A semi-empirical model of sunspot umbra

Chin.Astron.Astrophys.(l991)15/1,28-36

38 Pergamon Press plc Printed in Great Britain 0275-1062/91$10.00+.00

A translation of Acta Astron.Sin. (1990)31/3,276-283

A SEMI-EMPIRICAL MODEL OFSUNSPOT UMBRA1 DING Ming-de’

FANG Cheng’*’

'Department of Astronomy, Nanjing University 'CCAST (World Laboratory), Beijing, China Received

1989 May 9

ABSTRACT Based on data obtained with the Kitt Peak Observatory Solar Tower, we have constructed a semi-empirical model of sunspot umbra. Non-LTE calculation was used. The main features are: (1) The temperature structure of the corona and the transition region is similar to the VAL3C quiet sun model. (2) The chromosphere structure is characterized by a temperature plateau; the temperature in the upper layers iehigher than those of the quiet sun and the penumbra; (3) The minimum temperature is 3130 K, about 300 K less than that of the penumbra model. (4) the temperature structure of the photosphere is similar to those in previous models, while the micro-turbulent velocity is somewhat less, We find there is no radiative balance in the whole atmosphere. The net radiative loss rate has a maximum in the middle chromosphere and it is negative in the temperature minimum region.

Key words: The Sun-sunspotsloss

model atmosphere-radiative

1. INTRODUCTION Sunspots are an important phenomenon of solar activity. The existence of a strong magnetic field in a sunspot means that its atmosphere has an entirely different structure from the quiet sun. Many debates remain as regards sunspots such as their formation and of the the equilibrium of their atmosphere. Further investigations sunspot energy and dynamical process requires a clearer understanding of the structure of its atmosphere. Consequently, the construction of model sunspot atmosphere semi-empirically, with the help of observed spectra has become a popular subject. The study of sunspot models began in the fifties. Some of the early models which depended on slight observational data and simple assumptions have ceased to have any reference value (21. Following the appearance of high-resolution techniques and the continuing there has been a recent improvement in the theoretical analysis, surge of new models. They can be roughly divided into two classes:

Program supported

by the National

Natural

Science

Foundation

Umbra

Model

29

one uses the continuum spectrum, e.g., Refs. [2-51; the other uses spectral lines, as in 16-91. As the continuum is formed mostly in the photosphere, it does not enable us to derive the structure above the chromosphere. Again, if the number of spectral lines used is too small, it will lead to a high degree of non-uniqueness of the model. Thus, to ensure reliability, we should use a good number of spectral lines. In our previous paper [lo] we have successfully constructed a semi-empirical model for the atmosphere of sunspot penumbra using seven spectral lines. In this paper, we shall construct a model for the umbra under the same conditions.

2. OBSERVATIONALDATA On 1985 July 1, the solar disk showed an oval sunspot at p = 0.743. Its size was 10” x 14” and the umbra and the penumbra were clearly distinct. Using the 13.5-•etre vacuum spectrograph of Kitt Peak Observatory, USA, we obtained for the umbra seven line profiles (Ha, Hp, Cal1 H, K, X8498, X8542, X8662). A photoelectric diode and an EM1 9750 photomultiplier were used as detectors and the slit size was 3 x 0.3 mm (the solar image had a diameter of 760mm). Each profile was scanned 15 times and the average was taken. TABLE 1 gives the observational parameters.

TABLE 1

The Observational

LINE

IME (UT)

Ha Hb CaII H CaII K A 8498 A 8542 A 8662

20:33:18 20:42:27 21:09:48 21:06:35 20:19:22 20:24:38 20:28:34

Parameters

FILTER PRED PRED t 557 PRED t 557 PRED t 557 PRED PRED PRED

DEECTOR

SP. ORDER

diode EM1 9750 EM1 9750 EM1 9750 diode diode diode

The photoelectric observation has a high accuracy. The seeing at the time of observation was very good; by eye estimate, the trembling of the solar image was about l”, the sky was deep blue and the halo was very small, hence the scattered light (estimated to be about 1%) had a rather small effect on the observation. The final line profiles obtained are characterized by the following features: (1)

The wavelength range is quite extended; profile extends some f 3A to the wings.

(2)

There are no obvious

(3)

Strong

emission

from the core

the

asymmetries.

peaks are seen in the core of CaII

H and K.

30

3.

DING

& FANG

THEORY AND METHODOF MODELCALCULATION

We adopt the plane parallel static model; that is, the temperature T, the electron density ne, the micro-turbulent velocity vt are all one-dimenisonal functions of the column density III. For hydrogen and ionized calcium, we use non-LTE calculation. Some of the assumptions are: (1) According to the discussion in [lo], twist-free magnetic field has no effect hydrostatic equilibrium is well obtained

an axisymmetric, on the umbra atmosphere in this case.

and

(2) For hydrogen and ionized calcium, we used atmoic models with 5 bound states and one free state. Also, LTE treatment was applied to the energy levels 6 to 12 of hydrogen. In the calculation of the absorption coefficients of the different lines, we also included the contributions from H(f-f), H-(b-f) and H-(f-f). The atomic parameters were taken from (11-141. (3) The equation of radiative transfer was expressed in terms of second-order central differences. The boundary conditions were the same form as in 1151. (4) For the contributions to the electron density, in addition to we also included those from 10 metals that of the ionized hydrogen, and 9 non-metals under the LTE approximation. The various parameters were taken from [161. (5) As the sunspot temperature is relatively density is small, we can neglect the Stark line profile, then, we considered only the Van der Waals broadening and the resonance broadening parameters were taken from [13]

Fig.

1

low, and the electron broadening. For each Doppler broadening, the broadening. The various and [171.

Temperature profiles in the umbra (solid), penumbra and the VAL quiet sun (dot-dash) atmospheres

(dashed)

Umbra Model

31

The entire calculation proceeded as follows: for fixed initial model values T(a) and vr(a), we solved the equations of hydrostatic equilibrium, statistical equilibrium and radiative transfer by successive iteration, stopping the iteration when convergence was achieved. We then calculated the intensities of the various lines and by comparing with the observed intensities, we adjusted T(m) and vi(m), and repeated the calculation until the calculated and the observed intensities agreed. 4. THE IJWBRA MODELANDDISCUSSION After repeated adjustment and calculation we obtained our semi-empirical model for the atmosphere of sunspot umbra. TABLE2 lists in detail the variations of the various paremeters with height. Fig. 1 shows the tmperature as a function of the column density for the present umbra node1 , as well as for the penumbra model of [lOI and the quiet sun model of [lB]. The calculated and observed fine profiles are compared in Fig. 2. To eliminate the very strong self-reversal in the core of the calculated H and K lines, we used the usual Kneer and Mattig method 161 and with a final smoothing by a macroscopic turbulent velocity gauseian distribution of 8 km/s amplitude. The observations already showed the appearance of strong enission peaks in the cores of H and K. This led to the umbra temperature at the higher levels of the chromosphere being higher than the penumbra temperature. This is a very interesting phenomenon; it was qualitatively implicit in previous umbra models. The chromosphere temperature structure in sunspot models has always proved to be either one of two types, one is the “plateau” structure 12, 91, the other is the gradient” structure [6, S]. Lites and Skuaanich 191 have pointed out that the gradient structure cannot give agreement between calculation and observation for both the CaII and the HgII lines, while the plateau structure will lead to over-strong central self-reversals in the core of H and K lines. Since the central self-reversal can be decreased by suitably adjusting the siee of the turbulent velocity, we opted for the plateau structure with, however, a smaller width than in previous models. The temperature minimum region is the region of formation of minimum Hr and Kr. Calculations showed that the intensity of Hi and K1 is closely related to the value of I&n, while their distance from the core is related to the height of the region. The value of I’min we determined was 313OK, close to the result of Kneer and Mattig 161. The photospheric region has always been the focus of attention in because this is the region where the model sunspot construction, sunspot differs most from the quiet sun. The photosphere temperature profile in the present model does not differ much from velocity profile is the previous models, while the micro-turbulence more different. We found that the micro-turbulence exerts an extremely large effect on the half-widths of Cal1 lines

DING h FANG

32

TABLE 2

The Sunspot Usbra Model Atmspbere

m

d

(A-‘)

63--v

T

r..,

fi*

(cm-‘)

%

(cm-‘)

1

1727

1.160-07

141000

6.77

2.412-10

5.940.4.09

5.392+09

1

1722

1.240-07

a9100

6.77

2.421-10

9.042+09

s.209+09

3

1716

1.400-07

30000

6.76

Z.SOI-IO

1.562+10

1.419+10

4

1712

1.550-07

37000

6.76

2,662-10

2.0(14+10

1.s94+10

5

1710

1.640-O?

3zuno

6.75

2.800-10

2,393+10

z.t77+10

6

1706

1.780-07

28000

6.75

3.152--10

2.714+10

1.477+lo

7

1705

1.940-07

255ou

6.74

3.790-10

2.Y60+10

2.716+10

a

1615

3.340-07

24200

6.72

l.ZaZ-09

3.154i.10

2.909+10

9

1640

6.530-07

2JSOII

~0.65

3*u47-011

3.363+10

3.114+10

10

1614

8.520-07

22500

6.61

5.927-09

3.570+10

3.332+10

II

1611

*.7',0-_(,7

2,1,011

6.61)

8.495-O')

3.7110+10

3.593+10

12

1610

fi.9lO-07

IB5""

Ct.60

P.tbi--09

4.105+10

4.154+,il

13

1607

9.230--07

12300

6.59

1. ,9B--08

4.794+10

6.%12+IO

14

1606

9.530-07

10200

6.18

1.279-oa

3.804+10

9.255+10

15

1592

1.290-06

9500

6.10

1.788-08

3.1S3flO

1.092+11

16

1576

1.761-06

a500

6.39

2.034-00

2.076+10

1.363+11

I7

(347

z.a10-06

7650

6.34

z.iaa-oa

9.258+09

1.740+11

la

1465

7.140-06

7260

5.89

2.578-08

6.843+09

2.693.kI1

19

1402

I.l95-OS

7140

5.30

2.989-08

a.O65+09

3.785+1x

20

1304

2.362-05

7005

4.90

4.016-01

1.olo+lo

6.327+11

21

1211

4.212-05

68W

4.30

5,6U6-08

l.2a9-i.10

l.O'll+lt

22

1126

7.143--05

674"

3.75

11.303-08

1.48Y410

I.aIO+IL

23

1085

9.200-05

6645

3.51

9.87I-08

1.594+10

2.351+12

24

1006

1.505-04

6400

3.08

1.372-07

1.682+10

3.978+12

25

933

2.432-o)

6030

2.70

1.*0')--07

1.473i.10

6.a34+12

16

a66

3.90,--04

5550

2.37

2.252-Of

l.l63+10

1.195+13

27

806

6.230-04

,YYO

2.06

2.72o-07

a.197+09

2.132+13

28

753

9.920-04

43YO

1.77

3.175-07

3.530409

3.91)1+13

19

707

1.577-03

3900

1.52

l.O71-%7

7.9oat.09

6.988+13

30

666

2.504-03

3570

1.36

3.706-07

8.635409

1.217+14

31

628

3.973-03

3370

1.20

3.145-07

8.lla+o9

2.057+14

32

992

6.302-03

3240

I.oa

1.213-06

7.799+09

3.4oa+I4

33

558

9.992-03

3165

0.97

1.908-06

9.686+09

5.556+I4 s.911+14

34

524

1.584-02

3140

0.88

l.333--06

1.132+10

35

490

2.511-02

3130

0.80

i.424-06

1.555+10

1.421+15

36

457

3.Y60-02

3,4ll

0.74

1.348-05

2.244+10

2.251+,5

37

423

6.309-02

3150

0.71

!.968-05

3.24ai.w

3.560+15

38

390

9.99Y-02

3165

0.70

5.724-05

4.753.t.10

5.617+15

39

322

2.512-01

3210

0.70

1.638-04

1.045+11

l.392.t16

40

253

6.310-01

3285

0.70

2.057-03

2.395+11

3.417+16

41

217

1.000+00

3340

0.70

1.007-03

3.7oa+11

5.326+16

42

lal

I.585+00

3410

0.70

l.241-02

I.a52+11

8.271+16

43

144

2.512+00

3500

0.72

l*l45-02

9.476+11

1.277+17

44

106

3.981+00

3605

0.77

7.240-02

l.563+12

1.962+lf

45

66

6.310+00

3750

0.83

l.B54-01

2.739+12

2.9s5+17

46

25

1.000+01

3965

0.9l

5.295-01

5.386-l-12

4.465+17

47

3

1.259+01

4135

0.95

3.062-01

8.415+12

5.3a7+17

40

-33

1.77a+ol

4500

I.04

1.427+00

1.929+13

6.978+17

49

-72

2.512+01

5070

1.10

8.019+00

5.032+13

9.752+17

50

-l15

3.54*+01

if*@

1.16

1.620+01

l.079C14

1.loO+la

Umbra Model

(particularly the infrared lines). The adoption of vt =lkm/s in the previous models could be due to neglect of the effect on the wings. We found that, in the middle-upper layers of the VL should probably not exceed 0.8 km/s; of course, it photosphere, could increase a little with depth.

33

DING & FANG

34

Fig.

2

Comparison between the observed (solid) and calculated (dashed) line profiles of sunspot umbra

It should be pointed out that, in the construction of the semi-empirical model, arbitrary adjustment of T(B) and vt(m) had to be constantly made, involving a large number of adjustable parameters, hence the question of uniqueness of the resulting model has to be addresaed. In the present case, the observational data included as many as seven line profiles of various intensities and moreover, these lines have widths as large as f 3A; the new umbra model given here has succeeded in fitting all these profiles, hence we think the degree of non-uniqueness is small here. Our experience with the calculations tells us that a variation of 100K in temperature will alter the calculated profile by as much as 5-15X, depending on the line the region concerned, and this would be obvious. Hence we think the results obtained here can be relied upon to a large degree.

-4

‘2 LWm(gcm-*

1

II

I

Fig. 3 Net radiative loss rates for H, CaII and H- in the model umbra atmosphere. “TOTAL” is the sun of all three.

35

Usbra Model

Using the umbra sodel obtained, we further calculated the radiative loss rates for the various spectral lines and continuus regions, using the saae method as described in (101. Fig.3 shows the variation of the rates with x for hydrogen, ionized calcium and negative hydrogen. Fig. 3 shows that radiative equilibrius does not hold for the whole of the atmosphere. In the chroaosphere , in addition to CaII, H also sakes a large contribution to the radiative loss. This is somewhat different from the case of the penumbra [lo]. The total loss reaches a laximun at middle layers of the chromosphere and it falls quickly with increasing height. Avrett [19] thinks there must exist some mechanical energy that disperses upwards in order to maintain the energy balance. Both in the quiet sun model (181 and the penumbra model (lo], the radiative loss in the temperature minimum region was negative. This proved to be also the case in the present umbra model. This seems to be a rather firm conclusion within the present theoretical framework, but it is difficult to say whether this is due to some simplifying assumption in the models. About this, some authors (201 have asserted (1) that the effect of many weak lines were ignored in the calculation and their contribution could be positive and (2) that the solar atmosphere is not uniform and the results based on a uniform model may not be realistic. If the radiative loss is indeed negative, then it will mean that energy is absorbed in the radiative process, and it is difficult to find a suitable explanation both from theory and observation as to the form of transfer. This question must await further study.

5. CONCLUSIONS Cur conclusions

are

as follows:

1. Our model is based on seven lines profiles wavelength ranges, hence is has a high degree

with extended of uniqueness.

2. In our umbra model, the temperature distribution is characterixed by a plateau structure, but the width of the plateau is narrower than in previous models. The temperature in the upper layers of the chromosphere is higher than those in the penumbra and quiet sun models. In the photospheric layers, the micro-turbulence velocity is lower than in the previous models. 3. There is no radiative equilibrium in the umbra atmosphere and the radiative loss in the temperature minimum region is negative. Future work in model sunspot atmosphere includes taking into account the effects of partial frequency redistribution and intermediate turbulence field (211, the developing of “multi-elements” models corresponding to the fine structures of sunspots (22-241 and a gradual transition from kinematic to dynamic models.

36

DING

b FANG

ACKNOWLEGDEMENT We thank the US Optical Observatory for the support given to one of us (FANG Cheng) during his working visit at Tucson. We also thank Dr W. C. Livingston for many helpful discussions.

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