thickness estimates of spiral galaxies
TRANSCRIPT
Chin.Aatron.Astrophys. 5 (1982) 76-78 Pergamon Press. Printed in Great Britain
Act .Astrophys.Sin. 2 (19823 IO-34 0275-1062/~2~0~0076-o4$o7,5o/o
THICKNESS ESTIMATES OF SPIRAL GALAXIES
TONG Yi, WU Sheng-gu kpartment of Astronomy, Bei_jing paormal University, PENG Qiu-he Department of Astronomy, Nanjing ?Mversity.
Received 1981 February 18.
ABSTRACT. Using the rigorous solution for the perturbing potential obtained in Ref. /l/, which contains a non-wavy term, we found a relation between the thickness of the galaxy (H= Z/a> and the radius r ,, at which the arms start,oro = 7. From Ref. /2f, we selected 50 spiral galaxies, found their r0 values, hence their average thickness.
1. DERIVATION OF FORMULA
According to Ref. /l/, on the plane of symmetry (z =O) of a galaxy, the perturbing potential has the rigorous solution
where cy is the reciprocal of the half-thickness, and other symbols have their customary meanings. Consider the terms inside the square brackets. When or>>l, we can neglect second term in
J,(W) --&L*(W)= (-l)Yf$?(or)- $ (2)
similarly, when [klr>>l, we can neglect the second term in
&(14/r) -iH-,(lrilr) = (-l)W$'(I4/r) - -i; (3)
the
P-1
hz r(s+f>
( 1 r --m+----D 2 >( ) y Il+n*l
Under these conditions, (1) reduces to a uniquely aerlned wave function in step with the density wave,
where
Under the potential (4), spiral arms will form on the disk. Let us now further consider just how large /k/r or or must be for the last terms (the non
-wavy terms) to be negligible. It is only when these terms are small compared-to the preceding terms, that a unique wave solution of form (4) can be obtained, and spiral arms can form. Physically, it is only when the wavy portion of the potential is much greater than the non-wavy portion, that arms can result. Now, for large of, we need only consider the first term in the series in (2),
Thicknesses of Spiral Galaxies
r(+) - 1 + + $) w-’
77
Most galaxieshave 2 arms, so we take m=2. If we suppose that the non-wavy term becomes negligible when it is less than one-fifth of the wavy term.
2 then we find
oto& 7 Similarly we have
(7)
iklro>7 (8) This is to say, in the region where both these conditions are satisfied, spiral arms will be seen, the more clearly, the further out towards the edge; near the centre, where neither is satisfied, there will be no arms; in between the two regions where one of the two conditions is satisfied, arms may be possible but will not be very clear. This discussion is in agreement with the observations.
2. DISCUSSIONS ON THE SPIRAL ARMS OF THE MILKY WAY
We now take the Milky Way as an example and discuss the question of spiral arms further. The reciprocal half-thickness of the Milky Way at r=OKpc is cr=1.7Kpc-1. Hence form (7) we find ro24kpc. That is, spiral arms can appear outside the radius ofrl4Kpc, inside it, they cannot form because of the effect of thickness. generally take IkI e 1JW4.
Besides, for the Milky Way,we Then from (8) we get again &4Kpc. Thus, the starting positions
calculated from the two conditions turn out to be about the same; at this position, the arms may not yet be very clear or stable, and there may be outward ejection of matter. This is about the location of the u 3-kpc arm" . Thus, our theory satisfactorily explains of the location of the arms formation in the Milky Way, and changes the interpretation so far held about the 3-kpc arm being a material arm : we would assert that it is a density wave. We can now reverse the argument and apply it to other galaxies: from the observed locations of arms formation r0, we shall obtain their thicknesses using the relation or0=7.
3. AN ERROR ESTIMATE
First, we must point out that the value or=7 is a lower limit for the appearance of spiral arms. Consider a disk galaxy with the property that arms just hegin to (or fail to) form at the edge at radius R. Then according to (7), we shot~l~l have ff R = 7, and the oblateness of the system should then be H/2R=11?. But this is pr~'~is<~ly the oblateness of an E7 galaxy, and an E7galaxy is the closest to a disk $::rl:~xy without having any spiral arms. Therefore, or = 7 is a lower limit, for othcrwisc WC' wol~ld find spiral galaxies that are less flattened than some of the ellipticals. Having extablished the lower limit character, we now ask, can we take a larger value P The answer is yes, for example, if we take (non- wavy term) / (wavy term) = l/10, then we shnll have or, 29. - GJith a ratio l/10, the arms should be quite distinct, and there is no need to push any further in this direction, Thus, a range of l/5 - l/10 in the definition of small quantity corresponds to an error of -20X in the deduced thickness.
4. RESULTS
We selected 50 spiral galaxies from "The Hubble Atlas of Galaxies" 121. From each picture we find the radius (in mm.) at which the arms appear, convert it to the angular measure rO" with the scale provided, then using the distance D given in 131, find the true radius r0 in kpc. Lastly we find the thickness H from ~~2f~=2,&7 These values are all given in TABLE 1.
TABLE 1 shows that the great majority of the thicknesses we measured are aobout the same as that of the Milky Way; a large number are somewhat thinner. For different morphological types, the general tendency is that the Sa's are the thickest, the Sh's next, and the So's are the thinnest; the average values are, respectively 0.92 kpc, 0.77 kpc and 0.40 kpc.This tendency is in accordance with the evolution of galaxy morphology.
7%
TABLE 1.
NGC No.
% 3031
3368
35M
3623
371%
4594
4725
4736
4826
4941
7217
5 157
224
48%
891
106%
1097
13QQ
1832
I964
2841
2903
3351
3S21
3627
42s
43Q3
432L
4005
.5OSS
S194
324%
S333
6931
7314
;331
5,
253
S9R
628
1084
1637
24Q3
33S9
35s
5236
SB?
5962
6643
7604
7741
REFERENCES
mass 10'WQ
12.50
28.22
1.65
lo.8 23.61
30,oo
7.60
31.50
10.00
3.88
3.84
9.m
33.52
27.20
10,Qu
22.90
74.9
9.95
8.4
2.75
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26.24
16.22
5.77
9.m
57.62
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5.95
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16.42
6.90
6X.15
9.77
31.99
12.5
3.31
7.S
0.99
11.4
3.27
12.m
9.7Q
39.39
38.2
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3.6
5.51
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scale v /pm
3.69
1.84
1.88
5.34
5.32
1.68
I.27
3.46
7.34
2.66
2.76
1.1%
25.7
2.76
3.21
2.05
5.16
I.47
2.22
2.76
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3.69
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2.22
2.01
5.04
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4.49
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1.86
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3.69
1.17
5.34
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3.69
6.78
4.85
0.92
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239.85
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117.48
26.6
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[lj TONG Yi et al. Srtientia Sinica 10 (1981) 1233. [.?I A.Sandage, " The Hubble Atlas o?-Galaxies". -._ -.-
3,488
1.949
6.497
4.329
I.354
12.706
4.282
1.087
2.122
1.393
3.8%
3.089
6.018
2.79
2.629
1.487
i3.60
1.411
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I.686
2.296
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3.22
f.367
3.123
a.767
2.150
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0.387
3.630
I.223
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5.425
3.886
0.403
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5.656
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0.614
5.833
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0.506
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