Prospects for observing dynamically formed Binary Black Holes in the Local Universe with Gravitational Waves¶
Dongming Jin¶
- Research Supervisor
- Dr. Matthew Benacquista
- Committee Members
- Dr. Zdzislaw Musielak (co-chair)
- Dr. Manfred Cuntz
- Dr. Joseph Romano
- Dr. Sangwook Park
- Dr. Teviet Creighton
Graduate Advisor
- Dr. Qiming Zhang
April 27th, 2018
Overview¶
Populating GCs in the local universe¶
LoGal: galaxy within 30 Mpc
- 7,909 with GCs
- 1,037 has no measured luminosity
- 8,946 total
GClib: 662,772 GCs over 7,909 Galaxy
- manipulated LoGal.Ngc with different GC age
- assign random GC simulation
Simulating GCs¶
GClib models:
- $2\times5\times3\times3\times3 \times 10 = 324 \times 10 = 3,240$ simulations
- Node-hour: $150,000 + 90,000 = 240,000$
- tmp file: over 100 TB
GClib model diversity:
- 1,739 out of 662,772 are duplicated
- only half of the 3,240 GC simulations have one duplicated entry
Relativistic BBHs¶
BBHlib: extracted BBHs from 3,240 GC simulations
- 87,365 ejected BBHs
- 11,095 ejected BHBs
BBHdata: 17,883,760 BBHs with host galaxy info. + host GC info. + BBH info.
Prospects for Detection¶
- 86,939 BBH mergers + 17,883,760 BBH inspirals
Merger event rate (6.1.3)
- $\eta = N \frac{8 k_0}{3} f_\text{min}^{8/3} \left(\frac{30 \text{ Mpc}}{1 \text{ Gpc}} \right)^3 \left( \frac{13.5 \text{ Gyr}}{1 \text{ yr}} \right) \simeq 587 \text{ Gpc}^{-3} \text{ yr}^{-1}$
Localizationable BBHs: 14
PGC Dist M1 M2 Seperation Ecc
1628 0.787 3.158000 10.250000 0.807000 0.000000
1628 0.787 12.728000 19.122000 0.157281 0.852028
616 2.188 3.158000 10.250000 0.807000 0.000000
1628 0.787 3.158000 10.250000 0.807000 0.000000
1628 0.787 3.158000 10.250000 0.807000 0.000000
1628 0.787 3.158000 10.250000 0.807000 0.000000
1628 0.787 20.632000 137.730000 0.178258 0.477929
1628 0.787 21.424999 25.615999 0.165200 0.750944
1628 0.787 28.433001 26.503000 0.175125 0.813286
1628 0.787 3.158000 10.250000 0.807000 0.000000
131228 0.762 3.158000 10.250000 0.807000 0.000000
1628 0.787 3.158000 10.250000 0.807000 0.000000
1628 0.787 5.319600 10.276000 0.974700 0.000000
1627 0.813 5.319600 10.276000 0.974700 0.000000Introduction (chap 1)¶
Motivation¶
Objective¶
- GW from BBH formed in GCs -> Astrometry
- High performance computing: numerical simulations on superclusters
- Long baseline/timeline observation: time series analysis + multi-messanger
- Petascale data: systematic surveys, global multi-discipline collaborations
Background¶
Populating GCs in the local universe (chap 2)¶
GWGC for galaxy abundance (sec 2.1)¶
Harris catalog for GCpG model (sec 2.2)¶
GCpG models (sec 2.2)¶
Modeling GC populations (sec 2.3)¶
Populating GCs in the local universe (chap 2)¶
GWGC for galaxy abundance (sec 2.1)¶
Data prepare¶
- [=] load GWGC data: wrong data type
objectdue to~symbol => impute~as NaN and parse data type to float [=] check duplicates: explain pandas table, duplicated galaxies with Dist > 30 Mpc => drop
extract local galaxies within 30 Mpc and 3D view
spatial distribution and edge galaxies
check interesting features
Description
1- 7 I7 --- PGC [2,4715229]? Identifier from HYPERLEDA
(empty for globular clusters)
9- 36 A28 --- Name Common name of galaxy or globular
38- 46 F9.5 h RAhour Right ascension (J2000, decimal hours)
48- 55 F8.4 deg DEdeg Declination (J2000)
57- 60 F4.1 --- TT [-9,10]? Morphological type code (1)
62- 66 F5.2 mag Bmag ? Apparent blue magnitude
68- 74 F7.3 arcmin a ? Major diameter (arcmins)
76- 82 F7.3 arcmin e_a ? Error in major diameter (arcmins)
84- 90 F7.3 arcmin b ? Minor diameter (arcmins)
92- 98 F7.3 arcmin e_b ? Error in minor diameter (arcmins)
100-104 F5.3 --- b/a [0,1]? Ratio of minor to major diameters
106-110 F5.3 --- e_b/a ? Error on ratio of minor to major diameters
112-116 F5.1 deg PA [0,180]? Position angle of galaxy
(degrees from north through east)
118-123 F6.2 mag BMAG ? Absolute blue magnitude
125-131 F7.2 Mpc Dist ? Distance (Mpc)
133-138 F6.2 Mpc e_Dist ? error on Distance (Mpc)
140-143 F4.2 mag e_Bmag ? error on Apparent blue magnitude
145-148 F4.2 mag e_BMAG ? error on Absolute blue magnitudeIn [5]:
# 1. load GWGC data
GWGC=pd.read_csv(path_GWGC, delim_whitespace=True)
for key, n in Counter(GWGC.dtypes).most_common():
print("'%s' (count %d): \n - %s" % (key, n, '\n - '.join(x for x in GWGC.columns[GWGC.dtypes==key])))
In [6]:
# 1-. convert to numeric, impute '~' as NaN
for key in GWGC:
if key != 'Name':
GWGC[key]=pd.to_numeric(GWGC[key], errors='coerce')
# 2. check duplicates, duplicated galaxies with Dist > 30 Mpc
display(GWGC[GWGC.Name.duplicated(keep=False)] \
.loc[: ,['Name','Dist','err_Dist','RA','Dec','Type','App_Mag_B']])
Populating GCs in the local universe (chap 2)¶
GWGC for galaxy abundance (sec 2.1)¶
Data prepare¶
- [x] load GWGC data
[x] check duplicates
[=] extract local galaxies within 30 Mpc and 3D view: 8,946 galaxies out of 183,327, present in 3D plot
spatial distribution and edge galaxies: present in distplot
check interesting features
In [8]:
# 3. extract local galaxies within 30 Mpc and 3D view [slow]
GWGC['Mcat'] = 'Unknown'
GWGC['iMType'] = 0
GWGC.Mcat[GWGC.Type.between(-4,-3)] = 'E/S0'
GWGC.iMType[GWGC.Type.between(-4,-3)] = 6
GWGC.Mcat[GWGC.Type.between(0,11)] = 'S/Irr'
GWGC.iMType[GWGC.Type.between(0,11)] = 1
GWGC.Mcat[GWGC.Type.between(-6,-4)] = 'E'
GWGC.iMType[GWGC.Type.between(-6,-4)] = 2
GWGC.Mcat[GWGC.Type.between(-3,0)] = 'S0'
GWGC.iMType[GWGC.Type.between(-3,0)] = 3
LoGal = GWGC[GWGC.Dist<30]
if not flag_skip:
fig = figure(figsize=sfig_size)
ax = fig.add_subplot(111, projection='3d')
ax.scatter3D(GWGC.Dist*np.cos((GWGC.RA)/12*pi)*np.sin((90-GWGC.Dec)/180*pi),
GWGC.Dist*np.sin((GWGC.RA)/12*pi)*np.sin((90-GWGC.Dec)/180*pi),
GWGC.Dist*np.cos((90-GWGC.Dec)/180*pi),
alpha=0.1,s=2,edgecolors=None) # all GWGCs
ax.scatter3D(LoGal.Dist*np.cos((LoGal.RA)/12*pi)*np.sin((90-LoGal.Dec)/180*pi),
LoGal.Dist*np.sin((LoGal.RA)/12*pi)*np.sin((90-LoGal.Dec)/180*pi),
LoGal.Dist*np.cos((90-LoGal.Dec)/180*pi),alpha=0.1,c='g',s=2,edgecolors=None) # all LoGals
ax.view_init(20, 90)
r = [-100, 100]
for s, e in combinations(np.array(list(product(r, r, r))), 2):
if np.sum(np.abs(s-e)) == r[1]-r[0]:
ax.plot3D(*zip(s, e), color="k") # box to show orientation
ax.set_axis_off()
ax.set_xlim(left=-100,right=100)
ax.set_ylim(bottom=-100,top=100)
ax.set_zlim(-100,100)
ax.set_title('%d galaxies out of the %d galaxies are located in the Local Universe (within 30 Mpc)'
% (len(LoGal), len(GWGC)))
if flag_svfg:
fig.savefig(path.join(path_fig, 'GWGC3D.jpeg'),**params_svfg)
else:
ax = figure(figsize=sfig_size).add_subplot(111)
ax.imshow(imread(path.join(path_fig, 'GWGC3D.jpeg')))
ax.grid(False)
Populating GCs in the local universe (chap 2)¶
GWGC for galaxy abundance (sec 2.1)¶
Data prepare¶
- [x] load GWGC data
[x] check duplicates
[x] extract local galaxies within 30 Mpc and 3D view
[=] spatial distribution and edge galaxies: present in distplot
- D < 30 Mpc but D + $\Delta$D > 30 Mpc: 1,856 orange
- D > 30 Mpc but D - $\Delta$D < 30 Mpc: 3,015 blue
be cautious with bin in histgrams- galaxy morphological type, numerical Hubble stage, not well explained in paper or references therein: => a future topic
[-4, -3]: E/S0 => GCpG.iMType==6[0, 11]: S/Irr => GCpG.iMType==1[-6, -4): E => GCpG.iMType==2(-3, 0): S0 => GCpG.iMType==3NaN: Unknown, 0
- check interesting features
In [9]:
# 4. edge galaxies with morphology type, be cautious with bins in histograms
fig, ax = subplots(figsize=fig_size)
sns.distplot(GWGC.Dist,color='g',ax=ax) # NO. of galaxies at different Dist
ax.set_xlim([0, 180])
if 'GWGC_DT' not in locals():
GWGC_DT = GWGC.loc[:,['Dist','err_Dist','Mcat']].dropna()
GWGC_DT['Dcat'] = 0
GWGC_DT.Dcat[(GWGC_DT.Dist>30) & (GWGC_DT.Dist-GWGC_DT.err_Dist<30)] = -1
GWGC_DT.Dcat[(GWGC_DT.Dist<30) & (GWGC_DT.Dist+GWGC_DT.err_Dist>30)] = 1
sub_axes = axes([0.16, 0.56, 0.25, 0.25])
sub_axes.set_yticklabels([])
sub_axes.set_xlim([24, 38])
sub_axes2=sub_axes.twinx()
sub_axes2.set_yticklabels([])
sub_axes.grid(None)
sub_axes2.grid(None)
sns.distplot(GWGC_DT.Dist[GWGC_DT.Dcat==-1],kde=False,ax=sub_axes)
sns.distplot(GWGC_DT.Dist[GWGC_DT.Dcat==1],kde=False,ax=sub_axes)
sns.distplot(GWGC_DT.Dist[GWGC_DT.Dcat!=0],hist=False,ax=sub_axes2,color='y')
sub_axes = axes([0.6, 0.25, 0.25, 0.25])
sub_axes.set_xlabel('Morphology Type')
sub_axes.set_yticklabels([])
sub_axes.hist([GWGC_DT.Mcat[(GWGC_DT.Dcat==-1) & (GWGC_DT.Mcat!='Unknown')].values,
GWGC_DT.Mcat[(GWGC_DT.Dcat==1) & (GWGC_DT.Mcat!='Unknown')].values],
histtype='bar')
if flag_svfg:
savefig(path.join(path_fig,'Distance.jpeg'),**params_svfg)
Populating GCs in the local universe (chap 2)¶
GWGC for galaxy abundance (sec 2.1)¶
Data prepare¶
- [x] load GWGC data
[x] check duplicates
[x] extract local galaxies within 30 Mpc and 3D view
[x] spatial distribution and edge galaxies
[=] check interesting features: missing data
- statistics: present in pandas table
- missing data visualization: present in msnoplot
In [11]:
# 5-A. statistics
display(LoGal.loc[:,['Name','RA','Dec','Dist','Type','Abs_Mag_B','Abs_Mag_I']] \
.describe(include='all').loc[['count','unique','min','max','mean','std'],:])
In [12]:
# 5-B. visualization
img = msno.matrix(LoGal.loc[:, ['Name','RA','Dec','Dist','Type','Abs_Mag_B','Abs_Mag_I']], inline=False)
if flag_svfg:
img.savefig(path.join(path_fig, 'GWGC_NaN.jpeg'), **params_svfg)
Populating GCs in the local universe (chap 2)¶
Harris catalog for GCpG model (sec 2.2.1)¶
Data prepare¶
- load GCpG data
- check Morphology Type
check duplicates and NaNs
check interested features
Data explore¶
Description
0 1- 14 A14 --- Name Galaxy identifier
16- 27 A12 --- OName Other galaxy identifier
29- 38 F10.7 h RAhour ? Right Ascension in decimal hours (J2000)
40- 50 F11.7 deg DEdeg ? Declination in decimal degrees (J2000)
4 52- 56 A5 --- MType Morphology type
5 58- 63 F6.2 Mpc Dist ? NED distance; see Section 2
65- 69 F5.2 Mpc e_Dist ? Uncertainty in Dist
71- 83 A13 --- n_Dist Method used to determine Dist
85- 89 F5.3 mag Av NED foreground V band extinction
9 91- 97 F7.3 mag VMag [-25/-11]? Absolute visual magnitude (1)
99-104 F6.3 mag e_VMag [0.2/1.5]? Uncertainty in VMag
11 106-115 F10.3 mag KMag [-28/-14]? Absolute K band magnitude (2)
117-125 F9.3 mag e_KMag [0.1/0.3]? Uncertainty in KMag
127 A1 --- l_Ngc [>] Limit flag on Ngc
14 128-134 F7.1 --- Ngc [0/32500] Number of Globular Clusters (NGC)
136-142 F7.1 --- e_Ngc [0/20000] Uncertainty in Ngc
144-152 A9 --- r_Ngc Reference(s) for Ngc (see refs.dat file)
17 154-159 F6.2 km/s sigma [10/414]? Stellar velocity dispersion (3)
161-165 F5.2 km/s e_sigma ? Uncertainty in sigma
19 167-171 F5.2 kpc Reff [0.1/55]? Effective radius (4)
173-177 F5.2 kpc e_Reff ? Uncertainty in Reff
21 179-184 F6.3 [Msun] logMd [7.6/12.8]? Log of dynamical mass
186-191 F6.3 [Msun] e_logMd ? Uncertainty in logMd
23 193-197 F5.2 [Msun] logMGC ? Log of total globular cluster mass (5)
199-203 F5.2 [Msun] e_logMGC ? Uncertainty in logMGC
25 205-209 F5.2 [Msun] logMB ? Log of central supermassive black hole mass(6)
211-215 F5.2 [Msun] E_logMB ? Upper uncertainty in logMB
217-221 F5.2 [Msun] e_logMB ? Lower uncertainty in logMBPopulating GCs in the local universe (chap 2)¶
Harris catalog for GCpG model (sec 2.2.1)¶
Data prepare¶
- [=] load GCpG data
- [=] check Morphology Type:
MType: str=> to numeric, & parse in Harris manner, present in barplot [=] check duplicates and NaNs: explain pandas table, 418 galaxy enteries
- 0: Nan in
[VMag, KMag, Dist], MilkyWay => drop - 356: Nan in
[VMag, KMag], A1689-BCG, 770 Mpc => drop - 227,228: completely duplicated => drop 228
- 272,273: different data reduction => drop 272(higher uncertainty in data)
- 138: NaN in
logMGC, 3 inNgc=> data is dirty, check in 4-A-b
- 0: Nan in
check interested features
Data explore¶
In [14]:
# 1. load GCpG data
GCpG=pd.read_csv(path_GCpG)
# 2. MType to numeric, in Harris manner
GCpG['iMType'] = ''
GCpG['Mcat'] = ''
# E -> 2: 226(+18)
GCpG.iMType[GCpG.MType.str.contains(r'E')] = 2
GCpG.Mcat[GCpG.MType.str.contains(r'E')] = 'E'
# S0 -> 3: 75(+18)
GCpG.iMType[GCpG.MType.str.contains(r'S0')] = 3
GCpG.Mcat[GCpG.MType.str.contains(r'S0')] = 'S0'
# S/Irr -> 1: 103
GCpG.iMType[-(GCpG.MType.str.contains(r'E') | GCpG.MType.str.contains(r'S0'))] = 1
GCpG.Mcat[-(GCpG.MType.str.contains(r'E') | GCpG.MType.str.contains(r'S0'))] = 'S/Irr'
# E/S0 -> 6: 18
GCpG.iMType[GCpG.MType.str.contains(r'E') & GCpG.MType.str.contains(r'S0')] = 6
GCpG.Mcat[GCpG.MType.str.contains(r'E') & GCpG.MType.str.contains(r'S0')] = 'E/S0'
fig = figure(figsize=fig_size)
ax = fig.add_subplot(111)
GCpG.loc[:,['Name','MType','Mcat']].groupby(['Mcat','MType']).count() \
.unstack().plot(kind='bar',stacked=True,legend=False,rot=0, ax=ax)
text(0.8, 150,'%10s' % GCpG.loc[:,['Name','Mcat']].groupby('Mcat').count() \
.to_string().replace('Name','count'), **params_font)
text(1.8, 150, '%6s' % GCpG[GCpG.Mcat.isin(['S0','E/S0'])].loc[:,['Name','MType','Mcat']] \
.groupby(['Mcat','MType']).count().to_string().replace('Name','count').replace(' ', '%2s' % ' '), **params_font)
xlabel('Morphology Type')
if flag_svfg:
savefig(path.join(path_fig, 'MType.jpeg'),**params_svfg)
print('Our catalog contains 248 ellipticals, 93 S0\'s,'+
' and 81 spirals or irregulars, (Harris et. al. 2013).')
In [15]:
# 3. check duplicates and NaNs
display(GCpG.reindex([0,356,227,228,272,273,138]).loc[:,
['Name','RAhour','DEdeg','Dist','e_Dist','VMag','e_VMag','KMag',
'logMGC','Ngc','r_Ngc','sigma','Reff','logMd','logMB']])
if 0 in GCpG.index:
GCpG.drop([0,356,228,272],inplace=True)
Populating GCs in the local universe (chap 2)¶
Harris catalog for GCpG model (sec 2.2.1)¶
Data prepare¶
- [x] load GCpG data
- [x] check Morphology Type
[x] check duplicates and NaNs
[=] check interested features:
[=] statistics, present in 2 pandas tables
- tab1:
var(Ngc(6:0))significantly larger => addlogNgc(6:-inf) - tab2: explain unreasonable value in
Ngc, e_Ngc(0:3), logMGC(1:NaN), e_logMGC(2:0):Ngcderived fromlogMGC=> focus onNgc/logNgc(412/418)Ngc == 0:[7,51,87,88,271,304] => Ngc = 0.5 e_Ngc, avoid-inftemporatelye_Ngc == 0:[1,28,138] => e_Ngc = 0.5 Ngc, avoid-inftemporatelylogMGC, e_logMGC == NaN:[138] => linear interpolate NaNlogMGCfromlogNgc(412))e_logMGC ==0:[1,28] => leave alone
- tab1:
visualization
Data explore¶
In [17]:
# 4-A-a. statistics
display(GCpG.loc[:,['Name','VMag','KMag','Ngc','e_Ngc','r_Ngc',
'sigma','Reff','logMd','logMGC','Dist','logMB']] \
.describe(include='all').loc[['count','unique','min','max','mean','std'],:])
In [18]:
# 4-A-b. i: take logarithms -> logNgc; ii: fit NaN in logMGC and e_logMGC
display(GCpG.loc[(GCpG.loc[:,['Ngc','e_Ngc','logMGC','e_logMGC']]==0).sum(axis=1)==True,
['Name','Ngc','e_Ngc','logMGC','e_logMGC']])
if 'has_GC' not in locals(): # i
has_GC = (GCpG.Ngc > 0) # mark for logMGC interpolation
GCpG.Ngc[~has_GC] = 0.5 * GCpG.e_Ngc[~has_GC]
GCpG.e_Ngc[GCpG.e_Ngc==0] = 0.5 * GCpG.Ngc[GCpG.e_Ngc==0]
if 'logNgc' not in GCpG.columns:
GCpG['logNgc'] = GCpG.Ngc.apply(log10)
if 'nn_MGC' not in locals(): # ii
nn_MGC = GCpG.logMGC.notna() # update 138: logMGC NaN from Ngc/logNgc
LRsk = linear_model.LinearRegression()
LRsk.fit(GCpG.logNgc[has_GC & nn_MGC].values.reshape(-1,1),
GCpG.logMGC[has_GC & nn_MGC].values.reshape(-1,1))
GCpG.logMGC[~nn_MGC] = LRsk.predict(GCpG.logNgc[~nn_MGC].values.reshape(-1,1))
LRsk.fit(GCpG.e_Ngc[has_GC & nn_MGC].apply(log10).values.reshape(-1,1),
GCpG.e_logMGC[has_GC & nn_MGC].values.reshape(-1,1))
GCpG.e_logMGC[~nn_MGC] = LRsk.predict(GCpG.e_Ngc[~nn_MGC].apply(log10).values.reshape(-1,1))
Populating GCs in the local universe (chap 2)¶
Harris catalog for GCpG model (sec 2.2.1)¶
Data prepare¶
- [x] load GCpG data
- [x] check Morphology Type
[x] check duplicates and NaNs
[=] check interested features
[x] statistics, present in 2 pandas tables
[=] visualization
logNgc: present KDE in distplot- data completeness: represent in barplot
- spatial sparseness of
NaNdata: present in msnoplot
Data explore¶
In [19]:
# 4-B. visualization
fig = figure(figsize=fig_size)
ax = fig.add_subplot(211)
GCpG.logNgc.plot('kde',title='logNgc',ax=ax)
ax2 = fig.add_subplot(212)
GCpG.loc[:,GCpG.columns[GCpG.dtypes==float]].describe().loc['count',:].plot('bar', ax=ax2)
Out[19]:
In [20]:
img = msno.matrix(GCpG.loc[:, ['Name','VMag','KMag','Ngc','r_Ngc',
'sigma','Reff','logMd','Dist','logMB','logNgc']], inline=False)
if flag_svfg:
img.savefig(path.join(path_fig, 'GCpG_NaN.jpeg'), **params_svfg)
Populating GCs in the local universe (chap 2)¶
Harris catalog for GCpG model (sec 2.2.1)¶
Data prepare¶
Data explore¶
[=] explore correlations for data type in
float: present in pairplots with heatmap- properties:
'Dist', 'e_Dist', 'Av', 'VMag', 'e_VMag', 'KMag', 'e_KMag', 'Ngc', 'e_Ngc', 'sigma', 'e_sigma', 'Reff', 'e_Reff', 'logMd', 'e_logMd', 'logMGC', 'e_logMGC','logMB', 'E_logMB', 'e_logMB', 'logNgc'
- correlated =>
xxx & e_xxx: intuitive, not interestingAv & Dist:Distderived from NEDAv=> focus onDist(more intuitive)VMag & KMag(74:NaN): intuitive, LR in 2-A
- probably correlated =>
Dist & Ngc(6:0) & Reff(78:NaN)VMag & KMag(74:NaN) & sigma(147:NaN) & logMd(165:NaN) & logMGC(1:NaN) & logNgc(6:-inf)
- properties:
check correlated pairs [and interpolate values]
correlation ranking
In [21]:
# 1. explore correlations [slow]
def pp_heat(df, cor = 0.5, figname = 'pairplots.jpeg'):
'''
Input
df: pandas dataframe
Output
corr: df.corr()
pairplot with heatmap
'''
with sns.axes_style("white"):
pp = sns.pairplot(df.fillna(0), diag_kind='kde', markers='x', plot_kws={'s':15,'alpha':0.2})
corr = df.corr()
for i, j in zip(*triu_indices_from(pp.axes, 1)):
pp.axes[i, j].set_visible(False)
# pp.axes[i, j].figure.text((j+0.5)/len(corr),(len(corr)-i-0.5)/len(corr),
# str(round(corr.values[i, j],3)), **params_font)
for i, j in zip(*tril_indices_from(pp.axes, -1)):
if abs(corr.values[i, j]) < cor or isnan(corr.values[i, j]):
sns.despine(ax=pp.axes[i, j],top=True,right=True,left=True,bottom=True)
else:
sns.despine(ax=pp.axes[i, j],top=False,right=False,left=False,bottom=False)
for pos in ['top','right','left','bottom']:
pp.axes[i, j].spines[pos].set_edgecolor('gray')
pp.axes[i, j].figure.text((j+0.7)/len(corr),(len(corr)-i-0.3)/len(corr),
str(round(corr.values[i, j],3)), **params_font)
ax = pp.axes[0,0].figure.add_subplot(222)
sns.heatmap(corr, square=True, cmap=cmap, alpha=0.8, ax=ax)
savefig(path.join(path_fig,figname), **params_svfg)
return 0
if not flag_skip:
pp_heat(GCpG.loc[:,GCpG.columns[GCpG.dtypes==float][2:]], 0.5, 'pp_GCpG.jpeg')
else:
ax = figure(figsize=sfig_size).add_subplot(111)
ax.imshow(imread(path.join(path_fig, 'pp_GCpG.jpeg')))
ax.grid(False)
Populating GCs in the local universe (chap 2)¶
Harris catalog for GCpG model (sec 2.2.1)¶
Data prepare¶
Data explore¶
[x] explore correlations for data type in
float[=] check correlated pairs [and interpolate values]
[=]
VMag(418) & KMag(344+74): LR interpolate =>KMag(74:NaN), present in errorbar & OLS LR summaryDist & Ngc(6:0) & Reff(78:NaN)VMag(418)/KMag(344) & logMd(253)/sigma(271) & logMGC(417)/logNgc(412)
correlation ranking
In [22]:
# 2. check correlated pairs [and interpolate values]
# 2-A. `VMag` & `KMag`: linear regression
if 'nn_KMag' not in locals():
nn_KMag = GCpG.KMag.notna()
LRsk = linear_model.LinearRegression()
LRsk.fit(GCpG.VMag[nn_KMag].values.reshape(-1,1), GCpG.KMag[nn_KMag].values.reshape(-1,1))
GCpG.KMag[~nn_KMag] = LRsk.predict(GCpG.VMag[~nn_KMag].values.reshape(-1,1))
LRsk.fit(GCpG.e_VMag[nn_KMag].values.reshape(-1,1),
GCpG.e_KMag[nn_KMag].values.reshape(-1,1))
GCpG.e_KMag[~nn_KMag] = LRsk.predict(GCpG.e_VMag[~nn_KMag].values.reshape(-1,1))
LRols = sm.OLS(GCpG.KMag.values, GCpG.VMag.values).fit() # OLS(y, X)
print(LRols.summary2())
In [23]:
fig = figure(figsize=fig_size)
ax = fig.add_subplot(111)
ax.errorbar(GCpG.VMag[nn_KMag], GCpG.KMag[nn_KMag],
xerr=GCpG.e_VMag[nn_KMag], yerr=GCpG.e_KMag[nn_KMag], fmt='g+', **params_errbar)
ax.errorbar(GCpG.VMag[~nn_KMag], GCpG.KMag[~nn_KMag],
xerr=GCpG.e_VMag[~nn_KMag], yerr=GCpG.e_KMag[~nn_KMag], fmt='rx', **params_errbar)
ax.invert_xaxis()
ax.invert_yaxis()
ax.set_xlabel('VMag')
ax.set_ylabel('KMag')
legend(['existing KMag vs VMag: %d/%d' % (sum(nn_KMag),len(nn_KMag)),
'interpolated KMag vs VMag: %d/%d' % (sum(~nn_KMag),len(nn_KMag))])
if flag_svfg:
savefig(path.join(path_fig,'KMag_VMag.jpeg'), **params_svfg)
Populating GCs in the local universe (chap 2)¶
Harris catalog for GCpG model (sec 2.2.1)¶
Data prepare¶
Data explore¶
- [x] explore correlations for data type in
float [=] check correlated pairs [and interpolate values]
[x]
VMag(418) & KMag(344+74)[=]
Dist & Ngc(6:0) & Reff(78:NaN): no significant correlation => leave alone, present in pairplotVMag(418)/KMag(344) & logMd(253)/sigma(271) & logMGC(417)/logNgc(412)
correlation ranking
In [24]:
# 2-B. `Dist` & `Ngc` & `Reff`: no significant correlations
# pp = sns.pairplot(GCpG, vars= ['Dist','Ngc','Reff'], hue='Mcat', diag_kind= 'kde',
# plot_kws=params_hex, diag_kws={'lw':1}, size=3.2)
# for i, j in zip(*triu_indices_from(pp.axes, 1)):
# pp.axes[i, j].set_visible(False)
# upgrade to PairGrid https://seaborn.pydata.org/generated/seaborn.PairGrid.html
pp = sns.PairGrid(GCpG, vars= ['Dist','Ngc','Reff'], hue='Mcat',diag_sharey=False,size=3.2)
pp.map_lower(scatter, **params_hex)
pp.map_upper(sns.residplot)#, ci=97, scatter_kws={'s':15}, line_kws={'lw':1})
pp.map_diag(sns.kdeplot, lw=1)
pp.add_legend()
if flag_svfg:
savefig(path.join(path_fig,'pp_DNR.jpeg'), **params_svfg)
Populating GCs in the local universe (chap 2)¶
Harris catalog for GCpG model (sec 2.2.1)¶
Data prepare¶
Data explore¶
- [x] explore correlations for data type in
float [x] check correlated pairs [and interpolate values]
- [x]
VMag(418) & KMag(344+74) [x]
Dist & Ngc(6:0) & Reff(78:NaN)[=]
VMag(418)/KMag(344) & logMd(253)/sigma(271) & logMGC(417)/logNgc(412)[=]
logNgc(6:-inf) & logMGC(1:NaN):Ngc(6:0)deduct fromlogMGC(Harris et al. 2011), present in errorbar & OLS LR summary- =>
logNgc(412-1)LR interpolatelogMGC(1:NaN) - =>
logMGC(418-1-6)LR interpolateNgc/logNgc(6:0/-inf)
- =>
VMag/KMag (418/344+74) & logNgc/logMGC (412/417)logMd(253) & sigma(271) & logNgc/logMGC (412/417)
- [x]
correlation ranking
In [25]:
# 2-C-a. `logNgc` & `logMGC`: linear regression, interpolate logNgc from logMGC
LRsk = linear_model.LinearRegression()
LRsk.fit(GCpG.logMGC[has_GC].values.reshape(-1,1),
GCpG.logNgc[has_GC].values.reshape(-1,1))
GCpG.logNgc[~has_GC] = LRsk.predict(GCpG.logMGC[~has_GC].values.reshape(-1,1))
LRsk.fit(GCpG.e_logMGC[has_GC].values.reshape(-1,1),
GCpG.e_Ngc[has_GC].apply(log10).values.reshape(-1,1))
GCpG.e_Ngc[~has_GC] = 10**LRsk.predict(GCpG.e_logMGC[~has_GC].values.reshape(-1,1))
GCpG.Ngc[~has_GC] = 10**GCpG.logNgc[~has_GC]
LRols = sm.OLS(GCpG.logNgc.values, GCpG.logMGC.values).fit() # OLS(y, X)
print(LRols.summary2())
In [26]:
fig = figure(figsize=fig_size)
ax = fig.add_subplot(111)
ax.set_xlim([-1,6])
ax.set_ylim([4.5,10])
ax.errorbar(GCpG.logNgc[has_GC], GCpG.logMGC[has_GC],
xerr=log10(GCpG.e_Ngc[has_GC]), yerr=GCpG.e_logMGC[has_GC],
fmt='g+', **params_errbar)
ax.errorbar(GCpG.logNgc[~has_GC], GCpG.logMGC[~has_GC],
xerr=log10(GCpG.e_Ngc[~has_GC]), yerr=GCpG.e_logMGC[~has_GC],
fmt='rx', **params_errbar)
legend(['existing logNgc vs logMGC','interpolated logNgc vs logMGC'], loc=4)
ax.set_xlabel('logNgc')
ax.set_ylabel('logMGC')
if flag_svfg:
savefig(path.join(path_fig,'Ngc_MGC.jpeg'), **params_svfg)
Populating GCs in the local universe (chap 2)¶
Harris catalog for GCpG model (sec 2.2.1)¶
Data prepare¶
Data explore¶
[x] explore correlations for data type in
float[x] check correlated pairs [and interpolate values]
[x]
VMag(418) & KMag(344+74)[x]
Dist & Ngc(412+6) & Reff(78:NaN)[=]
VMag(418)/KMag(344) & logMd(253)/sigma(271) & logMGC(417)/logNgc(412)[x]
logNgc(412+6) & logMGC(417+1)[=]
VMag/KMag (418/344+74) & logNgc/logMGC (412/417): all complete- =>
VMag(original) vslogNgc(more intuitive) - => Luminosity model in sec 2.2.1, present in pairplot
- =>
logMd(253) & sigma(271) & logNgc/logMGC (412/417)
correlation ranking
In [27]:
# 2-C-b. `VMag(KMag) & logNgc(logMGC)`
pp = sns.pairplot(GCpG, vars=['VMag','KMag','logNgc','logMGC'], hue='Mcat',
plot_kws=params_scatter, diag_kind='kde', diag_kws={'lw':1})
for i, j in zip(*triu_indices_from(pp.axes, 1)):
pp.axes[i, j].set_visible(False)
pp.axes[0,0].invert_xaxis()
pp.axes[0,1].invert_xaxis()
pp.axes[1,0].invert_yaxis()
if flag_svfg:
savefig(path.join(path_fig,'pp_Luminosity.jpeg'), **params_svfg)
Populating GCs in the local universe (chap 2)¶
Harris catalog for GCpG model (sec 2.2.1)¶
Data prepare¶
Data explore¶
[x] explore correlations for data type in
float[x] check correlated pairs [and interpolate values]
[x]
VMag(418) & KMag(344+74)[x]
Dist & Ngc(6:0) & Reff(78:NaN)[=]
VMag(418)/KMag(344) & logMd(253)/sigma(271) & logMGC(417+1)/logNgc(412+6)[x]
logNgc(412+6) & logMGC(417+1)[x]
VMag/KMag (418/344+74) & logNgc/logMGC (412+6/417+1)[=]
logMd(253) & sigma(271) & logNgc/logMGC (412+6/417+1)- =>
logMd(253)= f(sigma(271),Reff(340)) vslogNgc(412+6):logMd(253) limiting term - => Dynamical mass model in sec 2.2.2, present in pairplot
- =>
correlation ranking
In [28]:
# 2-C-c. `logMd & sigma & logNgc(logMGC)`
pp = sns.pairplot(GCpG, vars=['logMd','sigma','logNgc','logMGC'], hue='Mcat',
plot_kws=params_scatter, diag_kind='kde', diag_kws={'lw':1})
for i, j in zip(*triu_indices_from(pp.axes, 1)):
pp.axes[i, j].set_visible(False)
if flag_svfg:
savefig(path.join(path_fig,'pp_Dynamical.jpeg'), **params_svfg)
Populating GCs in the local universe (chap 2)¶
Harris catalog for GCpG model (sec 2.2.1)¶
Data prepare¶
Data explore¶
[x] explore correlations for data type in
float[x] check correlated pairs [and interpolate values]
[x]
VMag & KMag[x]
Dist & Ngc & Reff[x]
VMag(KMag) & logMd (sigma) & logMGC(logNgc)
[=] correlation ranking: Random Forest with feature importance, explain in heatmap
VMag/KMag(344+74) vs logNgc(412+6)/logMGC(417+1)- impute median for NaN: =>
KMag & logMGC > KMag & logNgc > VMag & logNgc - impute mean for NaN: =>
KMag & logNgc > KMag & logMGC > VMag & logMGC - impute most frequent for NaN: =>
KMag & logNgc > KMag & logMGC > VMag & logMGC - 'normalize' by data availibility: 253/418(logMd) subset =>
KMag & Ngc > logMd & logMGC > VMag & logMGC, max correlation score drops - => possible selection effect
- => overfitting in interpolation
- => debating GCpG nature
- (color-blind/friendly seaborn colors and matplotlib)
In [30]:
# 3. correlation ranking, feature importance with Random Forest
## 3-A. impute median for NaN
## 3-B. impute mean for NaN
## 3-C. impute most frequent for NaN
## 3-D. 'normalize' by data avalibility 253/418
if 'strategies' not in locals():
features = ['Dist', 'e_Dist', 'Av', 'VMag', 'e_VMag', 'KMag', 'e_KMag', 'sigma', 'e_sigma',
'Reff', 'e_Reff', 'logMd', 'e_logMd', 'logMB', 'E_logMB', 'e_logMB']
targets = ['Ngc', 'e_Ngc', 'logNgc', 'logMGC', 'e_logMGC', 'iMType']
norm_feats = ['Dist', 'e_Dist', 'Av', 'VMag', 'e_VMag', 'KMag', 'e_KMag', 'sigma', 'e_sigma',
'Reff', 'e_Reff', 'logMd', 'e_logMd']
strategies = ['median', 'mean', 'most_frequent','norm']
def RF_NaN(df, feats, tags, strategy):
'''
Input
df: pandas dataframe
feats: list of feature colume names
tags: list of target colume names
strategy: str of strategy to deal with NaN
Output
df_rf: dataframe of Random Forest feature importances
'''
df_rf = pd.DataFrame(index=feats)
if strategy == 'norm':
X_rf = df.loc[:, feats].values
else:
X_rf = df.loc[:, feats].values
imp = Imputer(missing_values='NaN', strategy=strategy, axis=0)
X_rf = imp.fit_transform(X_rf)
RFreg = RandomForestRegressor()
for target in tags:
y_rf = df.loc[:, target].values
RFreg.fit(X_rf, y_rf)
df_rf[target] = RFreg.feature_importances_
return df_rf
axs = figure(figsize=sfig_size).subplots(2,2).ravel()
for i in range(len(strategies)):
if strategies[i] == 'norm':
df_rf = RF_NaN(GCpG.loc[:, norm_feats+targets].dropna(), norm_feats, targets, strategies[i])
axs[i].set_title("\'subset\' 253/418")
else:
df_rf = RF_NaN(GCpG, features, targets, strategies[i])
axs[i].set_title(strategies[i])
sns.heatmap(df_rf, cmap=cmap, alpha=0.8, ax=axs[i])
if flag_svfg:
savefig(path.join(path_fig,'RF_heat.jpeg'), **params_svfg)
Populating GCs in the local universe (chap 2)¶
GCpG models (sec 2.2)¶
Linear models: logMd vs logMGC/logNgc¶
[=] dynamical mass model
logMGC/logNgc$=\mathscr{LR}$(logMd)[=] scatter plot with LR: strong correlation, present in scatter + regplot
OLS LR summary
luminosity model $\mathscr{LR}$(
VMag/KMag)revisit with errorbar
GC SN model¶
In [32]:
# 1-A. `logMd vs logMGC/logNgc` in scatter
fig = figure(figsize=fig_size)
ax = fig.add_subplot(211)
ax2 = fig.add_subplot(212)
ax.set_xlim([7.5,13])
ax2.set_xlim([7.5,13])
ax2.set_ylim([-0.5, 5.5])
for i in range(len(GCpG.Mcat.unique())):
mask_MT = (GCpG.Mcat == GCpG.Mcat.unique()[i])
ax.scatter(GCpG.logMd[mask_MT], GCpG.logMGC[mask_MT],**markers[i])
sns.regplot(GCpG.logMd[mask_MT], GCpG.logMGC[mask_MT],
scatter=False, ci=None, line_kws={'lw':1}, ax=ax)
ax2.scatter(GCpG.logMd[mask_MT], GCpG.logNgc[mask_MT],**markers[i])
sns.regplot(GCpG.logMd[mask_MT], GCpG.logNgc[mask_MT],
scatter=False, ci=None, line_kws={'lw':1}, ax=ax2)
sns.regplot(GCpG.logMd,GCpG.logMGC, scatter=False, ci=ci, line_kws={'lw':1}, ax=ax)
sns.regplot(GCpG.logMd,GCpG.logNgc, scatter=False, ci=ci, line_kws={'lw':1}, ax=ax2)
ax.legend()
ax2.legend()
if flag_svfg:
savefig(path.join(path_fig,'LR_Mdyn.jpeg'), **params_svfg)
Populating GCs in the local universe (chap 2)¶
GCpG models (sec 2.2)¶
Linear models: logMd vs logMGC/logNgc¶
[=] dynamical mass model
logMGC/logNgc$=\mathscr{LR}$(logMd)[x] scatter plot with LR
[=] OLS LR summary: for
logMd(253): logMGC(0.996) > logNgc(0.923)
luminosity model $\mathscr{LR}$(
VMag/KMag)revisit with errorbar
GC SN model¶
In [33]:
# 1-B. `logMd vs logMGC/logNgc` in OLS regression
LRols = sm.OLS(GCpG.logNgc[GCpG.logMd.notna()].values,
GCpG.logMd[GCpG.logMd.notna()].values).fit() # OLS(y, X)
print(LRols.summary2())
In [34]:
LRols = sm.OLS(GCpG.logMGC[GCpG.logMd.notna()].values,
GCpG.logMd[GCpG.logMd.notna()].values).fit() # OLS(y, X)
print(LRols.summary2())
Populating GCs in the local universe (chap 2)¶
GCpG models (sec 2.2)¶
Linear models: VMag/KMag vs logNgc¶
[x] dynamical mass model $\mathscr{LR}$(
logMd)[=] luminosity model
logNgc$=\mathscr{LR}$(VMag/KMag)[=] scatter plot with LR: strong correlation, present in scatter + regplot
OLS LR summary
revisit with errorbar
GC SN model¶
In [35]:
# 2-A. `VMag/KMag vs logNgc` in scatter
fig = figure(figsize=fig_size)
ax = fig.add_subplot(211)
ax2 = fig.add_subplot(212)
ax.set_xlim([-12, -27])
ax2.set_xlim([-12, -27])
for i in range(len(GCpG.Mcat.unique())):
mask_MT = (GCpG.Mcat == GCpG.Mcat.unique()[i])
ax.scatter(GCpG.VMag[mask_MT], GCpG.logNgc[mask_MT],**markers[i])
sns.regplot(GCpG.VMag[mask_MT], GCpG.logNgc[mask_MT],
scatter=False, ci=None, line_kws={'lw':1}, ax=ax)
ax2.scatter(GCpG.KMag[mask_MT], GCpG.logNgc[mask_MT],**markers[i])
sns.regplot(GCpG.KMag[mask_MT], GCpG.logNgc[mask_MT],
scatter=False, ci=None, line_kws={'lw':1}, ax=ax2)
sns.regplot(GCpG.VMag,GCpG.logNgc, scatter=False, ci=ci, line_kws={'lw':1}, ax=ax)
sns.regplot(GCpG.KMag,GCpG.logNgc, scatter=False, ci=ci, line_kws={'lw':1}, ax=ax2)
ax.legend()
ax2.legend()
if flag_svfg:
savefig(path.join(path_fig,'LR_Mag.jpeg'), **params_svfg)
Populating GCs in the local universe (chap 2)¶
GCpG models (sec 2.2)¶
Linear models: VMag/KMag vs logNgc¶
[x] dynamical mass model $\mathscr{LR}$
(logMd)[x] luminosity model
logNgc=$\mathscr{LR}$(VMag/KMag)[x] scatter plot with LR
[=] OLS LR summary
VMag(0.866) < KMag(0.869) < logMd(0.923/0.996)- recall
logMd(253/418), KMag(344/418)interpolated fromVMag - => might be selection effect:
logMdrequires good photometry data - => overfitting in interpolation of
KMag
revisit
GC SN model¶
In [36]:
# 2-B. `VMag/KMag vs logNgc` in OLS linear regression
LRols = sm.OLS(GCpG.logNgc.values, GCpG.VMag.values).fit() # OLS(y, X)
print(LRols.summary2())
In [37]:
LRols = sm.OLS(GCpG.logNgc.values, GCpG.KMag.values).fit() # OLS(y, X)
print(LRols.summary2())
Populating GCs in the local universe (chap 2)¶
GCpG models (sec 2.2)¶
Linear models:¶
[x] dynamical mass model
logMGC/logNgc=$\mathscr{LR}$(logMd)[x] luminosity model
logNgc=$\mathscr{LR}$(VMag/KMag)[=] revisit with errorbar, present in 2 errorbars
- [=] dynamical mass model: limited by data completeness (253/418)
- luminosity model
GC SN model¶
In [38]:
# 3-A. revisit `logMd vs logMGC/logNgc` with errorbar
fig = figure(figsize=fig_size)
ax = fig.add_subplot(111)
ax2 = ax.twinx()
ax.set_xlim([7.5,13])
for i in range(len(GCpG.Mcat.unique())):
mask_MT = (GCpG.Mcat == GCpG.Mcat.unique()[i])
ax.errorbar(GCpG.logMd[mask_MT], GCpG.logMGC[mask_MT],
xerr=GCpG.e_logMd[mask_MT], yerr=GCpG.e_logMGC[mask_MT], **fmts[i])
ax2.errorbar(GCpG.logMd[mask_MT], GCpG.logNgc[mask_MT],
xerr=GCpG.e_logMd[mask_MT], yerr=GCpG.e_Ngc[mask_MT].apply(log10), **fmts[i])
ax.set_ylabel('logMGC $[M_\odot]$')
ax2.set_ylabel('logNgc')
ax.set_xlabel('logMd $[M_\odot]$')
sns.regplot(GCpG.logMd,GCpG.logMGC, scatter=False, ci=ci, line_kws={'lw':1}, ax=ax)
sns.regplot(GCpG.logMd,GCpG.logNgc, scatter=False, ci=ci, line_kws={'lw':1}, ax=ax2)
ax.legend()
if flag_svfg:
savefig(path.join(path_fig,'LR_Mdyn_err.jpeg'), **params_svfg)
LRols = sm.OLS(GCpG.logNgc.values, GCpG.KMag.values).fit() # OLS(y, X)
print(LRols.summary2())
Populating GCs in the local universe (chap 2)¶
GCpG models (sec 2.2)¶
Linear models: logMd vs logMGC/logNgc and VMag/KMag vs logNgc¶
[x] dynamical mass model
logMGC/logNgc=$\mathscr{LR}$(logMd)[x] luminosity model
logNgc=$\mathscr{LR}$(VMag/KMag)[=] revisit with errorbar, present in 2 errorbars
- [x] dynamical mass model (253/418)
- [=] luminosity model: no better with huge uncertainties
GC SN model¶
In [39]:
# 3. revisit `VMag/KMag vs logNgc` with errorbar
fig = figure(figsize=fig_size)
ax = fig.add_subplot(111)
ax2 = ax.twiny()
ax.set_xlim([-12, -27])
ax2.set_xlim([-12, -27])
sub_axes = axes([0.15, 0.42, 0.4, 0.4])
for i in range(len(GCpG.Mcat.unique())):
mask_MT = (GCpG.Mcat == GCpG.Mcat.unique()[i])
ax.errorbar(GCpG.VMag[mask_MT], GCpG.logNgc[mask_MT],
xerr=GCpG.e_VMag[mask_MT], yerr=GCpG.e_Ngc[mask_MT].apply(log10), **fmts[i])
ax2.errorbar(GCpG.KMag[mask_MT], GCpG.logNgc[mask_MT],
xerr=GCpG.e_KMag[mask_MT], yerr=GCpG.e_Ngc[mask_MT].apply(log10), **fmts[i])
sub_axes.scatter(GCpG.VMag[mask_MT], GCpG.logNgc[mask_MT], **markers[i])
sns.regplot(GCpG.VMag,GCpG.logNgc, scatter=False, ci=ci, line_kws={'lw':1}, ax=ax)
sns.regplot(GCpG.KMag,GCpG.logNgc, scatter=False, ci=ci, line_kws={'lw':1}, ax=ax2)
ax.set_ylabel('logNgc')
ax.set_xlabel('VMag')
ax2.set_xlabel('KMag')
sub_axes.set_xlim([-10, -26])
sub_axes.grid(False)
sub_axes.axis('off')
sub_axes.legend()
sub_axes.set_title('Fig. 3, Rodriguez et al. (2015)')
if flag_svfg:
savefig(path.join(path_fig,'LR_Mag_err.jpeg'), **params_svfg)
In [40]:
LRols = sm.OLS(GCpG.logNgc.values, GCpG.KMag.values).fit() # OLS(y, X)
print(LRols.summary2())
Populating GCs in the local universe (chap 2)¶
GCpG models (sec 2.2)¶
Linear models: bothare messy and trustless¶
GC SN model¶
- [=] the $N_\text{GC}$ per unit of parent galaxy luminosity
VMag, normalized to $M_V$ = −15, present in scatter- $S_N \equiv N_\text{GC} \times 10^{0.4(M_V^T + 15)}$
- $N_\text{GC} = S_N / 10^{0.4(M_V^T + 15)}$
[=] LOWESS regression for $S_N$ vs
VMag, present in ploterror comparison
In [42]:
# 1. compute GC SN and represent
if 'Sn' not in GCpG.columns:
GCpG['Sn'] = GCpG.Ngc*10**(0.4*(GCpG.VMag+15.0))
GCpG['e_Sn'] = GCpG.e_Ngc*10**(0.4*(GCpG.VMag+15.0))
fig = figure(figsize=fig_size)
ax = fig.add_subplot(111)
ax.set_ylim([-2,40])
ax.set_xlim([-10,-25])
ax.set_xlabel("VMag")
ax.set_ylabel("GC SN")
for i in range(len(GCpG.Mcat.unique())):
mask_MT = (GCpG.Mcat == GCpG.Mcat.unique()[i])
ax.scatter(GCpG.VMag[mask_MT], GCpG.Sn[mask_MT], **markers[i])
ax.set_title('Fig. 10, Harris et al. (2013)')
# 2. lowess regression
lowess = sm.nonparametric.lowess(GCpG.Sn, GCpG.VMag, frac=0.36) # lowess(y, X)
f_Sn = interp1d(lowess[:,0],lowess[:,1],bounds_error=False)
ax.plot(GCpG.VMag.sort_values(), f_Sn(GCpG.VMag.sort_values()))
ax.plot(GCpG.VMag.sort_values(), f_Sn(GCpG.VMag.sort_values()),'*', markersize=16,
markevery=len(GCpG)-1,label='VMag $\in$ [%.2f,%.2f]' % (GCpG.VMag.max(), GCpG.VMag.min()))
ax.legend(loc=0)
if flag_svfg:
savefig(path.join(path_fig, 'GCSN.jpeg'), **params_svfg)
LRols = sm.OLS(GCpG.logNgc.values, GCpG.KMag.values).fit() # OLS(y, X)
In [43]:
print(LRols.summary2())
Populating GCs in the local universe (chap 2)¶
GCpG models (sec 2.2)¶
Linear models: bothare messy and trustless¶
GC SN model¶
- [x] the $N_\text{GC}$ per unit of parent galaxy luminosity
VMag- $S_N \equiv N_\text{GC} \times 10^{0.4(M_V^T + 15)}$
- $N_\text{GC} = S_N / 10^{0.4(M_V^T + 15)}$
[x] LOWESS regression for $S_N$ vs
VMag[=] error comparison, zoom in view, justificationin errorbar
- => error suppressed on high luminosity end: a few noisy outliners less-weighted
- => fair performance on low luminosity end: low influence as the base $M_V$ will be low
- => well represented in the most range
- => truncated by
VMagrange[-11.17, -24.19]: lower-bound
In [45]:
# 3. error comparison, zoomed in
fig = figure(figsize=fig_size)
ax = fig.add_subplot(111)
ax.axis([-10,-25,-2,40])
ax.set_xlabel("VMag")
ax.set_ylabel("GC SN")
axins = zoomed_inset_axes(ax, 3, loc=9)
axins.axis([-18, -20, -0.2, 4.8])
for i in range(len(GCpG.Mcat.unique())):
mask_MT = (GCpG.Mcat == GCpG.Mcat.unique()[i])
ax.scatter(GCpG.VMag[mask_MT], GCpG.Sn[mask_MT], alpha=0.8, **markers[i])
ax.errorbar(GCpG.VMag[mask_MT], GCpG.Sn[mask_MT],
xerr=GCpG.e_VMag[mask_MT], yerr=GCpG.e_Sn[mask_MT], **fmts[i])
axins.scatter(GCpG.VMag[mask_MT], GCpG.Sn[mask_MT], alpha=0.8, **markers[i])
axins.errorbar(GCpG.VMag[mask_MT], GCpG.Sn[mask_MT],
xerr=GCpG.e_VMag[mask_MT], yerr=GCpG.e_Sn[mask_MT], **fmts[i])
ax.legend(GCpG.Mcat.unique())
ax.add_patch(patches.Rectangle((-20,-0.2),2,5,linewidth=1,edgecolor='k',facecolor='none'))
axins.add_patch(patches.Rectangle((-19.99,-0.19),1.98,4.96,linewidth=1,edgecolor='k',facecolor='none'))
ax.set_title('Fig. 10, Harris et al. (2013)')
axins.grid(False)
lowess = sm.nonparametric.lowess(GCpG.Sn, GCpG.VMag, frac=0.36) # lowess(y, X)
f_Sn = interp1d(lowess[:,0],lowess[:,1],bounds_error=False)
ax.plot(GCpG.VMag.sort_values(), f_Sn(GCpG.VMag.sort_values()))
axins.plot(GCpG.VMag.sort_values(), f_Sn(GCpG.VMag.sort_values()))
_, pp1,pp2 = mark_inset(ax,axins,loc1=1, loc2=1, color='k')
pp1.loc1=1
pp1.loc2=2
pp2.loc1=3
pp2.loc2=4
if flag_svfg:
savefig(path.join(path_fig, 'GCSN_err.jpeg'), **params_svfg)
In [46]:
LRols = sm.OLS(GCpG.logNgc.values, GCpG.KMag.values).fit() # OLS(y, X)
print(LRols.summary2())
Populating GCs in the local universe (chap 2)¶
Modeling GC populations (sec 2.3)¶
[=] bandpass conversion:
B/I/KinGWGCfrom White et al. (2011) toV/Kin GCSN model from Harris et al. (2013)[=] GWGC luminosities,
Abs_Mag_B/I(7,877/5,606), App_Mag_B/I/K(7,877/5,606/2,995), and err_Abs/App_Mag_B/I/K:- count statistics, present in pandas table
- completeness, msnoplot, lots of NaNs
explore correlations
fit conversion constant
choose most probable value
revert to Ngc
GC distribution
In [47]:
# 1-A. statistics of GWGC luminosity
display(LoGal.loc[:,LoGal.columns[LoGal.columns.str.contains('Mag')]] \
.describe().loc[['count','min','max','mean','std'],:].transpose().merge(
LoGal.loc[:,LoGal.columns[LoGal.columns.str.contains('Mag')]].isna().sum(axis=0).to_frame('NaNs'),
left_index=True,right_index=True).transpose())
In [48]:
img = msno.matrix(LoGal.loc[:,LoGal.columns[LoGal.columns.str.contains('Mag')]], inline=False)
if flag_svfg:
img.savefig(path.join(path_fig, 'GWGC_mags_NaN.jpeg'), **params_svfg)
Populating GCs in the local universe (chap 2)¶
Modeling GC populations (sec 2.3)¶
[=] bandpass conversion:
B/I/KinGWGCfrom White et al. (2011) toV/Kin GCSN model from Harris et al. (2013)[x] GWGC luminosities
[=] explore correlations, explain in pairplots with heatmap
- correlated
- =>
xxx & err_xxx: intuitive, not interesting - =>
abs_x & app_x: b'cos $app\_x-abs\_X=5\log_{10}(d)-5$ - =>
err_Abs_x & err_App_x: of course, pass
- =>
- all bands are monochromatic (AB magnitude system)
- => $ M_A-M_\odot=-2.5\log10(\frac{L_s}{L_\odot})=M_B-M_\odot $
- correlated
fit conversion constant
choose most probable value
revert to Ngc
GC distribution
In [49]:
# 1-B. correlations between bands
if not flag_skip:
corr = pp_heat(LoGal.loc[:,LoGal.columns[LoGal.columns.str.contains('Mag')]], 0.8, 'pp_GWGC_mags.jpeg')
else:
ax = figure(figsize=sfig_size).add_subplot(111)
ax.imshow(imread(path.join(path_fig, 'pp_GWGC_mags.jpeg')))
ax.grid(False)
Populating GCs in the local universe (chap 2)¶
Modeling GC populations (sec 2.3)¶
[=] bandpass conversion:
B/I/KinGWGCfrom White et al. (2011) toV/Kin GCSN model from Harris et al. (2013)[x] GWGC luminosities
[x] explore correlations
[=] fit conversion constant, 197 galaxies in both catalogs, inspect in mag/color scatter
[Abs_Mag_B (7,877), err_Abs_Mag_B (7,877)] > [Abs_Mag_I (5,606), err_Abs_Mag_I (0)]: => prefer B over I for Ngc estimationB/I in GWGC (197/148) vs V/K in GCSN=> almost homogenous, prefer B for fitB/I in GWGC (197/148) vs V-K in GCSN=> color/Mcat didn't help (unequal axis)
choose most probable value
revert to Ngc
GC distribution
In [51]:
# 1-C. fit conversion constant, color scatter
idx_GCpG = GCpG.Name.isin(LoGal.Name)
idx_LoGal = LoGal.Name.isin(GCpG.Name)
nullfmt = NullFormatter() # no labels
left, width = 0.1, 0.65
bottom, height = 0.1, 0.65
bottom_h = left_h = left + width + 0.02
rect_main = [left, bottom, width, height]
rect_histx = [left, bottom_h, width, 0.2]
rect_histy = [left_h, bottom, 0.2, height]
figure(figsize=sfig_size)
ax_histx = axes(rect_histx)
ax_histy = axes(rect_histy)
ax_histx.xaxis.set_major_formatter(nullfmt)
ax_histy.yaxis.set_major_formatter(nullfmt)
ax_main = axes(rect_main)
ax_main.set_xlabel('VMag in GCpG from Harris et al. (2013)')
ax_main.set_ylabel('BMag in GWGC from White et al. (2011)')
ax_main.scatter(GCpG.VMag[idx_GCpG], LoGal.Abs_Mag_B[idx_LoGal],
c=GCpG.iMType[idx_GCpG], s=40*(2+log2(30/GCpG.Dist[idx_GCpG])), cmap=cmap)
ax_main.errorbar(GCpG.VMag[idx_GCpG], LoGal.Abs_Mag_B[idx_LoGal],
xerr=GCpG.e_VMag[idx_GCpG], yerr=LoGal.err_Abs_Mag_B[idx_LoGal],
fmt='+', elinewidth=3, alpha=0.6)
xx, yy = mgrid[ax_main.get_xlim()[0]:ax_main.get_xlim()[1]:100j,
ax_main.get_ylim()[0]-0.5:ax_main.get_ylim()[1]+0.5:100j]
positions = vstack([xx.ravel(), yy.ravel()])
values = vstack([GCpG.VMag[idx_GCpG].values, LoGal.Abs_Mag_B[idx_LoGal].values])
kernel = st.gaussian_kde(values)
zz = reshape(kernel(positions).T, xx.shape)
cfset = ax_main.contourf(xx, yy, zz, cmap='Greens', alpha=0.3) # Contourf plot
cset = ax_main.contour(xx, yy, zz, colors='gray', alpha=0.7) # Contour plot
ax_main.clabel(cset, inline=1, fontsize=16)
ax_main.legend(['Galaxy in both catalog: %d' % idx_GCpG.sum()], fontsize=16)
ax_main.set_xlim([-23,-16])
ax_main.set_ylim([-22,-15])
ax_histx.set_xlim(ax_main.get_xlim())
ax_histy.set_ylim(ax_main.get_ylim())
binwidth = 0.25
xymax = max([max(fabs(GCpG.VMag[idx_GCpG])), max(fabs(LoGal.Abs_Mag_B[idx_LoGal]))])
lim = (int(xymax/binwidth) + 1) * binwidth
bins = arange(-lim, lim + binwidth, binwidth)
_ = ax_histx.hist(GCpG.VMag[idx_GCpG], bins=bins)
_ = ax_histy.hist(LoGal.Abs_Mag_B[idx_LoGal], bins=bins, orientation='horizontal')
if flag_svfg:
savefig(path.join(path_fig,'joint_mags.jpeg'), **params_svfg)
In [52]:
# 1-C. fit conversion constant, color scatter
figure(figsize=sfig_size)
ax_histx = axes(rect_histx)
ax_histy = axes(rect_histy)
ax_histx.xaxis.set_major_formatter(nullfmt)
ax_histy.yaxis.set_major_formatter(nullfmt)
ax_main = axes(rect_main)
ax_main.set_xlabel('VMag-KMag in GCpG from Harris et al. (2013)')
ax_main.set_ylabel('BMag in GWGC from White et al. (2011)')
ax_main.scatter(GCpG.VMag[idx_GCpG]-GCpG.KMag[idx_GCpG], LoGal.Abs_Mag_B[idx_LoGal],
c=GCpG.iMType[idx_GCpG], s=40*(2+log2(30/GCpG.Dist[idx_GCpG])), cmap=cmap)
ax_main.errorbar(GCpG.VMag[idx_GCpG]-GCpG.KMag[idx_GCpG], LoGal.Abs_Mag_B[idx_LoGal],
xerr=GCpG.e_VMag[idx_GCpG]-GCpG.e_KMag[idx_GCpG], yerr=LoGal.err_Abs_Mag_B[idx_LoGal],
fmt='+', elinewidth=3, alpha=0.6)
xx, yy = mgrid[ax_main.get_xlim()[0]:ax_main.get_xlim()[1]:100j,
ax_main.get_ylim()[0]-0.5:ax_main.get_ylim()[1]+0.5:100j]
positions = vstack([xx.ravel(), yy.ravel()])
values = vstack([GCpG.VMag[idx_GCpG].values-GCpG.KMag[idx_GCpG].values,
LoGal.Abs_Mag_B[idx_LoGal].values])
kernel = st.gaussian_kde(values)
zz = reshape(kernel(positions).T, xx.shape)
cfset = ax_main.contourf(xx, yy, zz, cmap='Greens', alpha=0.3) # Contourf plot
cset = ax_main.contour(xx, yy, zz, colors='gray', alpha=0.7) # Contour plot
ax_main.clabel(cset, inline=1, fontsize=16)
ax_main.legend(['Galaxy in both catalog: %d' % idx_GCpG.sum()], fontsize=16)
ax_main.set_xlim([1,4])
ax_main.set_ylim([-22,-14])
ax_histx.set_xlim(ax_main.get_xlim())
ax_histy.set_ylim(ax_main.get_ylim())
binwidth = 0.25
xymax = max([max(fabs(GCpG.VMag[idx_GCpG])), max(fabs(LoGal.Abs_Mag_B[idx_LoGal]))])
lim = (int(xymax/binwidth) + 1) * binwidth
bins = arange(-lim, lim + binwidth, binwidth)
_ = ax_histx.hist(GCpG.VMag[idx_GCpG]-GCpG.KMag[idx_GCpG], bins=bins)
_ = ax_histy.hist(LoGal.Abs_Mag_B[idx_LoGal], bins=bins, orientation='horizontal')
if flag_svfg:
savefig(path.join(path_fig,'joint_color.jpeg'), **params_svfg)
Populating GCs in the local universe (chap 2)¶
Modeling GC populations (sec 2.3)¶
[=] bandpass conversion:
B/I/KinGWGCfrom White et al. (2011) toV/Kin GCSN model from Harris et al. (2013)[x] GWGC luminosities
[x] explore correlations
[x] fit conversion constant:
B/I in GWGC (197/148) vs V/K in GC SNon 197 shared galaxies[=] choose most probable value, based on kdeplot
- =>$\mathrm{VMag}_\max = -20.515828946699045$
- =>$\mathrm{BMag}_\max = -19.627615047946385$
- =>$\mathrm{IMag}_\max = -21.008287606264027$
revert to Ngc
GC distribution
In [53]:
# 1-D. most probable value from kdeplot
f_joint, (ax1, ax2, ax3) = subplots(3, sharex=True, sharey=True, figsize=sfig_size)
ax1.set_xlim([-25,-10])
p_v = sns.kdeplot(GCpG.VMag[idx_GCpG], shade=True, ax=ax1, color='g')
x,y = p_v.get_lines()[0].get_data()
V_max = x[y.argmax()]
ax1.vlines(V_max, 0, y.max(), linestyle='dashed')
ax1.legend(['$\mathrm{VMag}_{\max}$ = %.2f, from Harris et al. (2013)' % V_max])
p_b = sns.kdeplot(LoGal.Abs_Mag_B[idx_LoGal], shade=True, ax=ax2, color='b')
x,y = p_b.get_lines()[0].get_data()
B_max = x[y.argmax()]
ax2.vlines(B_max, 0, y.max(), linestyle='dashed')
ax2.legend(['$\mathrm{BMag}_{\max}$ = %.2f, from White et al. (2011)' % B_max])
p_i = sns.kdeplot(LoGal.Abs_Mag_I[idx_LoGal], shade=True, ax=ax3, color='r')
x,y = p_i.get_lines()[0].get_data()
I_max = x[y.argmax()]
ax3.vlines(I_max, 0, y.max(), linestyle='dashed')
ax3.legend(['$\mathrm{IMag}_{\max}$ = %.2f, from White et al. (2011)' % I_max])
f_joint.subplots_adjust(hspace=0)
if flag_svfg:
savefig(path.join(path_fig,'mags_max.jpeg'), **params_svfg)
Populating GCs in the local universe (chap 2)¶
Modeling GC populations (sec 2.3)¶
[x] bandpass conversion:
B/I/KinGWGCfrom White et al. (2011) toV/Kin GCSN model from Harris et al. (2013)[x] GWGC luminosities
[x] explore correlations
[x] fit conversion constant:
B/I in GWGC (197/148) vs V/K in GC SNon 197 shared galaxies[x] choose most probable value, based on kdeplot
- =>$\mathrm{VMag}_\max = -20.515828946699045$
- =>$\mathrm{BMag}_\max = -19.627615047946385$
- =>$\mathrm{IMag}_\max = -21.008287606264027$
[=] revert to Ngc:
- convert
BMag/IMag in GWGCtoVMag in GCSN: B to V (7,877) + I to V (extra 32) = 7,909/8,946 - apply $N_\text{GC} = S_N / 10^{0.4(M_V^T + 15)}$
- result statistics and 5 'Interpolated Galaxy Samples', present in pandas table
- => lower-bound GC Population is
658,910, over 8,946 galaxies within 30 Mpc - => 1,037 (11.59%) GWGC galaxy not in count
- => 130 (1.45%) extra galaxy not counted due to GC SN truncation
- => lowess fit is
modest
- => lower-bound GC Population is
- convert
GC distribution
In [54]:
# 2. revert to Ngc
# 2-A. convert BMag/IMag to VMag
LoGal['VMag']=pd.Series(LoGal.Abs_Mag_B + V_max - B_max, index=LoGal.index)
LoGal['e_VMag']=pd.Series(LoGal.err_Abs_Mag_B
- GCpG[idx_GCpG].e_VMag.median()
+ LoGal[idx_LoGal].err_Abs_Mag_B.median())
_ = LoGal.Abs_Mag_B.isnull() & LoGal.Abs_Mag_I.notnull()
LoGal.VMag[_] = LoGal.Abs_Mag_I[_] + V_max - I_max
# 2-B. apply GCSN to get Ngc
LoGal['Ngc'] = pd.Series(map(lambda x: 0 if isnan(f_Sn(x))
else round(f_Sn(x)/10**(0.4*(x+15))), LoGal.VMag),
index=LoGal.index)
LoGal['e_Ngc'] = pd.Series(map(lambda x: LoGal.Ngc[x] if isnan(f_Sn(LoGal.VMag[x]-LoGal.e_VMag[x]))
else abs(LoGal.Ngc[x] - (f_Sn(LoGal.VMag[x]-LoGal.e_VMag[x]) /
10**(0.4*(LoGal.VMag[x]-LoGal.e_VMag[x]+15)))),
LoGal.index), index=LoGal.index)
# 2-C. statistics of result and 5 'Interpolated Galaxy Samples'
# print('Total GC Population in the Local Universe (30 Mpc) is %d' % LoGal.Ngc.sum())
_ = ['Name', 'RA', 'Dec', 'VMag', 'Dist', 'Abs_Mag_B', 'Abs_Mag_I', 'Ngc','e_Ngc', 'iMType']
display(pd.concat([LoGal.loc[:,_].describe(include='all').loc[['count','min','max','mean','std']],
LoGal.loc[:,_].sample(5,random_state=427)]).loc[:,_])
Populating GCs in the local universe (chap 2)¶
Modeling GC populations (sec 2.3)¶
[x] bandpass conversion:
B/I/KinGWGCfrom White et al. (2011) toV/Kin GCSN model from Harris et al. (2013)[x] GWGC luminosities
[x] explore correlations
[x] fit conversion constant:
B/I in GWGC (197/148) vs V/K in GC SNon 197 shared galaxies[x] choose most probable value, based on kdeplot
- =>$\mathrm{VMag}_\max = -20.515828946699045$
- =>$\mathrm{BMag}_\max = -19.627615047946385$
- =>$\mathrm{IMag}_\max = -21.008287606264027$
[x] revert to Ngc: 7,909/8,946 with Ngc
- => lower-bound GC Population is
658,910, over 8,946 galaxies within 30 Mpc - => 1,037 (11.59%) GWGC galaxy not in count
- => 130 (1.45%) extra galaxy not counted due to GC SN truncation
- => lowess fit is
modest
- => lower-bound GC Population is
[=] GC distribution
- present in all-sky plot
In [56]:
# 2-D. all-sky zoom in view
fig = figure(figsize=fig_size)
ax = fig.add_subplot(111, projection="mollweide")
ax.set_xticklabels(['2h','4h','6h','8h','10h','12h','14h','16h','18h','20h','22h'])
ax.grid(True)
vmin = 0.1
host_GC = LoGal.Ngc > 0
# RA: [0h-24h] -> [-pi,pi]; Dec: [-90°,90°] -> [-pi/2,pi/2]
ax.scatter((LoGal.RA[host_GC]-12)/12*pi, LoGal.Dec[host_GC]/180*pi,
s=5*(2+log2(30./LoGal.Dist[host_GC])),c=LoGal.Ngc[host_GC],alpha=0.5,edgecolors=None,
norm=LogNorm(vmin=vmin,vmax=LoGal.Ngc.max()),cmap=cmap)
ax.scatter((LoGal.RA[-host_GC]-12)/12*pi, LoGal.Dec[-host_GC]/180*pi,
s=2*log2(30./LoGal.Dist[-host_GC]),c='k',alpha=0.5,edgecolors=None)
axins = zoomed_inset_axes(ax, 3.2, loc=5, bbox_to_anchor=(1, 0.5), bbox_transform=ax.figure.transFigure)
axins.axis([-0.35, 0.3, -0.1, 0.4])
cax = axins.scatter((LoGal.RA[host_GC]-12)/12*pi, LoGal.Dec[host_GC]/180*pi,
s=5*(2+log2(30./LoGal.Dist[host_GC])),c=LoGal.Ngc[host_GC],alpha=0.5,edgecolors=None,
norm=LogNorm(vmin=vmin,vmax=LoGal.Ngc.max()),cmap=cmap)
axins.scatter((LoGal.RA[-host_GC]-12)/12*pi,LoGal.Dec[-host_GC]/180*pi,
s=2*log2(30./LoGal.Dist[-host_GC]),c='k',alpha=0.5,edgecolors=None)
axins.set_xticklabels([])
axins.set_yticklabels([])
cbar = fig.colorbar(cax, orientation='horizontal',fraction=0.07,anchor=(0.5,0.0))
ax.add_patch(patches.Rectangle((-0.35,-0.1),0.65,0.5,linewidth=1,edgecolor='k',facecolor='none'))
axins.add_patch(patches.Rectangle((-0.349,-0.096),0.646,0.495,linewidth=1,edgecolor='k',facecolor='none'))
anno_kws = {'width':1, 'headwidth':1, 'headlength':1, 'shrink':0.05}
ax.annotate('',xy=(2.44,0.80),xytext=(-0.48,0.4),arrowprops=anno_kws)
ax.annotate('',xy=(2.50,-.84),xytext=(-0.48,-0.07),arrowprops=anno_kws)
ax.set_title('Total GC Population in the Local Universe (30 Mpc) is %d' % LoGal.Ngc.sum())
if flag_svfg:
savefig(path.join(path_fig, 'all-sky.jpeg'), **params_svfg)
Populating GCs in the local universe (chap 2)¶
GWGC for galaxy abundance (sec 2.1): 8,946 galaxies within 30 Mpc¶
Harris catalog for GCpG model (sec 2.2.1): linear model can be improved¶
GCpG models (sec 2.2): a modest GC SN model¶
Modeling GC populations (sec 2.3): 658,910 GCs in 7,909 out of the galaxies within 30 Mpc¶
| Name | RA | Dec | VMag | Dist | Abs_Mag_B | Abs_Mag_I | Ngc | e_Ngc | iMType | |
|---|---|---|---|---|---|---|---|---|---|---|
| count | 8946 | 8946.000000 | 8946.000000 | 0.0 | 8946.000000 | 7877.000000 | 5606.000000 | 0.0 | 0.0 | 8946.000000 |
| min | NaN | 0.005940 | -89.334590 | NaN | 0.010000 | -22.590000 | -25.520000 | NaN | NaN | 0.000000 |
| max | NaN | 23.996680 | 86.131800 | NaN | 29.986000 | -4.970000 | -1.680000 | NaN | NaN | 6.000000 |
| mean | NaN | 10.973250 | 3.455383 | NaN | 19.011495 | -16.210734 | -17.158776 | NaN | NaN | 0.916387 |
| std | NaN | 5.277991 | 34.889882 | NaN | 6.616150 | 2.451796 | 2.627928 | NaN | NaN | 0.936047 |
| 168135 | SDSSJ142852.74+123454.0 | 14.481330 | 12.581660 | NaN | 28.958000 | NaN | NaN | NaN | NaN | 0.000000 |
| 168242 | SDSSJ144213.99+333517.4 | 14.703880 | 33.588320 | NaN | 24.528000 | NaN | NaN | NaN | NaN | 0.000000 |
| 146627 | SDSSJ080937.39+413526.6 | 8.160380 | 41.590710 | NaN | 11.625000 | -13.280000 | -12.870000 | NaN | NaN | 0.000000 |
| 24031 | UGC06251 | 11.223940 | 53.595040 | NaN | 18.450000 | -16.830000 | NaN | NaN | NaN | 1.000000 |
| 171592 | SDSSJ151605.06+561614.7 | 15.268080 | 56.270820 | NaN | 9.875000 | -15.740000 | -16.790000 | NaN | NaN | 0.000000 |
In [57]:
ax = figure(figsize=(16,20)).add_subplot(111)
ax.imshow(imread('Fig_D/mont.png'))
ax.grid(False)
_ = ax.axis('off')