- Email: [email protected]
A determination of the K-correction for quasars
Pergamon Press.
5 (1981) 174-182 -27 11980) 368-378
Chin.Astron.Astrophys.
Acta Astron. Sinica
Printed in Great Britain Ol46-6364/81/020174-09$07.50/O
A DETERMINATIONOF THE K-CORRECTION FOR QUASARS *
Xiao
Xing-hua,
Cao Sheng-lin
Cheng Fu-zhen [iniversity
Beijing (bservatory,and Yang Lan-tian
Tang Xiao-ying
Received
1979 September
continuum
line
widths
spectral
indices
emission
line
for
A knowledge
of
in elucidating the
the
continuum
example,
Grewing
and Woltjer’s
indices being
of
calculate
[l]
of
of
using
al
[3], of
have
the
[4]
College
Hubble
found
lines.
the
spectrum
of
nature.
In all
these
portion
presence
of
optical energy
index continuum
curve
studies,
however,
the
2
continuum,
is
important
the
study
quasars,
candle
strong
Setti
was further
quasars of
of
different
redshifts
continuum
sample
of
for
10 objects.
of
the
the
number of
of
assumption
between
for giving
quasars
from
properties
this
K-correction.
a large
as a standard
correlation
2,
However,
spectra
Under the
diagrams.
a strong
in
and radio
a mean optical
intrinsic
the
spectral
against
continuum
a synthesized
radio
plotted
the
by the
and a
determined
of
studied
who analysed
component
lJ,B,Vcolours
gives
continuum
made difficult
their
Baldwin
Teachers
B-V and U-B, corrected
these
the
line
iteratively
are
and their
obtained
et
emission
the
features
[2]
an examination
width
Z between
Many people
and Lamla
idea
K-correction,
prcperties
has been
lines.
cosmological,
and the
to
emission
by Netzer in
used
characteristic
spectrum
emission
the
are
redshifts,
The colors
355 quasars. of
an emission
two components
Eliminating
then
their
and wide
developed
is
made up of
and intensities,
portion
mean relations. which
is
These
component.
from
Huazhong
12
ABSTRACT The K-correction
paper
Beijing Normal University,
of Science and Technology,
luminosity
size
is
rather
restricted. Burbidge’s
1977 Catalog
of
1978,
contains
1038 quasars.
radio
spectral
indices).
First,
we shall
and flat
spectrum
find
the
objects.
Secondly,
we shall
find
curves
we shall
find
quasar
* Figs.
for
the
1 and 2 in in
this
the
the
different
the
abridged
original
after
of
the
after
continuum
emission
been
use
lines the
color-redshift spectrum.
with
data
and color
we shall
determining
radio-spectrum
have
redshift
material,
corrected
translation.
supplemented
355 have
this
mean intensity
colours,
separately
[5],
Of these,
To exploit
on the
ions
Quasars
separately effect
the
measurements
statistical
of
curves.
Lastly,
up to
end of (270
for the
steep-spectrum
emission
Thirdly,
we shall
derive
3,4,5
re-labelled
using the
lines these K-correct-
quasars.
omitted
and Figs.
with
methods.
1,2,3
K-Correction
The apparent
magnitude
of
a quasar
comes
mi p - 2.5 log where I(u)
is
photometric
the
band
Ib(vI and the
It
is
-i,
into
i representing
emission
the
line
that
statistics
of
lines,
the
their
sample
equivalent
the
following LINE
CIV
1549
CIII MgII
U, B,
S;(v)
v,
I(v) is
V.
is
the
sensitivity
made up of
the
the
main characteristics
which
the
consists
of
mainly
of
as we could
large In each
W.
effect
for
of
quasars
the
emission
the
line
[19-381
a determination
case
O,
of
of
a I-distribution
the
8
x2
W
17.1
62A
d.f.
91
1.48
0.584
10
9.2
38
31
110
1.86
0.490
10
14.3
57
41
the
fit
is
W and the
Lp
10
significant
standard
Mean Equivalent
LINE
of
and the
line For
frequency to
inten3 emission
distribution apply,
Widths
given
and Intensities
1026
and the
at 5% level,
deviation
5SA
by the
ci
last
two columns
give
P-distribution.
Emission
Lines
of
quasars
L, 1216
OVI 1034
-_ SPECTRA
steep
flat
steep
all
-t
40
s
3.2
=
flat
all
steep
50
36
257
166
157
1
14
3.3
2.9
4.0
1
6
4
6
j
flat
1
all
--
2
NUMBER =
42
i=
NV 1240
LINE
SiV 1397 i- OIV 1406
ov 1549
_SPECTfti
steep
1
flat
steep
1 all
1 flat
1 all
steep
/
flat
1 all
-w
83
68
70
a
2.0
1.2
1.8
11
12
93
N~BER -
very
we must look
U(W’)
2798
1
presence
spectra.
was observed
1909
Table
the
lines,
widths
0.304
of
the
x,8 h
size
show that
of
background
(x G 0)
values
value
is
on quasar
1.25
expected
continuum
Ieiv):
110
The X2-values
curve
(2)
sufficiently
Sample
(1)
frequency
statistically
I,ivi,
f
with
one of
width
formula
dv,
as much material
is
the
I,(v).
To eliminate
We collected
sities.
from
intensities
Ib(Y) f
well-known
Ie iv).
strong
I(Y) &(Y) I at the optical
flux
I(v) -
the
175
Quasars
THE EMISSION-LINE PORTION OF THE ~-CORRECTION
1.
of
for
K-Correction
176
for
Tahle LINE
1 (contd.)
He11 1640
I steep
SPECTRA
all
steep
40
NUllBER LINE
[NcV] 3426
SPECTRA
80
63
0.7
0.7
1 .o
0.72
0.9
7
22
61
16
35
107
steep
flat
steep ___~
all
-w
70
s
1.56
!IUPlBER -l-
30
50
35
1.2
2.5
1.1
1
4
4
1
SPECTRA
steep
1 flat
flat
30
25
33
0.4
1.4
0.43
2.2
2
13
18
I
1
3
Hg 4861
H, 4340 + [OIIIJ 4363
Ha 4102
LINE
steep
1 all
63
[NeIII] 3869 + HI 3968
[OII] 3727 all
all
flat
39
13
flat
steep
steep ---
53
51
0.5
all
I
0.6
40
s
flat
I
---
2
MgII 2798
[OIII] 1909
flat
iP
Quasars
steep
all
flat
flat
all
-~~
-~____
w B
60
38
29
30
31
0.3
0.3
0.32
0.40
0.57
1
3
NUMBER
5
11
86
92 0.5
28
86
0.9
0.9
8
44
17
= LINE
[OIII] 4959
SPEC’TRA steep
LOIII] 5007
flat
steep
all
_~~___ w
23
44
P
0.92
1.25
18 emission a
spectrum optical than
lines
objects
(a > 0.5),
resolving the
power
types.
Generally
objects
than
4100A.
The table
a certain speaking,
in the
having
66
80
530
462
1.4
2.5
2.26
2.2
11
9
25
shows
equivalent
given
objects
no better
the
mean intensity
1026A and 6563A. in TABLE I,
(a < 0.5)
than
10
and all
?A to lOA,
the mean equivalent
separately
for
objects.
we used
steepBecause
only
those
the
wider
w and F.
difference lines
flat-spectrum
also
between
7
ti is
in Angstroms.
B are
flat-spectrum is
135
-ngths
wav
intensit.
of
all
1.3
rest
involved
calculation
TABLE 1 reveals
some lines
with
mean relative
flat
---
mean equivalent width to the continuum.
and the
15A in
steep
all
28
1
3 D is the relative
For
flat
~-
NUIBER
width
H. 6563
that widths
in
shorter objects, the in
line than
intensity 41OOA are
while
emission the
range
reverse lines of
between stronger is
the
seriously hundreds
different
spectral
in
the
steep-spectrum
case
for
lines
affect of
radio
quasar
angstroms
longer photometry,
and intensities
than
K-Correction
times
the
background.
when we study
several
the
continuum.
The U, B, I/data vations
of
Mt.
Johnson-Morgan
Amf
In the
first
the
this
paper
and Palomar.
is
come mostly
The
177
important
from
photometric
to
the
system
Ibfvl
and
correct
60-, [39]
for
IOO-, is
the
emission
and ZOO-inch
about
the
same
lines
obser-
as
line
to
continuum
Table
correction
for
the
lines
is
given
by
(4)
we put Ih
(w) = 1,
and we took
I,(V)
from
the
data
of
TABLE I,
After we found the be an isosceles triangle in the rest-frame. (21 we applied (4) once more to (v), as described below, spectrum I
b
These
corrections.
2
emission
’
approximation
final
IQ(v), the
=
each
approximate give
in
it
Therefore,
Quasars
system.
When we know
regarding
used
Wilson
for
The
“emission-line
Emission-Line
T
lteep
Portion
T
Spectra(-Am)
of
Flat
are
K-corrections”
the
K-Correction
Spectra
(-
for
in TABLE 2.
Quasars
T All
Anz)
given
Spectra
(-Am)
Z
lJ
B
V
0.00
0.15
0.16
0.04
0.05
0.21
0.07
0.12
0.17
0.04
0.05
0.12
0.13
0.19
0.11
0.17
0.28
0.09
0.16
0.21
0.10
0.16
0.15
0.23
0.10
0.15
0.31
0.12
0.14
0.24
0.15
0.10
0.15
0.19
0.08
0.13
0.24
0.08
0.13
0.19
0.20
0.12
0.18
0.17
0.10
0.13
0.19
0.10
0.15
0.16
0.25
0.14
0.16
0.16
0.12
0.10
0.18
0.12
0.13
0.16
--
u
B
u
V
B
V
0.30
0.15
0.13
0.14
0.12
0.07
0.07
0.12
0.10
0.15
0.35
0.13
0.11
0.14
0.11
0.06
0.17
0.11
0.09
0.14
0.40
0.03
0.10
0.15
0.02
0.04
0.15
0.02
0.08
0.14
0.45
0
0.16
0.15
0
0.11
0.12
0
0.12
0.13
0.50
0
0.16
0.19
0
0.13
0.12
0
0.13
0.15
0.55
0.01
0.15
0.19
0.01
0.12
0.10
0.01
0.12
0.15
0.60
0.05
0.13
0.16
0.05
0.11
O.L)S
0.05
0.11
0.12
0.65
0.07
0.12
0.12
0.07
0.09
0.06
0.07
0.09
0.09
0.70
0.10
0.11
0.08
0.10
0.09
0.04
0.10
0.09
0.06
0.75
0.11
0.07
0.07
0.12
0.06
0.03
0.12
0.06
0.05
0.80
0.13
0.05
0.10
0.13
0.04
0.07
0.13
0.04
0.08
0.85
0.17
0.03
0.18
0.17
0.02
0.14
0.17
0.02
0.15
0.90
0.20
0.01
0.19
0.20
0.01
0.15
0.20
0.01
0.15
0.95
0.25
0.01
0.18
0.24
0.01
0.14
0.25
0.01
0.14
1 .oo
0.29
0.01
0.16
0.27
0.01
0.12
0.27
0.01
0.12
1.05
0.27
0.03
0.12
0.25
0.03
0.10
0.25
0.03
0.10
1.10
0.30
0.08
0.09
0.26
0.08
0.08
0.27
0.08
0.08
1.15
0.34
0.12
0.07
0.30
0.12
0.06
0.32
0.12
0.06
1.20
0.38
0.13
0.04
0.33
0.13
0.03
0.36
0.13
0.03
1.25
0.41
0.13
0.03
0.35
0.13
0.02
0.40
0.13
0.02
1.30
0.44
0.14
0.02
0.37
0.14
0.02
0.42
0.14
0.02
1.35
0.45
0.14
0
0.35
0.14
0
0.38
0.14
0
1.40
0.51
0.15
0
0.38
0.15
0
0.44
0.15
0
1.45
0.60
0.19
0
0.43
0.19
0
0.46
0.19
0
1.50
0.62
0.23
0
0.41
0.22
0
0.50
0.22
0
1.55
0.71
0.25
0.01
0.47
0.24
0.01
0.56
0.24
0.01
178
K-Correction for Quasars
Table 2 (contd.) tee?
2. Figs.
1
(-am)
All Spectra (-Am>
Z
u
B
Y
u
B
v
u
B
V
1.60
0.79
0.33
0.03
0.53
0.39
0.03
0.62
0.31
0.03
1.65
0.85
0.36
0.09
0.57
0.32
0.09
0.66
0.33
0.09
1.70
0.90
0.37
0.15
0.61
0.33
0.15
0.69
0.34
0.15
1.75
0.90
0.37
0.17
0.61
0.32
0.17
0.66
0.34
0.17
1.80
0.93
0.37
0.18
0.64
0.32
0.18
0.68
0.34
0.18
1.85
0.96
0.38
0.17
0.65
0.32
0.17
0.69
0.36
0.17 0.16
1.90
0.98
0.41
0.16
0.68
0.24
0.16
0.70
0.40
1.95
0.97
0.40
0.14
0.69
0.31
0.14
0.72
0.39
0.14
2.00
0.92
0.42
0.14
0.66
0.32
0.14
0.69
0.40
0.14
2.05
0.84
0.44
0.14
0.61
0.32
0.14
0.64
0.41
0.14
2.10
0.72
0.52
0.17
0.54
0.37
0.17
0.55
0.45
0.17
2.15
0.60
0.58
0.22
0.46
0.41
0.21
0.48
0.56
0.21
2.20
0.54
0.68
0.25
0.44
0.47
0.24
0.45
0.55
0.24
2.25
0.41
0.79
0.31
0.34
0.54
0.29
0.36
0.62
0.29
2.30
0.37
0.87
0.37
0.35
0.59
0.34
0.35
0.67
0.34
2.35
0.35
0.90
0.42
0.34
0.62
0.38
0.34
0.69
0.39
2.40
0.35
0.95
0.43
0.35
0.65
0.38
0.35
0.72
0.39
2.45
0.35
0.96
0.42
0.35
0.65
0.37
0.35
0.72
0.38
2.50
0.35
0.95
0.41
0.35
0.64
0.35
0.35
0.70
0.37
2.55
0.35
0.94
0.40
0.35
0.63
0.34
0.35
0.69
0.37
2.60
0.34
0.91
0.40
0.34
0.61
0.33
0.34
0.67
0.38
2.65
0.32
0.89
0.40
0.32
0.60
0.32
0.32
0.65
0.39
2.70
0.24
0.85
0.40
0.24
0.58
0.31
0.24
0.62
0.40
2.75
0.11
0.82
0.38
0.11
0.56
0.29
0.11
0.60
0.40
2.80
0.07
0.78
0.38
0.07
0.54
0.28
0.07
0.58
0.39
2.85
0.05
0.76
0.37
0.05
0.54
0.26
0.05
0.56
0.37
2.90
0.02
0.77
0.41
0.02
0.58
0.30
0.02
0.60
0.41
2.95
0
0.75
0.51
0
0.58
0.34
0
0.59
0.44
3 .oo
0.71
0.53
0.56
0.41
0.57
0.50
3.05
0.67
0.72
0.53
0.48
0.55
0.57
3.10
0.62
0.83
0.51
0.56
0.52
0.64
3.15
0.57
0.92
0.48
0.63
0.49
0.70
3.20
0.53
1 .oo
0.45
0.69
0.46
0.75
3.25
0.48
1.04
0.42
0.72
0.43
0.78
3.30
0.44
1.08
0.39
0.75
0.40
0.80
3.35
0.40
1.09
0.37
0.76
0.37
0.81
3.40
0.36
I .08
0.34
0.74
0.34
0.79
3.45
0.33
1.05
0.31
0.62
0.31
0.77
3.50
0.30
1.01
0.29
0.69
0.29
0.73
3.55
0.27
0.96
0.26
0.65
0.27
0.69
MEAN OPTICAL
A
Tlat Spectra
Spectra<-Am)
CONTINUUM SPECTRUM FOR QUASARS
and 2 show two colour-redshift diagrams for 355 quasars. The colours have been
corrected for (i) the galactic absorption and (ii) the "emission-line portion of the Kcorrections" of TABLE 2.
Correction (ii) distinguishes the present diagrams from previous
color-redshift diagrams such as given in [39]. We note that quasars of different radio
K-Correction
spectra
appear
summarized
to
follow
the
same trends
by the
quadratic
B-V
-
0.0187 + 0.3914 z -
U-B
-
-0.8195
Application
of
the
F-test
regression
in the
these
179
Quasars
figures.
These
trends
are
adequately
curves, 0.08297
-I- 0.3247 Z -
shows
for
22,
0.09054 22.
mean relations
cl -
kO.03
u -
kO.05
to
(51 (6
he highly
significant,
[30l
B-1. 1.0-
.
0.8-
.
.
.
-_0
I
Fig.
Figs. are
1 and 2. steep
cirlces
Eliminating continuum
spectrum are
for
the
power
over
crosses a.
the
two equations This
3.
2
diagrams
quasars.
Fig. Fitting
objects;
mean values
2 between
spectrum
Colour-redshift
is
are
for flat
spectrum curves
(5)
gives
and (6)
shown in Fig.
the various
355 quasars.
The fitting
A mean continuum
law I = v-' to
I
spectrum ranges
Filled
objects; (5)
open
and (6)
effectively
circles
are
a mean optical
3.
for
shown.
quasars.
gives
9001L-IlOOK,
a -
0.98;
29OOb;-38OOh;,
a -
0.73;
llOO~--29006;,
a -
1.30;
38OOA-55OOb;,
a -
0.1.
K-Correction
180
Table
T 0.05
Flat
Kv
Z
0.01
K-Correction
T
Spectra
Steep
3
0
-0.t9
0 -0.01
0 0.01
for
0
-0.25
Quasars
for
Quasars
-r
Spectra
All
KB
KB
0 0
Spectra
0
0
0
-0.08
-0.21
-3.03
KU
0 0.01
-0.33
-0.03
-0.0’1
-0.29
-0.06
-0.04
-0.31
-0.04
-0.08
-0.29
-0;08
-0.01
-0.03
-0.30
-0.06
-0.11
-0.30
-0.12
-0.04
-0.05
-0.33
-0.05
-0.13
-0.34
-0.12
-0.06
-0.06
-0.25
-0.04
-0.13
-0.36
-0.11
-0.06
-0.03
-0.39
-0.04
-0.11
-0.39
-0.11
-0.04
-0.40
-0.03
-0.01
-0.42
-0.11
0.06
0.11
-0.40
-0.11
0.01
-0.44
-0.16
0.08
-0.21
0.12
-0.42
-0.13
0.02
-0.48
-0.17
0.09
-0.53
-0.20
0.13
-0.41
-0.12
0.01
-0.49
-0.16
0.10
0.60
-0.51
-0.18
0.10
-0.40
-0.11
0.00
-0.47
-0.15
0.07
0.65
-0.48
-0.15
0.09
-0.39
-0.0’1
-0.02
-0.45
-0.13
0.06
0.7c
-0.45
-0.15
0.07
-0.38
-0.08
-0.03
-0.43
-0.12
0.04
0.75
-0.45
-o.to
0.07
-0.38
-0.04
-0.04
-0.43
-0.08
0.03
0.80
-0.49
-0.08
0.07
-0.43
-0.02
-0.03
-0.47
-0.06
0.04
-0.05
O.lf
-0.28
-0.08
0.15
-0.29
-0.11
o.zc
-0.31
-0.16
0.25
-0.34
-0.15
0.3c
-0.35
-0.14
0.35
-0.39
-0.14
0.4c
-0.43
-0.14
0.08
0.45
-0.46
-0.21
0.5a
-0.52
0.55
0.00
0.85
-0.58
-0.05
0.04
-0.51
0.01
-0.06
-0.55
-0.03
0.01
0.90
-0.59
-0.02
0.03
-0.52
0.03
-0.07
-0.55
-0.01
0.00
0.95
-0.57
-0.01
-0.01
-0.50
0.04
-0.10
-0.53
-0.00
1.00
-0.55
-0.04
-0.48
0.05
-0.12
-0.51
0.01
-0.05
1.05
-0.51
-0.01
-0.46
0.04
-0.08
-0.49
0.00
-0.01
1.10
-0.47
-0.15
-0.02
-0.43
0.00
-0.03
-0.46
0.04
-0.02
1.15
-0.45
-0.08
-0.05
-0.41
-0.03
-0.11
-0.44
0.07
-0.06
0.00
0 .oo
-0.04
1.20
-0.41
-0.08
-0.07
-0.37
-0.03
-0.12
-0.40
-0.07
-0.08
1.25
-0.40
-0.07
-0.09
-0.36
-0.02
-0.13
-0.39
-0.06
-0.11
1.30
-0.38
-0.07
-0.11
-0.35
-0.02
-0.14
-0.38
-0.06
-0.12
1.35
-0.35
-0.06
-0.11
-0.32
-0.01
-0.11
-0.35
-0.05
-0.07
1.40
-0.34
-0.05
-0.16
-0.31
-0.13
-0.34
-0.04
-0.12
I .45
-0.33
-0.08
-0.24
-0.30
-0.03
-0.17
-0.33
-0.07
-0.13
1.50
-0.33
-0.11
-0.25
-0.30
-0.05
-0.14
-3.33
-0.09
-0.16
1.55
-0.36
-0.t2
-0.33
-0.31
-0.06
-0.19
-0.34
-0.10
-0.21
1.60
-0.34
-0.19
-0.40
-0.31
-0.20
-0.24
-0.34
-0.16
-0.26
1.65
-0.40
-0.21
-0.45
-0.37
-0.12
-0.27
-0.40
-0.17
-0.29
0.00
1.70
-0.44
-0.21
-0.49
-0.41
-0.12
-0.30
-0.44
-0.17
-0.31
1.75
-0.46
-0.20
-0.47
-0.43
-0.10
-0.28
-0.46
-0.16
-0.26
1.80
-0.46
-0.19
-0.48
-0.43
-0.09
-0.29
-0.46
-0.15
-0.26
1.85
-0.44
-0.19
-0.50
-0.41
-0.08
-0.29
-0.44
-0.16
-0.26
1.90
-0.41
-0.21
-0.53
-0.38
+o.ot
-0.31
-0.41
-0.19
-0.26
1.95
-0.38
-0.19
-0.49
-0.35
-0.05
-0.31
-0.38
-0.17
-0.27
2.00
-0.38
-0.20
-0.43
-0.35
-0.05
-0.27
-0.38
-0.17
-0.23
2.05
-0.37
-0.21
-0.25
-0.34
-0.04
-0.22
-0.37
-0.17
-0.18
2.10
-0.40
-0.28
-0.22
-0.37
-0.08
-0.14
-0.30
-0.20
-0.08
2.15
-0.43
-0.33
-0.10
-0.39
-0.11
-0.06
-0.42
-0.30
-0.01
2.20
-0.46
-0.42
-0.03
-0.42
-0.16
-0.03
-0.45
-0.28
0.03
K-Correction for Quasars
181
Table 3 (contd.)
-r
Steep Spectra KV
2
2.25 2.30 2.35 2.40 2.45 2.50 2.55 2.60 2.65 2.70 2.75 2.80 2.85 2.98 2.95 3.00 3.05 3.10 3.15 3.20 3.25 3.30 3.35
KU
-0.51 -0.57 -0.60 -0.60 -0.59 -0.57 -0.55 -0.55 -0.54 -0.53 -0.50 -0.49 -0.48 -0.50 -0.60 -0.61 -0.69 -0.90 -0.98 -1.04 -1.08 -1.11 -1.11 -1.10 3.4a 3.45 -1.07 3.5t -1.02 3.51 -0.96
3.
Flat Spectra
-0.52 -0.59 -0.60 -0.64 -.0.65 -0.63 -0.60 -0.57 -0.54 -0.49 -0.46 -0.42 -0.40 -0.40 -0.38 -0.33 -0.29 -0.24 -0.18 -0.14 -0.10 -0.06 -0.03 0.01 0.04 0.07 0.10
+0.11 0.15 0.17 0.17 0.17 0.17 0.17 0.18 0.20 0.28 0.41 0.45 0.47 0.49 0.50 0.49 0.49 0.49 0.48 0.48 0.47 0.47 0.47 0.47 0.47 0.47 0.47
KV
-0.46 -0.51 -0.53 -0.52 -0.51 -0.48 -0.46 -0.45 -0.43 -0.41 -0.38 -0.36 -0.34 -0.36 -0.40 -0.46 -0.52 -0.60 -0.66 -0.71 -0.73 -0.75 -0.75 -0.73 -0.61 -0.67 -0.62
KU
KV
KB
-0.22 -0.26 -0.27 -0.29 -0.29 -0.26 -0.24 -0.22 -0.20 -0.17 -0.15 -0.13 -0.13 -0.14 -0.14 -0.11 -0.08 -0.06 -0.02 -0.01 0.01 0.04 0.05 0.08 0.11 0.13 0.16
All Spectra
i-
0.08 0.07 0.08 0.07 0.07 0.07 0.07 0.08 0.10 0.18 0.31 0.35 0.37 0.39 0.40 0.39 0.39 0.39 0.38 0.38 0.37 0.37 0.37 0.37 0.37 0.37 0.37
-0.49 -0.54 -0.57 -0.56 -0.55 -0.53 -0.52 -0.53 -0.53 -0.53 -0.52 -0.50 -0.48 -0.50 -0.53 -0.58 -0.64 -0.75 -0.76 -0.80 -0.82 -0.83 -0.83 -0.81 -0.79 -0.75 -0.69
-0.34 -0.38 -0.38 -0.40 -0.40 -0.36 -0.34 -0.32 -0.29 -0.25 -0.23 -0.21 -0.19 -0.22 -0.21 -0.19 -0.17 -0.14 -0.10 -0.07 -0.04 -0.01 0.01 0.04 0.07 0.09 0.11
0.13 0.17 0.14 0.15 0.14 0.14 0.14 0.15 0.17 0.25 0.38 0.42 0.44 0.46 0.47 0.46 0.46 0.46 0.45 0.45 0.44 0.44 0.44 0.44 0.44 0.44 0.44
K-CORRECTIONS FOR QUASARS
The currently used formula for the K-correction is K = -0.751og
(1 + 2).
There are two problems. i)
(7)
It neglects the effect of emission lines and this is unreason-
able; ii) It was obtained using the mean radio spectral index o = 0.7.
Great caution must
be exercised in so extending the index to the optical range. Having now determined a mean optical continuum spectrum, we can now find the continuum portion of the K-correction according to the formula - 2.5 log (1 f z>, (81 0 suffixes Y and 0 referred to the rest and observed frequencies respectively. K = - 2.5 log +
where
Adding the values given by (8) to the values in TABLE 2 gives the total K-corrections for quasars. The results are given in TABLE 3.
Acknowledgement
We thank Professors Fang Li-zhi and Qu Qin-yue for guidance.
K-Correction
182
for
Quasars
REFERENCES
r11 [21 r31 r41 151 [61 r71 i-81 r91 [lOI 1111 Cl21 [I31 [I41 [I51 [I61 [I71 [I81 [I91 [201 [2Il [=I t231 [241 1251 [261 [271 [281 r291 1301 r311 [321 [331 [341 r351 [361 [371 [381 [391 [40]
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M. and
E. Landa, 2. Bstroph, 68(1968),
G. and L. Woltjer, H., et, al., 1.
Ann. New
York,
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N. E. A. S., 184(1978),
473. S., 224(1973),
8.
21.
Baldwin, J. A., Ap. J., 2l4(1977), 679. Burbi‘dge, G. R., et al., Ap. J. Supp& 33(1977), 113. Wright, A. E., et aL, Ap. J., n1(1977), L113. Ckmer, P. 8. and M. G. Smith, -%P. J., Zl3(1977), 607. Smith, E E., et al., Ap. J., 215(1977), 427. Osmer, P. S., & M. G. Smith, BP. J., 215(1977), L47. Schmidt, M., Ap. J., 217(1977), 358. Hoag, A. A., & M. G. Smith, BP. J., 2l7(197ij, 362. Osmer, P. S., Ap. J., 218(1977), L8Y. Peterson, B. A., & J. J. Condon, Ap. .I., 219 f 1978), Ll. Sramek, R A., & D. W. Weedman, Ap. J., 221(1978), 468. Wills, D., & R. Lpnds, Ap. J. Suppl, 36(1978), 317. Smith, A. L., A. Ap., 27(1977), 205. Adam, G., A. Ap., Suppl, 31(1978), 151. Savage,
A., et al., df. N. E. A. 8., 183(1978).
473.
Browne, I. W. A., et al., hf. N. R. A. S., 173(1975), 87. Browne, I. W. A., et al., M. N. R. A. S., 177(1976), 77. Browne, I. W. A., et al., M. N. R. A. S., 1?9(1977), 65. Savage, A., et al., M. N. R. A. S., 183(1978), 473. Oke, J. B., Ap. J., 141(1965), 10. Oke, J. B., Ap. J., 145(1966), 668. Oke, J. B., Ap. J., 161(1970), L17. Baldwin, J. A, et al., Ap. J., 185(1973), 739. Baldwin, J. A., et al., Ap. J., i95(1975), 269. Baldwin, J. A., et al., Ap. J., 206(1976). L83. Peterson, B. A., et al., Ap. J., 207(1976), L5. Osmer, P. S., & hf. G. Smith, Ap. J., 210(1976), 267. Peterson, B. A., et al., Ap. J., 211(1977), Lll5. Osmer, P. 5, & M. G. Smith, Ap. J., 2l3(1977), 607. Baldwin, J. A., Ap. J., 214(1977), 679. Smith, H. E, et al., Ap. J., 215(1977), 427. Osmer, P. S., & M. 0. &ith, Ap. J., 215(1977), L47. wer, P. S., Ap. J., 218(1977), L89. Peterson, B. A., et al., Ap. J., 219(1978), Ll. Baldwin, J. A., Ap. J., 226(1978), 8. Sandage, A., BP. J., 178(1972), 25. Department of Mathematics, Shanghai Normal University, Analysis and Experimental Design” (in Chinese).
“Regression
5 (1981) 174-182 -27 11980) 368-378
Chin.Astron.Astrophys.
Acta Astron. Sinica
Printed in Great Britain Ol46-6364/81/020174-09$07.50/O
A DETERMINATIONOF THE K-CORRECTION FOR QUASARS *
Xiao
Xing-hua,
Cao Sheng-lin
Cheng Fu-zhen [iniversity
Beijing (bservatory,and Yang Lan-tian
Tang Xiao-ying
Received
1979 September
continuum
line
widths
spectral
indices
emission
line
for
A knowledge
of
in elucidating the
the
continuum
example,
Grewing
and Woltjer’s
indices being
of
calculate
[l]
of
of
using
al
[3], of
have
the
[4]
College
Hubble
found
lines.
the
spectrum
of
nature.
In all
these
portion
presence
of
optical energy
index continuum
curve
studies,
however,
the
2
continuum,
is
important
the
study
quasars,
candle
strong
Setti
was further
quasars of
of
different
redshifts
continuum
sample
of
for
10 objects.
of
the
the
number of
of
assumption
between
for giving
quasars
from
properties
this
K-correction.
a large
as a standard
correlation
2,
However,
spectra
Under the
diagrams.
a strong
in
and radio
a mean optical
intrinsic
the
spectral
against
continuum
a synthesized
radio
plotted
the
by the
and a
determined
of
studied
who analysed
component
lJ,B,Vcolours
gives
continuum
made difficult
their
Baldwin
Teachers
B-V and U-B, corrected
these
the
line
iteratively
are
and their
obtained
et
emission
the
features
[2]
an examination
width
Z between
Many people
and Lamla
idea
K-correction,
prcperties
has been
lines.
cosmological,
and the
to
emission
by Netzer in
used
characteristic
spectrum
emission
the
are
redshifts,
The colors
355 quasars. of
an emission
two components
Eliminating
then
their
and wide
developed
is
made up of
and intensities,
portion
mean relations. which
is
These
component.
from
Huazhong
12
ABSTRACT The K-correction
paper
Beijing Normal University,
of Science and Technology,
luminosity
size
is
rather
restricted. Burbidge’s
1977 Catalog
of
1978,
contains
1038 quasars.
radio
spectral
indices).
First,
we shall
and flat
spectrum
find
the
objects.
Secondly,
we shall
find
curves
we shall
find
quasar
* Figs.
for
the
1 and 2 in in
this
the
the
different
the
abridged
original
after
of
the
after
continuum
emission
been
use
lines the
color-redshift spectrum.
with
data
and color
we shall
determining
radio-spectrum
have
redshift
material,
corrected
translation.
supplemented
355 have
this
mean intensity
colours,
separately
[5],
Of these,
To exploit
on the
ions
Quasars
separately effect
the
measurements
statistical
of
curves.
Lastly,
up to
end of (270
for the
steep-spectrum
emission
Thirdly,
we shall
derive
3,4,5
re-labelled
using the
lines these K-correct-
quasars.
omitted
and Figs.
with
methods.
1,2,3
K-Correction
The apparent
magnitude
of
a quasar
comes
mi p - 2.5 log where I(u)
is
photometric
the
band
Ib(vI and the
It
is
-i,
into
i representing
emission
the
line
that
statistics
of
lines,
the
their
sample
equivalent
the
following LINE
CIV
1549
CIII MgII
U, B,
S;(v)
v,
I(v) is
V.
is
the
sensitivity
made up of
the
the
main characteristics
which
the
consists
of
mainly
of
as we could
large In each
W.
effect
for
of
quasars
the
emission
the
line
[19-381
a determination
case
O,
of
of
a I-distribution
the
8
x2
W
17.1
62A
d.f.
91
1.48
0.584
10
9.2
38
31
110
1.86
0.490
10
14.3
57
41
the
fit
is
W and the
Lp
10
significant
standard
Mean Equivalent
LINE
of
and the
line For
frequency to
inten3 emission
distribution apply,
Widths
given
and Intensities
1026
and the
at 5% level,
deviation
5SA
by the
ci
last
two columns
give
P-distribution.
Emission
Lines
of
quasars
L, 1216
OVI 1034
-_ SPECTRA
steep
flat
steep
all
-t
40
s
3.2
=
flat
all
steep
50
36
257
166
157
1
14
3.3
2.9
4.0
1
6
4
6
j
flat
1
all
--
2
NUMBER =
42
i=
NV 1240
LINE
SiV 1397 i- OIV 1406
ov 1549
_SPECTfti
steep
1
flat
steep
1 all
1 flat
1 all
steep
/
flat
1 all
-w
83
68
70
a
2.0
1.2
1.8
11
12
93
N~BER -
very
we must look
U(W’)
2798
1
presence
spectra.
was observed
1909
Table
the
lines,
widths
0.304
of
the
x,8 h
size
show that
of
background
(x G 0)
values
value
is
on quasar
1.25
expected
continuum
Ieiv):
110
The X2-values
curve
(2)
sufficiently
Sample
(1)
frequency
statistically
I,ivi,
f
with
one of
width
formula
dv,
as much material
is
the
I,(v).
To eliminate
We collected
sities.
from
intensities
Ib(Y) f
well-known
Ie iv).
strong
I(Y) &(Y) I at the optical
flux
I(v) -
the
175
Quasars
THE EMISSION-LINE PORTION OF THE ~-CORRECTION
1.
of
for
K-Correction
176
for
Tahle LINE
1 (contd.)
He11 1640
I steep
SPECTRA
all
steep
40
NUllBER LINE
[NcV] 3426
SPECTRA
80
63
0.7
0.7
1 .o
0.72
0.9
7
22
61
16
35
107
steep
flat
steep ___~
all
-w
70
s
1.56
!IUPlBER -l-
30
50
35
1.2
2.5
1.1
1
4
4
1
SPECTRA
steep
1 flat
flat
30
25
33
0.4
1.4
0.43
2.2
2
13
18
I
1
3
Hg 4861
H, 4340 + [OIIIJ 4363
Ha 4102
LINE
steep
1 all
63
[NeIII] 3869 + HI 3968
[OII] 3727 all
all
flat
39
13
flat
steep
steep ---
53
51
0.5
all
I
0.6
40
s
flat
I
---
2
MgII 2798
[OIII] 1909
flat
iP
Quasars
steep
all
flat
flat
all
-~~
-~____
w B
60
38
29
30
31
0.3
0.3
0.32
0.40
0.57
1
3
NUMBER
5
11
86
92 0.5
28
86
0.9
0.9
8
44
17
= LINE
[OIII] 4959
SPEC’TRA steep
LOIII] 5007
flat
steep
all
_~~___ w
23
44
P
0.92
1.25
18 emission a
spectrum optical than
lines
objects
(a > 0.5),
resolving the
power
types.
Generally
objects
than
4100A.
The table
a certain speaking,
in the
having
66
80
530
462
1.4
2.5
2.26
2.2
11
9
25
shows
equivalent
given
objects
no better
the
mean intensity
1026A and 6563A. in TABLE I,
(a < 0.5)
than
10
and all
?A to lOA,
the mean equivalent
separately
for
objects.
we used
steepBecause
only
those
the
wider
w and F.
difference lines
flat-spectrum
also
between
7
ti is
in Angstroms.
B are
flat-spectrum is
135
-ngths
wav
intensit.
of
all
1.3
rest
involved
calculation
TABLE 1 reveals
some lines
with
mean relative
flat
---
mean equivalent width to the continuum.
and the
15A in
steep
all
28
1
3 D is the relative
For
flat
~-
NUIBER
width
H. 6563
that widths
in
shorter objects, the in
line than
intensity 41OOA are
while
emission the
range
reverse lines of
between stronger is
the
seriously hundreds
different
spectral
in
the
steep-spectrum
case
for
lines
affect of
radio
quasar
angstroms
longer photometry,
and intensities
than
K-Correction
times
the
background.
when we study
several
the
continuum.
The U, B, I/data vations
of
Mt.
Johnson-Morgan
Amf
In the
first
the
this
paper
and Palomar.
is
come mostly
The
177
important
from
photometric
to
the
system
Ibfvl
and
correct
60-, [39]
for
IOO-, is
the
emission
and ZOO-inch
about
the
same
lines
obser-
as
line
to
continuum
Table
correction
for
the
lines
is
given
by
(4)
we put Ih
(w) = 1,
and we took
I,(V)
from
the
data
of
TABLE I,
After we found the be an isosceles triangle in the rest-frame. (21 we applied (4) once more to (v), as described below, spectrum I
b
These
corrections.
2
emission
’
approximation
final
IQ(v), the
=
each
approximate give
in
it
Therefore,
Quasars
system.
When we know
regarding
used
Wilson
for
The
“emission-line
Emission-Line
T
lteep
Portion
T
Spectra(-Am)
of
Flat
are
K-corrections”
the
K-Correction
Spectra
(-
for
in TABLE 2.
Quasars
T All
Anz)
given
Spectra
(-Am)
Z
lJ
B
V
0.00
0.15
0.16
0.04
0.05
0.21
0.07
0.12
0.17
0.04
0.05
0.12
0.13
0.19
0.11
0.17
0.28
0.09
0.16
0.21
0.10
0.16
0.15
0.23
0.10
0.15
0.31
0.12
0.14
0.24
0.15
0.10
0.15
0.19
0.08
0.13
0.24
0.08
0.13
0.19
0.20
0.12
0.18
0.17
0.10
0.13
0.19
0.10
0.15
0.16
0.25
0.14
0.16
0.16
0.12
0.10
0.18
0.12
0.13
0.16
--
u
B
u
V
B
V
0.30
0.15
0.13
0.14
0.12
0.07
0.07
0.12
0.10
0.15
0.35
0.13
0.11
0.14
0.11
0.06
0.17
0.11
0.09
0.14
0.40
0.03
0.10
0.15
0.02
0.04
0.15
0.02
0.08
0.14
0.45
0
0.16
0.15
0
0.11
0.12
0
0.12
0.13
0.50
0
0.16
0.19
0
0.13
0.12
0
0.13
0.15
0.55
0.01
0.15
0.19
0.01
0.12
0.10
0.01
0.12
0.15
0.60
0.05
0.13
0.16
0.05
0.11
O.L)S
0.05
0.11
0.12
0.65
0.07
0.12
0.12
0.07
0.09
0.06
0.07
0.09
0.09
0.70
0.10
0.11
0.08
0.10
0.09
0.04
0.10
0.09
0.06
0.75
0.11
0.07
0.07
0.12
0.06
0.03
0.12
0.06
0.05
0.80
0.13
0.05
0.10
0.13
0.04
0.07
0.13
0.04
0.08
0.85
0.17
0.03
0.18
0.17
0.02
0.14
0.17
0.02
0.15
0.90
0.20
0.01
0.19
0.20
0.01
0.15
0.20
0.01
0.15
0.95
0.25
0.01
0.18
0.24
0.01
0.14
0.25
0.01
0.14
1 .oo
0.29
0.01
0.16
0.27
0.01
0.12
0.27
0.01
0.12
1.05
0.27
0.03
0.12
0.25
0.03
0.10
0.25
0.03
0.10
1.10
0.30
0.08
0.09
0.26
0.08
0.08
0.27
0.08
0.08
1.15
0.34
0.12
0.07
0.30
0.12
0.06
0.32
0.12
0.06
1.20
0.38
0.13
0.04
0.33
0.13
0.03
0.36
0.13
0.03
1.25
0.41
0.13
0.03
0.35
0.13
0.02
0.40
0.13
0.02
1.30
0.44
0.14
0.02
0.37
0.14
0.02
0.42
0.14
0.02
1.35
0.45
0.14
0
0.35
0.14
0
0.38
0.14
0
1.40
0.51
0.15
0
0.38
0.15
0
0.44
0.15
0
1.45
0.60
0.19
0
0.43
0.19
0
0.46
0.19
0
1.50
0.62
0.23
0
0.41
0.22
0
0.50
0.22
0
1.55
0.71
0.25
0.01
0.47
0.24
0.01
0.56
0.24
0.01
178
K-Correction for Quasars
Table 2 (contd.) tee?
2. Figs.
1
(-am)
All Spectra (-Am>
Z
u
B
Y
u
B
v
u
B
V
1.60
0.79
0.33
0.03
0.53
0.39
0.03
0.62
0.31
0.03
1.65
0.85
0.36
0.09
0.57
0.32
0.09
0.66
0.33
0.09
1.70
0.90
0.37
0.15
0.61
0.33
0.15
0.69
0.34
0.15
1.75
0.90
0.37
0.17
0.61
0.32
0.17
0.66
0.34
0.17
1.80
0.93
0.37
0.18
0.64
0.32
0.18
0.68
0.34
0.18
1.85
0.96
0.38
0.17
0.65
0.32
0.17
0.69
0.36
0.17 0.16
1.90
0.98
0.41
0.16
0.68
0.24
0.16
0.70
0.40
1.95
0.97
0.40
0.14
0.69
0.31
0.14
0.72
0.39
0.14
2.00
0.92
0.42
0.14
0.66
0.32
0.14
0.69
0.40
0.14
2.05
0.84
0.44
0.14
0.61
0.32
0.14
0.64
0.41
0.14
2.10
0.72
0.52
0.17
0.54
0.37
0.17
0.55
0.45
0.17
2.15
0.60
0.58
0.22
0.46
0.41
0.21
0.48
0.56
0.21
2.20
0.54
0.68
0.25
0.44
0.47
0.24
0.45
0.55
0.24
2.25
0.41
0.79
0.31
0.34
0.54
0.29
0.36
0.62
0.29
2.30
0.37
0.87
0.37
0.35
0.59
0.34
0.35
0.67
0.34
2.35
0.35
0.90
0.42
0.34
0.62
0.38
0.34
0.69
0.39
2.40
0.35
0.95
0.43
0.35
0.65
0.38
0.35
0.72
0.39
2.45
0.35
0.96
0.42
0.35
0.65
0.37
0.35
0.72
0.38
2.50
0.35
0.95
0.41
0.35
0.64
0.35
0.35
0.70
0.37
2.55
0.35
0.94
0.40
0.35
0.63
0.34
0.35
0.69
0.37
2.60
0.34
0.91
0.40
0.34
0.61
0.33
0.34
0.67
0.38
2.65
0.32
0.89
0.40
0.32
0.60
0.32
0.32
0.65
0.39
2.70
0.24
0.85
0.40
0.24
0.58
0.31
0.24
0.62
0.40
2.75
0.11
0.82
0.38
0.11
0.56
0.29
0.11
0.60
0.40
2.80
0.07
0.78
0.38
0.07
0.54
0.28
0.07
0.58
0.39
2.85
0.05
0.76
0.37
0.05
0.54
0.26
0.05
0.56
0.37
2.90
0.02
0.77
0.41
0.02
0.58
0.30
0.02
0.60
0.41
2.95
0
0.75
0.51
0
0.58
0.34
0
0.59
0.44
3 .oo
0.71
0.53
0.56
0.41
0.57
0.50
3.05
0.67
0.72
0.53
0.48
0.55
0.57
3.10
0.62
0.83
0.51
0.56
0.52
0.64
3.15
0.57
0.92
0.48
0.63
0.49
0.70
3.20
0.53
1 .oo
0.45
0.69
0.46
0.75
3.25
0.48
1.04
0.42
0.72
0.43
0.78
3.30
0.44
1.08
0.39
0.75
0.40
0.80
3.35
0.40
1.09
0.37
0.76
0.37
0.81
3.40
0.36
I .08
0.34
0.74
0.34
0.79
3.45
0.33
1.05
0.31
0.62
0.31
0.77
3.50
0.30
1.01
0.29
0.69
0.29
0.73
3.55
0.27
0.96
0.26
0.65
0.27
0.69
MEAN OPTICAL
A
Tlat Spectra
Spectra<-Am)
CONTINUUM SPECTRUM FOR QUASARS
and 2 show two colour-redshift diagrams for 355 quasars. The colours have been
corrected for (i) the galactic absorption and (ii) the "emission-line portion of the Kcorrections" of TABLE 2.
Correction (ii) distinguishes the present diagrams from previous
color-redshift diagrams such as given in [39]. We note that quasars of different radio
K-Correction
spectra
appear
summarized
to
follow
the
same trends
by the
quadratic
B-V
-
0.0187 + 0.3914 z -
U-B
-
-0.8195
Application
of
the
F-test
regression
in the
these
179
Quasars
figures.
These
trends
are
adequately
curves, 0.08297
-I- 0.3247 Z -
shows
for
22,
0.09054 22.
mean relations
cl -
kO.03
u -
kO.05
to
(51 (6
he highly
significant,
[30l
B-1. 1.0-
.
0.8-
.
.
.
-_0
I
Fig.
Figs. are
1 and 2. steep
cirlces
Eliminating continuum
spectrum are
for
the
power
over
crosses a.
the
two equations This
3.
2
diagrams
quasars.
Fig. Fitting
objects;
mean values
2 between
spectrum
Colour-redshift
is
are
for flat
spectrum curves
(5)
gives
and (6)
shown in Fig.
the various
355 quasars.
The fitting
A mean continuum
law I = v-' to
I
spectrum ranges
Filled
objects; (5)
open
and (6)
effectively
circles
are
a mean optical
3.
for
shown.
quasars.
gives
9001L-IlOOK,
a -
0.98;
29OOb;-38OOh;,
a -
0.73;
llOO~--29006;,
a -
1.30;
38OOA-55OOb;,
a -
0.1.
K-Correction
180
Table
T 0.05
Flat
Kv
Z
0.01
K-Correction
T
Spectra
Steep
3
0
-0.t9
0 -0.01
0 0.01
for
0
-0.25
Quasars
for
Quasars
-r
Spectra
All
KB
KB
0 0
Spectra
0
0
0
-0.08
-0.21
-3.03
KU
0 0.01
-0.33
-0.03
-0.0’1
-0.29
-0.06
-0.04
-0.31
-0.04
-0.08
-0.29
-0;08
-0.01
-0.03
-0.30
-0.06
-0.11
-0.30
-0.12
-0.04
-0.05
-0.33
-0.05
-0.13
-0.34
-0.12
-0.06
-0.06
-0.25
-0.04
-0.13
-0.36
-0.11
-0.06
-0.03
-0.39
-0.04
-0.11
-0.39
-0.11
-0.04
-0.40
-0.03
-0.01
-0.42
-0.11
0.06
0.11
-0.40
-0.11
0.01
-0.44
-0.16
0.08
-0.21
0.12
-0.42
-0.13
0.02
-0.48
-0.17
0.09
-0.53
-0.20
0.13
-0.41
-0.12
0.01
-0.49
-0.16
0.10
0.60
-0.51
-0.18
0.10
-0.40
-0.11
0.00
-0.47
-0.15
0.07
0.65
-0.48
-0.15
0.09
-0.39
-0.0’1
-0.02
-0.45
-0.13
0.06
0.7c
-0.45
-0.15
0.07
-0.38
-0.08
-0.03
-0.43
-0.12
0.04
0.75
-0.45
-o.to
0.07
-0.38
-0.04
-0.04
-0.43
-0.08
0.03
0.80
-0.49
-0.08
0.07
-0.43
-0.02
-0.03
-0.47
-0.06
0.04
-0.05
O.lf
-0.28
-0.08
0.15
-0.29
-0.11
o.zc
-0.31
-0.16
0.25
-0.34
-0.15
0.3c
-0.35
-0.14
0.35
