 +/- 1.1	117.0 +/- 9.9	-0.2 +/- 0.7
R = ( 1.8" - 2.8")	7.2 +/- 1.0	86.5 +/- 8.3	0.1 +/- 0.7
R = ( 2.4" - 3.6")	6.5 +/- 1.2	90.2 +/- 10.8	-0.6 +/- 0.8
R = ( 2.8" - 4.4")	3.9 +/- 1.4	45.2 +/- 34.1	0.4 +/- 1.2
R = ( 3.4" - 5.0")	4.5 +/- 2.7	46.6 +/- 48.6	0.5 +/- 2.2

Model:

Bin Extent	V1 (km/s)	Phase (PA + V)	V0 (km/s)
R = ( 0.0" - 0.6")	4.3 +/- 2.0	93.9 +/- 27.9	-0.3 +/- 1.5
R = ( 0.6" - 1.3")	7.1 +/- 1.2	100.4 +/- 9.5	-0.1 +/- 0.8
R = ( 1.2" - 2.0")	5.9 +/- 1.1	124.5 +/- 9.4	-0.1 +/- 0.7
R = ( 1.8" - 2.8")	8.0 +/- 1.1	90.2 +/- 7.5	-0.1 +/- 0.7
R = ( 2.4" - 3.6")	7.8 +/- 1.2	93.6 +/- 8.8	-0.5 +/- 0.8
R = ( 2.8" - 4.4")	4.2 +/- 1.7	10.9 +/- 28.4	0.5 +/- 1.2
R = ( 3.4" - 5.0")	4.9 +/- 2.8	60.5 +/- 39.3	0.9 +/- 2.0

9 bins case: Tables of Fit Information

Unsym data:

Bin Extent	V1 (km/s)	Phase (PA + V)	V0 (km/s)
R = ( 0.0" - 0.5")	5.5 +/- 2.2	112.4 +/- 25.9	2.7 +/- 1.6
R = ( 0.5" - 1.1")	7.8 +/- 1.5	105.2 +/- 10.9	4.4 +/- 1.1
R = ( 0.9" - 1.6")	6.7 +/- 1.2	121.1 +/- 9.6	-0.9 +/- 0.8
R = ( 1.4" - 2.2")	6.1 +/- 1.2	102.0 +/- 11.4	-2.0 +/- 0.9
R = ( 1.8" - 2.7")	7.2 +/- 1.2	89.5 +/- 9.2	0.1 +/- 0.8
R = ( 2.3" - 3.3")	8.6 +/- 1.4	105.7 +/- 8.2	-0.6 +/- 0.9
R = ( 2.6" - 3.9")	6.5 +/- 1.5	16.3 +/- 17.1	-4.4 +/- 1.2
R = ( 3.1" - 4.5")	4.0 +/- 3.4	114.3 +/- 37.3	-3.4 +/- 1.7
R = ( 3.4" - 5.0")	5.5 +/- 4.1	81.1 +/- 31.6	-0.0 +/- 2.3

Sym data:

Bin Extent	V1 (km/s)	Phase (PA + V)	V0 (km/s)
R = ( 0.0" - 0.5")	4.0 +/- 2.2	100.0 +/- 35.6	-0.4 +/- 1.6
R = ( 0.5" - 1.1")	6.5 +/- 1.5	93.7 +/- 13.0	0.1 +/- 1.1
R = ( 0.9" - 1.6")	5.5 +/- 1.2	119.0 +/- 11.6	0.0 +/- 0.8
R = ( 1.4" - 2.2")	5.7 +/- 1.2	104.4 +/- 12.1	-0.2 +/- 0.9
R = ( 1.8" - 2.7")	7.5 +/- 1.2	86.8 +/- 8.8	-0.0 +/- 0.8
R = ( 2.3" - 3.3")	6.6 +/- 1.4	93.0 +/- 11.3	0.2 +/- 0.9
R = ( 2.6" - 3.9")	5.7 +/- 1.5	63.3 +/- 19.6	-0.3 +/- 1.2
R = ( 3.1" - 4.5")	4.2 +/- 2.5	86.4 +/- 47.8	-0.8 +/- 1.7
R = ( 3.4" - 5.0")	4.1 +/- 3.0	41.7 +/- 58.5	0.7 +/- 2.3

Model:

Bin Extent	V1 (km/s)	Phase (PA + V)	V0 (km/s)
R = ( 0.0" - 0.5")	3.2 +/- 2.2	89.0 +/- 44.5	-0.9 +/- 1.7
R = ( 0.5" - 1.1")	8.0 +/- 1.5	103.6 +/- 10.7	0.2 +/- 1.1
R = ( 0.9" - 1.6")	6.7 +/- 1.2	119.9 +/- 9.5	-0.1 +/- 0.8
R = ( 1.4" - 2.2")	6.0 +/- 1.2	101.7 +/- 11.7	-0.5 +/- 0.9
R = ( 1.8" - 2.7")	7.4 +/- 1.2	89.8 +/- 8.9	0.4 +/- 0.8
R = ( 2.3" - 3.3")	8.2 +/- 1.4	101.7 +/- 8.7	0.2 +/- 0.9
R = ( 2.6" - 3.9")	6.1 +/- 1.4	33.3 +/- 18.9	-0.5 +/- 1.2
R = ( 3.1" - 4.5")	5.0 +/- 3.4	117.0 +/- 27.6	-0.6 +/- 1.6
R = ( 3.4" - 5.0")	5.4 +/- 3.5	63.0 +/- 36.7	0.7 +/- 2.2

Equally distributing points

I've also tried a few approaches in which I distribute the points equally into the shells. I've plotted the results for 10/20/25 below:

	10 points	20 points	25 points
Raw Data
Sym Data
Model

Manually distributing points

And lastly, here's a few attempts where I am distributing the points sort of manually to get decent coverage in each "shell." I think that one of these might strike the right balance of point coverage for each shell, but it's essentially a trade off for whether or not we want the "corners" exlucded from either the outer most or second outermost shell. Regardless, as long as the bin choices are "reasonable," it seems like we're getting somewhat consistent results/trends: