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Understanding Mass Transfer in Extremely
Low Mass WD Binaries
David Kaplan (UW-Milwaukee), Lars Bildsten (KITP), Justin Steinfadt (UCSB➞ATT Govt. Solutions)
Center for Gravitation, Cosmology, & Astrophysics
Tuesday, April 17, 12
How To Form AM CVn• Or, what happens to double WDs with
Extremely Low Mass (ELM; <0.2 M☉) He WDs
• SDSS finding many of these (Kilic, Brown, ...)
• Stay bright (large & hot) for Gyr due to stably burning H shell (Panei et al. 2007)
• 10-3 to 10-2 M⊙ of H
• Many found in tight orbits with other WDs (or pulsars), will reach contact in < 10 Gyr (Badenes, Brown, Kawka, Kilic, Mullally, Steinfadt, Vennes, ...)
The ELM Survey IV 3
TABLE 1X-RAY OBSERVATIONS OF ELM WDS
Name ObsID Dist NH Expos Count rate LX(pc) (cm−2) (ks) (cts s−1) (ergs s−1)
SDSS J082212.57+275307.4 12352 430 3.5! 1020 2.0 < 2.2! 10−3 < 2.2! 1029SDSS J084910.13+044528.7 12354 930 4.1! 1020 10.9 < 4.2! 10−4 < 2.0! 1029
NOTE. — 99% confidence X-ray count rate upper limits for two ELM WDs from Chandra X-rayobservations. Count rate limit is in 0.3-6 keV band and LX limit is in 0.5-6 keV band.
FIG. 1.— Spectral fits (red solid lines) to the composite spectra of our targets (jagged lines, left panels) and to the flux-normalized line profiles (right panels).The composite spectrum of J1056+6536 suffers from flux calibration problems and is not shown in the left panels.
where the model using the continuum shape is not a goodmatch to the observations. We use the results from the Balmerline profile fitting for this star. This model agrees remarkablywell with the spectral energy distribution based on the SDSSphotometry.Figure 2 compares the best-fit Teff and logg measurements
for our targets against the predicted evolutionary sequencesfrom Panei et al. (2007). Based on these tracks, our targetshave M = 0.17 − 0.40 M!; some of them are more massivethan predicted from the relatively noisy SDSS spectroscopydata. The physical parameters of all seven systems discussedin this paper are presented in Table 3. The age and distanceestimates are somewhat uncertain for M ! 0.17 M! objects,because many of them fall in the gap between 0.17 and 0.18M! He-core WD tracks. Panei et al. (2007) and Kilic et al.(2010a) argue that diffusion-induced hydrogen-shell flashestake place forM > 0.17M!, which yield small hydrogen en-velopes. Hence, lower mass objects have massive hydrogenenvelopes, larger radii, lower surface gravities, and longercooling times. The inconsistency between the observed pa-rameters for Teff " 10,000 K and logg # 6 objects and thePanei et al. (2007, Figure 2) models makes accurateWDmassand luminosity estimates difficult for them. Fortunately, massand luminosity change very little over the range of effectivetemperature and surface gravity sampled by these WDs. WeadoptM = 0.17M! andMg $ 8 mag for these objects.All seven targets show significant velocity variations with
peak-to-peak velocity amplitudes of 120-640 km s−1 and 1-18
h orbital periods. Figure 3 shows the observed radial veloc-ities and the best fit orbits for our targets. We present the
0.1600.1700.171 0.1800.1870.2030.225
0.249
0.3060.333
0.3840.416
FIG. 2.— Surface gravity versus effective temperature of the observedWDs (filled points) in the ELM Survey, compared with predicted tracksfor He WDs with 0.16–0.42 M! (Panei et al. 2007). The dashed and dot-ted lines show solar metallicity and halo metallicity (Z=0.001) models ofSerenelli et al. (2001, 2002) for 0.17 M! WDs, respectively. The seven newsystems presented in this paper are shown as red points. Triangles show theELMWD companions to PSR J1012+5307 and J1911−5958A.
Known ELM WDs from Kilic et al. (2012)
Tuesday, April 17, 12
0.01 0.1 1
0.4
0.6
0.8
1
1.2
1.4
1.6
Period (days)
Tot
alM
ass
(M!
)tmerge= 10Gyr100 Myr1 Myr
KPD 1930+2752
NLTT 11748
SDSS J0106CSS 41177
SDSS J0651SDSS J1840
Single LineDouble LinesdBPSR< 0.20 M!
First eclipsing
Second eclipsing
Extremely short P
Pulsations
Pulsars
Tuesday, April 17, 12
What Happens at Contact?•Marsh et al (‘04):
•Detailed mass transfer, including stability
•But: used cold EOS for both accretor (CO WD) and donor (He WD)
•OK for accretor
•But what about for donor?
Disc-fed accretion: always stable
Direct-impact accretion: always unstable
Direct-impact
accretion:
can be sta
bilized by sy
nchronization
Tuesday, April 17, 12
0 0.01 0.02 0.03 0.04 0.0510−10
10−5
100
105
1010
Radius (R!)
Den
sity
(g/cm
2)orX
<0.2 M⊙He WDs are Large• Steinfadt et al.: eclipse observations of
NLTT 11748 show R≈0.04 R☉ for M≈0.16 M☉
• Cold EOS: 0.02 R☉ (e.g., Eggleton)
• Outer portions are not degenerate O
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160 200 240
0.06
0.07
!P (s)
d P
!0.04 !0.02 0.00 0.02 0.04 0.06
0.94
0.96
0.98
1.00
Phase (cycles)
Rel
ativ
eFlu
x160 200 240
0.03
0.04
!S (s)
d S
0.46 0.48 0.50 0.52 0.54 0.56
0.94
0.96
0.98
1.00
Phase (cycles)
Rel
ativ
eFlu
x
2009-12-222010-01-082010-02-03
Core (degenerate)
Envelope (non-degen)
0.16 M⊙ w/ 3e-3 M⊙ envelope
density
H fraction
Also see Driebe et al. (’98); Sarna et al. (’00); Serenelli et al. (’01); Bassa et al. (’03 & ’06); van Kerkwijk et al. (’05), ...
Tuesday, April 17, 12
Our Calculation• Orbital evolution of ELM He WD
+ CO WD (0.7-1.0 M☉)
• He WD follows models of Steinfadt et al. (2010): used to determine (dR/dM, X) as mass is stripped
• Follow mass transfer & orbital evolution
• Consider disk/direct impact
• Stability of accreted matter to burning
• Do not deal with synchronizations torques
• Range of envelope masses (i.e., ages) for each core mass
0.1 0.11 0.12 0.13 0.14 0.15 0.160.02
0.03
0.04
0.05
0.06
0.07
0.08
Donor Mass (M!)DonorRadius(R
!)
0.15/2.120.15/2.510.15/2.900.15/3.300.15/3.690.15/4.090.15/4.500.15/4.910.15/5.30M strip=Menv
X=0.01
cold EOS
Kaplan et al. (2012)
0.15 M⊙ core, with (2-5)x10-3 M⊙ envelope
Tuesday, April 17, 12
Response to Mass Loss• Normal WD: R~M-⅓
• Mass transfer stable if
• is -⅓ for a normal WD, so need mass ratio < ⅔
• ELM He WD: • Outer layers not degenerate, ζ≫1
• Disk accretion guaranteed stable; even without disk, more stable systems (reduce Ṁ, as in D’Antona et al. or Deloye & Roelofs)
⇣ ⌘ d logR
d logM
Mdonor
Maccrete
<5
6+
⇣
2
Tuesday, April 17, 12
• Early evolution: prolonged period of hydrogen accretion
• Ṗ just set by GR
• Once on accretor, is H burning stable? pp only!
5
10
P(m
in)
10ï1210ï1110ï10
|P|(
s/s)
10ï410ï310ï2
!M
(M!)
10ï810ï710ï6
|M|(
M!/yr)
10ï2
10ï1
100
X
1 0.1 0.01 0.00110ï1100101
!He
T ime Before Period Minimum (Myr)
0.15 M☉
0.20 M☉
Kaplan et al. (2012)
Tuesday, April 17, 12
3 4 5 6 7 8 9 10 2010ï11
10ï10
10ï9
10ï8
10ï7
10ï6
10ï5
0.10.3
1
3
0.1
0.3
1
3
Orbital Period (min)
Mas
sT
rans
fer
Rat
e(M
!/yr
)
dNe unstable
Stable He
Stable H
0.125/3.000.200/1.80
X=0.99X=0.9X=0.5X=0.1X=0.01X=0.001
Low-mass (large) donor: accretion starts early, remains disk-fed & stable
High-mass (small) donor: accretion starts late, enters direct-impact, but stays dynamically stable
High Ṁ: stable H or He burning: accreted matter stays on
Stable burning calculations still preliminary (based on Nomoto et al. 2007 & Iben & Tutukov 1989)
Late evolution: ζ determines Ṁ, follows classical result
0.8 M⊙ accretor
Kaplan et al. (2012)
Higher Ṁ: only Ṁ<Ṁedd stays on
Tuesday, April 17, 12
3 4 5 6 7 8 9 10 2010ï11
10ï10
10ï9
10ï8
10ï7
10ï6
10ï5
Orbital Period (min)
Mas
sT
rans
fer
Rat
e(M
!/yr
)
dNe unstable
Stable He
Stable H
Ma = 0 .7 M! 0.125/3.000.125/6.000.150/2.120.150/5.300.175/1.900.175/5.100.200/1.800.200/4.90
X=0.99X=0.9X=0.5X=0.1X=0.01X=0.001
3 4 5 6 7 8 9 10 2010ï11
10ï10
10ï9
10ï8
10ï7
10ï6
10ï5
Orbital Period (min)
Mas
sT
rans
fer
Rat
e(M
!/yr
)
dNe unstable
Stable He
Stable H
Ma = 0 .8 M! 0.125/3.000.125/6.000.150/2.120.150/5.300.175/1.900.175/5.100.200/1.800.200/4.90
X=0.99X=0.9X=0.5X=0.1X=0.01X=0.001
3 4 5 6 7 8 9 10 2010ï11
10ï10
10ï9
10ï8
10ï7
10ï6
10ï5
Orbital Period (min)
Mas
sT
rans
fer
Rat
e(M
!/yr
)
dNe unstable
Stable He
Stable H
Ma = 0 .9 M! 0.125/3.000.125/6.000.150/2.120.150/5.300.175/1.900.175/5.100.200/1.800.200/4.90
X=0.99X=0.9X=0.5X=0.1X=0.01X=0.001
3 4 5 6 7 8 9 10 2010ï11
10ï10
10ï9
10ï8
10ï7
10ï6
10ï5
Orbital Period (min)
Mas
sT
rans
fer
Rat
e(M
!/yr
)
dNe unstable
Stable He
Stable H
Ma = 1 .0 M! 0.125/3.000.125/6.000.150/2.120.150/5.300.175/1.900.175/5.100.200/1.800.200/4.90
X=0.99X=0.9X=0.5X=0.1X=0.01X=0.001
Vary accretor mass:
Kaplan et al. (2012)
Tuesday, April 17, 12
3 4 5 6 7 8 9 10
102
103
104
105
Stable H
LEdd
Ma = 0 .7 M!
4 5 6 7 8 9 10
102
103
104
105
Stable HLEdd
Ma = 0 .8 M!
4 5 6 7 8 9 10101
102
103
104
105
Stable HLEdd
Ma = 1 .0 M!
0.125/3.000.125/6.000.150/2.120.150/5.300.175/1.900.175/5.100.200/1.800.200/4.90
3 4 5 6 7 8 9 10101
102
103
104
105
Stable HLEdd
Ma = 0 .9 M!
Nuc
lear
Bur
ning
Lum
inos
ity
(L!
)
Orbital Period (min)
Changing H fraction reduces luminosity
Kaplan et al. (2012)
time
Tuesday, April 17, 12
0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 1 1.05 1.10
0.05
0.1
0.15
0.2
0.25
Accretor Mass (M!)
Don
orM
ass
(M!
)
DiscïfedDirect Impact
Stability: many more systems avoid merger
Marsh et al. d
isk/direct limit
We see no dynamically unstable systems
More disk-fed
Kaplan et al. (2012)
range in Menv
Tuesday, April 17, 12
0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 1 1.05 1.10
0.05
0.1
0.15
0.2
0.25
Accretor Mass (M!)
Don
orM
ass
(M!
)
DiscïfedDirect Impact
Stability: many more systems avoid merger
Marsh et al. d
isk/direct limit
We see no dynamically unstable systems
More disk-fed
Kaplan et al. (2012)
range in Menv
Tuesday, April 17, 12
Direct-Impact of H-Rich Material• HM Cancri (5.4 min binary): 0.27
M☉& 0.55 M☉ (Roelofs et al.; Israel et al., Cropper et al.; Strohmayer et al.)
• Ṗ<0 but Lx (Ṁ) puzzlingly low
• D’Antona et al. (2006): mass transfer will start from non-degenerate H envelope
• Changes orbital evolution, since dR/dM<0
• Largely can explain orbital evolution, luminosity of HM Cnc
• All metals sedimented out of H layer, somewhat out of He (Hebert et al.)
mass transfer from the now pure helium WD.4 The mass-transferrate M is much smaller, up to a factor 10, when the period is de-creasing. The period derivative is shown in Figure 2,wherewe seethat the value of Porb for RX J0806+15 is in the range provided byourmodels. The Porb of RX J1914+24 (V407Vul) ismuch smaller(in spite of the larger X-ray luminosity). It would be better ex-plained with a positive Porb and stronger mass-transfer rate, butthe exploratory aim of these computations does not pretend to lookfor a precise fit of the Porb, and we do not fully explore the wholerange of space parameters for this kind of evolution. Taken atface value, the results might indicate that RX J1914+24 is stillbeginning the mass-transfer phase. The onset of mass transfer isshown in Figure 2 for sequence 4, and it occurs at an orbital pe-riod slightly shorter than the period of V407 Vul.
From the bottom panel of Figure 1, we see that the mass-lossrate at 5.4 minutes for the lower branch (decreasing orbital periods)is in the range 2 ; 10!8 to 5 ; 10!8 M" yr!1, while it is #2:5 ;10!7 M" yr!1 for the upper branch (increasing period). Conse-quently, the standard expected X-ray luminosity would be reducedfrom 2 ; 1035ergs s!1 to#2 ; 1034 to 4 ; 1034 ergs s!1. Note thattwo points must be investigated in more detail before we push fur-ther the interpretation of X-ray luminosity:
1. The role of compressional heating (Bildsten et al. 2006).Although the mass-transfer rates we find are quite smaller thanthose predicted by GR for a fully degenerate helium donor, a rateof #2 ; 10!8 M" yr!1 would still raise the primaryWD luminos-ity toMv # 8, pushing the distance to#4 kpc and raising the ob-servedX-ray luminosity to#3 ; 1034 ergs s!1, perfectly consistentwith our newmass-transfer rates. However, the modeling of com-pressional heating should be extended to smaller primary WDmasses (like the 0.35M" cases studied here) to assess the effectof the peculiar geometry of nonYspherically symmetric accretionon WDs with very extended nondegenerate envelopes (only the
0.65 and 1.05 M" cases are reported in Bildsten et al. 2006).Further studies of the system may better constrain the primaryTeA and luminosity, and then the system distance.2. The X-ray luminosity. This is a fraction of the accretion
luminosity, and this latter depends on which fraction of the masslost is actually accreted onto the primary WD. There is a smallrange of the parameter space for which the X-ray luminosity ofRX J0806+15 is also compatible with the large mass-loss rate of
4 Similar computations were first done by Fedorova & Ergma (1989) in the con-text of the decreasing orbital period of the ultrashort X-ray binary MXB 1820-30having Porb $ 11:4 minutes.
Fig. 1.—Evolution of radius, mass, and mass-loss rate along sequences: 1 (dash-dotted line), 2 (dashed line), and 3 (dotted line). The periods of RX J0806+15 andRX J1914+24 are indicated as vertical segments. [See the electronic edition of theJournal for a color version of this figure.]
Fig. 2.—Period derivative vs. orbital period for the sequences: 1 (dash-dottedline), 2 (dashed line), 3 (dotted line), and 4 (solid line). Period derivatives for thetwo DDWDs are from Reinsch (2005). [See the electronic edition of the Journalfor a color version of this figure.]
Fig. 3.—Timescale of evolution P/P vs. orbital period for sequence 2. Thelower portion of the curve corresponds to decreasing Porb, the upper curve to in-creasing Porb.
D’ANTONA ET AL.1432 Vol. 653
HM Cnc
V407 Vul
Tuesday, April 17, 12
Direct-Impact of H-Rich Material• HM Cancri (5.4 min binary): 0.27
M☉& 0.55 M☉ (Roelofs et al.; Israel et al., Cropper et al.; Strohmayer et al.)
• Ṗ<0 but Lx (Ṁ) puzzlingly low
• D’Antona et al. (2006): mass transfer will start from non-degenerate H envelope
• Changes orbital evolution, since dR/dM<0
• Largely can explain orbital evolution, luminosity of HM Cnc
• All metals sedimented out of H layer, somewhat out of He (Hebert et al.)
mass transfer from the now pure helium WD.4 The mass-transferrate M is much smaller, up to a factor 10, when the period is de-creasing. The period derivative is shown in Figure 2,wherewe seethat the value of Porb for RX J0806+15 is in the range provided byourmodels. The Porb of RX J1914+24 (V407Vul) ismuch smaller(in spite of the larger X-ray luminosity). It would be better ex-plained with a positive Porb and stronger mass-transfer rate, butthe exploratory aim of these computations does not pretend to lookfor a precise fit of the Porb, and we do not fully explore the wholerange of space parameters for this kind of evolution. Taken atface value, the results might indicate that RX J1914+24 is stillbeginning the mass-transfer phase. The onset of mass transfer isshown in Figure 2 for sequence 4, and it occurs at an orbital pe-riod slightly shorter than the period of V407 Vul.
From the bottom panel of Figure 1, we see that the mass-lossrate at 5.4 minutes for the lower branch (decreasing orbital periods)is in the range 2 ; 10!8 to 5 ; 10!8 M" yr!1, while it is #2:5 ;10!7 M" yr!1 for the upper branch (increasing period). Conse-quently, the standard expected X-ray luminosity would be reducedfrom 2 ; 1035ergs s!1 to#2 ; 1034 to 4 ; 1034 ergs s!1. Note thattwo points must be investigated in more detail before we push fur-ther the interpretation of X-ray luminosity:
1. The role of compressional heating (Bildsten et al. 2006).Although the mass-transfer rates we find are quite smaller thanthose predicted by GR for a fully degenerate helium donor, a rateof #2 ; 10!8 M" yr!1 would still raise the primaryWD luminos-ity toMv # 8, pushing the distance to#4 kpc and raising the ob-servedX-ray luminosity to#3 ; 1034 ergs s!1, perfectly consistentwith our newmass-transfer rates. However, the modeling of com-pressional heating should be extended to smaller primary WDmasses (like the 0.35M" cases studied here) to assess the effectof the peculiar geometry of nonYspherically symmetric accretionon WDs with very extended nondegenerate envelopes (only the
0.65 and 1.05 M" cases are reported in Bildsten et al. 2006).Further studies of the system may better constrain the primaryTeA and luminosity, and then the system distance.2. The X-ray luminosity. This is a fraction of the accretion
luminosity, and this latter depends on which fraction of the masslost is actually accreted onto the primary WD. There is a smallrange of the parameter space for which the X-ray luminosity ofRX J0806+15 is also compatible with the large mass-loss rate of
4 Similar computations were first done by Fedorova & Ergma (1989) in the con-text of the decreasing orbital period of the ultrashort X-ray binary MXB 1820-30having Porb $ 11:4 minutes.
Fig. 1.—Evolution of radius, mass, and mass-loss rate along sequences: 1 (dash-dotted line), 2 (dashed line), and 3 (dotted line). The periods of RX J0806+15 andRX J1914+24 are indicated as vertical segments. [See the electronic edition of theJournal for a color version of this figure.]
Fig. 2.—Period derivative vs. orbital period for the sequences: 1 (dash-dottedline), 2 (dashed line), 3 (dotted line), and 4 (solid line). Period derivatives for thetwo DDWDs are from Reinsch (2005). [See the electronic edition of the Journalfor a color version of this figure.]
Fig. 3.—Timescale of evolution P/P vs. orbital period for sequence 2. Thelower portion of the curve corresponds to decreasing Porb, the upper curve to in-creasing Porb.