-0.39
-0.14
0.4c
-0.43
-0.14
0.08
0.45
-0.46
-0.21
0.5a
-0.52
0.55
0.00
0.85
-0.58
-0.05
0.04
-0.51
0.01
-0.06
-0.55
-0.03
0.01
0.90
-0.59
-0.02
0.03
-0.52
0.03
-0.07
-0.55
-0.01
0.00
0.95
-0.57
-0.01
-0.01
-0.50
0.04
-0.10
-0.53
-0.00
1.00
-0.55
-0.04
-0.48
0.05
-0.12
-0.51
0.01
-0.05
1.05
-0.51
-0.01
-0.46
0.04
-0.08
-0.49
0.00
-0.01
1.10
-0.47
-0.15
-0.02
-0.43
0.00
-0.03
-0.46
0.04
-0.02
1.15
-0.45
-0.08
-0.05
-0.41
-0.03
-0.11
-0.44
0.07
-0.06
0.00
0 .oo
-0.04
1.20
-0.41
-0.08
-0.07
-0.37
-0.03
-0.12
-0.40
-0.07
-0.08
1.25
-0.40
-0.07
-0.09
-0.36
-0.02
-0.13
-0.39
-0.06
-0.11
1.30
-0.38
-0.07
-0.11
-0.35
-0.02
-0.14
-0.38
-0.06
-0.12
1.35
-0.35
-0.06
-0.11
-0.32
-0.01
-0.11
-0.35
-0.05
-0.07
1.40
-0.34
-0.05
-0.16
-0.31
-0.13
-0.34
-0.04
-0.12
I .45
-0.33
-0.08
-0.24
-0.30
-0.03
-0.17
-0.33
-0.07
-0.13
1.50
-0.33
-0.11
-0.25
-0.30
-0.05
-0.14
-3.33
-0.09
-0.16
1.55
-0.36
-0.t2
-0.33
-0.31
-0.06
-0.19
-0.34
-0.10
-0.21
1.60
-0.34
-0.19
-0.40
-0.31
-0.20
-0.24
-0.34
-0.16
-0.26
1.65
-0.40
-0.21
-0.45
-0.37
-0.12
-0.27
-0.40
-0.17
-0.29
0.00
1.70
-0.44
-0.21
-0.49
-0.41
-0.12
-0.30
-0.44
-0.17
-0.31
1.75
-0.46
-0.20
-0.47
-0.43
-0.10
-0.28
-0.46
-0.16
-0.26
1.80
-0.46
-0.19
-0.48
-0.43
-0.09
-0.29
-0.46
-0.15
-0.26
1.85
-0.44
-0.19
-0.50
-0.41
-0.08
-0.29
-0.44
-0.16
-0.26
1.90
-0.41
-0.21
-0.53
-0.38
+o.ot
-0.31
-0.41
-0.19
-0.26
1.95
-0.38
-0.19
-0.49
-0.35
-0.05
-0.31
-0.38
-0.17
-0.27
2.00
-0.38
-0.20
-0.43
-0.35
-0.05
-0.27
-0.38
-0.17
-0.23
2.05
-0.37
-0.21
-0.25
-0.34
-0.04
-0.22
-0.37
-0.17
-0.18
2.10
-0.40
-0.28
-0.22
-0.37
-0.08
-0.14
-0.30
-0.20
-0.08
2.15
-0.43
-0.33
-0.10
-0.39
-0.11
-0.06
-0.42
-0.30
-0.01
2.20
-0.46
-0.42
-0.03
-0.42
-0.16
-0.03
-0.45
-0.28
0.03
K-Correction for Quasars
181
Table 3 (contd.)
-r
Steep Spectra KV
2
2.25 2.30 2.35 2.40 2.45 2.50 2.55 2.60 2.65 2.70 2.75 2.80 2.85 2.98 2.95 3.00 3.05 3.10 3.15 3.20 3.25 3.30 3.35
KU
-0.51 -0.57 -0.60 -0.60 -0.59 -0.57 -0.55 -0.55 -0.54 -0.53 -0.50 -0.49 -0.48 -0.50 -0.60 -0.61 -0.69 -0.90 -0.98 -1.04 -1.08 -1.11 -1.11 -1.10 3.4a 3.45 -1.07 3.5t -1.02 3.51 -0.96
3.
Flat Spectra
-0.52 -0.59 -0.60 -0.64 -.0.65 -0.63 -0.60 -0.57 -0.54 -0.49 -0.46 -0.42 -0.40 -0.40 -0.38 -0.33 -0.29 -0.24 -0.18 -0.14 -0.10 -0.06 -0.03 0.01 0.04 0.07 0.10
+0.11 0.15 0.17 0.17 0.17 0.17 0.17 0.18 0.20 0.28 0.41 0.45 0.47 0.49 0.50 0.49 0.49 0.49 0.48 0.48 0.47 0.47 0.47 0.47 0.47 0.47 0.47
KV
-0.46 -0.51 -0.53 -0.52 -0.51 -0.48 -0.46 -0.45 -0.43 -0.41 -0.38 -0.36 -0.34 -0.36 -0.40 -0.46 -0.52 -0.60 -0.66 -0.71 -0.73 -0.75 -0.75 -0.73 -0.61 -0.67 -0.62
KU
KV
KB
-0.22 -0.26 -0.27 -0.29 -0.29 -0.26 -0.24 -0.22 -0.20 -0.17 -0.15 -0.13 -0.13 -0.14 -0.14 -0.11 -0.08 -0.06 -0.02 -0.01 0.01 0.04 0.05 0.08 0.11 0.13 0.16
All Spectra
i-
0.08 0.07 0.08 0.07 0.07 0.07 0.07 0.08 0.10 0.18 0.31 0.35 0.37 0.39 0.40 0.39 0.39 0.39 0.38 0.38 0.37 0.37 0.37 0.37 0.37 0.37 0.37
-0.49 -0.54 -0.57 -0.56 -0.55 -0.53 -0.52 -0.53 -0.53 -0.53 -0.52 -0.50 -0.48 -0.50 -0.53 -0.58 -0.64 -0.75 -0.76 -0.80 -0.82 -0.83 -0.83 -0.81 -0.79 -0.75 -0.69
-0.34 -0.38 -0.38 -0.40 -0.40 -0.36 -0.34 -0.32 -0.29 -0.25 -0.23 -0.21 -0.19 -0.22 -0.21 -0.19 -0.17 -0.14 -0.10 -0.07 -0.04 -0.01 0.01 0.04 0.07 0.09 0.11
0.13 0.17 0.14 0.15 0.14 0.14 0.14 0.15 0.17 0.25 0.38 0.42 0.44 0.46 0.47 0.46 0.46 0.46 0.45 0.45 0.44 0.44 0.44 0.44 0.44 0.44 0.44
K-CORRECTIONS FOR QUASARS
The currently used formula for the K-correction is K = -0.751og
(1 + 2).
There are two problems. i)
(7)
It neglects the effect of emission lines and this is unreason-
able; ii) It was obtained using the mean radio spectral index o = 0.7.
Great caution must
be exercised in so extending the index to the optical range. Having now determined a mean optical continuum spectrum, we can now find the continuum portion of the K-correction according to the formula - 2.5 log (1 f z>, (81 0 suffixes Y and 0 referred to the rest and observed frequencies respectively. K = - 2.5 log +
where
Adding the values given by (8) to the values in TABLE 2 gives the total K-corrections for quasars. The results are given in TABLE 3.
Acknowledgement
We thank Professors Fang Li-zhi and Qu Qin-yue for guidance.
K-Correction
182
for
Quasars
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