D’ANTONA ET AL.1432 Vol. 653
HM Cnc
V407 Vul
mass transfer from the now pure helium WD.4 The mass-transferrate M is much smaller, up to a factor 10, when the period is de-creasing. The period derivative is shown in Figure 2,wherewe seethat the value of Porb for RX J0806+15 is in the range provided byourmodels. The Porb of RX J1914+24 (V407Vul) ismuch smaller(in spite of the larger X-ray luminosity). It would be better ex-plained with a positive Porb and stronger mass-transfer rate, butthe exploratory aim of these computations does not pretend to lookfor a precise fit of the Porb, and we do not fully explore the wholerange of space parameters for this kind of evolution. Taken atface value, the results might indicate that RX J1914+24 is stillbeginning the mass-transfer phase. The onset of mass transfer isshown in Figure 2 for sequence 4, and it occurs at an orbital pe-riod slightly shorter than the period of V407 Vul.
From the bottom panel of Figure 1, we see that the mass-lossrate at 5.4 minutes for the lower branch (decreasing orbital periods)is in the range 2 ; 10!8 to 5 ; 10!8 M" yr!1, while it is #2:5 ;10!7 M" yr!1 for the upper branch (increasing period). Conse-quently, the standard expected X-ray luminosity would be reducedfrom 2 ; 1035ergs s!1 to#2 ; 1034 to 4 ; 1034 ergs s!1. Note thattwo points must be investigated in more detail before we push fur-ther the interpretation of X-ray luminosity:
1. The role of compressional heating (Bildsten et al. 2006).Although the mass-transfer rates we find are quite smaller thanthose predicted by GR for a fully degenerate helium donor, a rateof #2 ; 10!8 M" yr!1 would still raise the primaryWD luminos-ity toMv # 8, pushing the distance to#4 kpc and raising the ob-servedX-ray luminosity to#3 ; 1034 ergs s!1, perfectly consistentwith our newmass-transfer rates. However, the modeling of com-pressional heating should be extended to smaller primary WDmasses (like the 0.35M" cases studied here) to assess the effectof the peculiar geometry of nonYspherically symmetric accretionon WDs with very extended nondegenerate envelopes (only the
0.65 and 1.05 M" cases are reported in Bildsten et al. 2006).Further studies of the system may better constrain the primaryTeA and luminosity, and then the system distance.2. The X-ray luminosity. This is a fraction of the accretion
luminosity, and this latter depends on which fraction of the masslost is actually accreted onto the primary WD. There is a smallrange of the parameter space for which the X-ray luminosity ofRX J0806+15 is also compatible with the large mass-loss rate of
4 Similar computations were first done by Fedorova & Ergma (1989) in the con-text of the decreasing orbital period of the ultrashort X-ray binary MXB 1820-30having Porb $ 11:4 minutes.
Fig. 1.—Evolution of radius, mass, and mass-loss rate along sequences: 1 (dash-dotted line), 2 (dashed line), and 3 (dotted line). The periods of RX J0806+15 andRX J1914+24 are indicated as vertical segments. [See the electronic edition of theJournal for a color version of this figure.]
Fig. 2.—Period derivative vs. orbital period for the sequences: 1 (dash-dottedline), 2 (dashed line), 3 (dotted line), and 4 (solid line). Period derivatives for thetwo DDWDs are from Reinsch (2005). [See the electronic edition of the Journalfor a color version of this figure.]
Fig. 3.—Timescale of evolution P/P vs. orbital period for sequence 2. Thelower portion of the curve corresponds to decreasing Porb, the upper curve to in-creasing Porb.
D’ANTONA ET AL.1432 Vol. 653
Tuesday, April 17, 12
Conclusions
• Detailed calculation of mass transfer for ELM He WD and CO WD
• Hot, non-degenerate envelope prolongs period of stable mass transfer, significantly expands systems that will avoid merger
• Nout/Nin~4 just based on Ṗ; weight by Ṁ increases outgoing bias
• If stable burning, that could change luminosity
• Long period of stable H burning kept He WD hot, explains high-entropy AM CVn donors?
• Still need to do fully self-consistent calculation of ζ and burning stability for these accretion rates and compositions (MESA)
Tuesday, April 17, 12
![Leitthema - Springer · (2002) [58]x x x x x x x x Brune (2002) [23]x xx xx x xx Burmester (2014) [24]x x x Butollo (2012) [25]x xx x xx xx Casal (2005) [26]xx x xx x Claassen (2005)](https://img.pdfslide.net/doc/110x75/605f28310469a1434626bf30/leitthema-springer-2002-58x-x-x-x-x-x-x-x-brune-2002-23x-xx-xx-x-xx-burmester.jpg)
