Impact of tangled magnetic fields on star formation

Philipp Girichidis

Christoph Federrath (Monash)Robi Banerjee (Hamburg)Ralf Klessen (Heidelberg)

Anthony Whitworth (Cardiff)

Institute for Theoretical Astrophysics, University of HeidelbergHamburger Sternwarte

16 February 2012

Introduction Importance of the initial conditions Impact of magnetic fields

Outline

1 Introduction

2 Importance of the initial conditions

3 Impact of magnetic fields

Introduction Importance of the initial conditions Impact of magnetic fields

Star-forming Region

Arzoumanian et al. (2011)

very complex morphology

filamentary structure

turbulent motions

Konyves et al. (2010)

critical / supercritical BE sphere

trans- / supersonic velocitydispersion

difficult to use as initial conditionsuse highly simplified conditions (not realistic)

Introduction Importance of the initial conditions Impact of magnetic fields

Core Density Profile (Observation)

Pirogov 2009

ρ ∝ r−p, p = 1.6± 0.3innermost part flattenslinear plot!

10−19

10−18

10−17

10−16

10−15

10−14

1000 10000

ρ[g

cm−3]

r [AU]

λJ/2

TH

BE

PL15

PL20

Introduction Importance of the initial conditions Impact of magnetic fields

Hidden Fragmentation

Massive dense cores in Cygnus X (Bontemps et al. 2010)

3.5 mm and 1.5 mm observations, resolution limit: 1700 AU

mass: 84M�, size: 20, 000 AU

Introduction Importance of the initial conditions Impact of magnetic fields

Hidden Fragmentation?

Massive dense cores in Cygnus X (Bontemps et al. 2010)

mass: 58M�, size: 12, 000 AU

Introduction Importance of the initial conditions Impact of magnetic fields

Turbulence in the ISM

complex gas motions with turbulent characterMac Low & Klessen (2004), Elemegreen & Scalo (2004),McKee & Ostriker (2007)

distinction between modes(simulations: Schmidt et al. (2009), Federrath et al. (2010)):

compressive modes ~∇× ~v = ~0solenoidal modes ~∇ · ~v = 0measurements via the widths of the pdfconnect widths to modes σ2 = ln(1 + b2M2)with b = 1/3 : solenoidal modes & b = 1 : compressive modes

Introduction Importance of the initial conditions Impact of magnetic fields

Collapse Time Scalesdensity

radius

tff

tff

logdensity

log radius

tfftff

≈ 1

ρ ≈ const

ρ ∝ r−p

tfftff

=√

3−p3

Introduction Importance of the initial conditions Impact of magnetic fields

Collapse Time Scalesdensity

radius

tff

tff

logdensity

log radius

tfftff

≈ 1

ρ ≈ const

ρ ∝ r−p

strong impact

of turbulence

tfftff

=√

3−p3

Introduction Importance of the initial conditions Impact of magnetic fields

Cloud Morphology

Uniform density Bonnor-Ebert Power-law ρ ∝ r−1.5

t = 48 kyr

N∗ = 429

t = 32 kyr

N∗ = 302

t = 31 kyr

N∗ = 308

– distinct subclusters– long collapse time– IMF in qualitativeagreement

– one central cluster– IMF in qualitativeagreement

– compact cluster– central massive star

Girichidis (2011)

Introduction Importance of the initial conditions Impact of magnetic fields

Cloud Morphology II

different initial random seed ⇒ completely different result

Power-law ρ ∝ r−1.5 seed 1 Power-law ρ ∝ r−1.5 seed 2PL15-m-1 t = 24 kyr

Nsink = 1

PL15-m-2 t = 31 kyr

Nsink = 308

one central cluster

IMF in qualitativeagreement

one single massive star

Introduction Importance of the initial conditions Impact of magnetic fields

Pure Impact of Turbulence (random seed)

Combined Effects

combination of global collapse and local collapse

combination of local turbulent decay and accretion driventurbulence

⇒ analyse initial turbulence separately (periodic box)

random seed 1 random seed 2

converging region in the centre diverging region in the centre

Introduction Importance of the initial conditions Impact of magnetic fields

Observations of Magnetic Fields II

early observations of B field suggest large scale fieldGoodman et al. (1990), Ward-Thompson et al. (2000),Crutcher et al. (2004)

Recent observations of B field on small scales (Kirk 2006)

low-mass region (∼ 1M�, 0.05 pc)

strength: 10− 30µG

Introduction Importance of the initial conditions Impact of magnetic fields

Observations of Magnetic Fields II

Observations of magnetic fieldson small scales (Beuther 2010)

high-mass region:∼ 100− 1000M�, 0.2 pc

strength: B ≈ 14 mG

Introduction Importance of the initial conditions Impact of magnetic fields

Theoretical small scale structure

B-field amplification via small-scale dynamo (Kazantsev 1968,Subramanian 1997, Brandenburg & Subramanian 2005,Schober 2012)

with small scale turbulenceThe Astrophysical Journal, 731:62 (16pp), 2011 April 10 Federrath et al.

10-2

10-1

100

101

102

103

104

105

106

107

1 10 100 1000 10000

P(B)

k k

k-2

k3.0!=12!=10!= 8!= 6!= 4!= 2!= 1!= 0

Figure 5. Magnetic energy spectra in the fixed frame of reference, i.e., thespectra are normalized to the initial Jeans wavenumber, kJ0. The time evolutionin the collapse regime shows that the magnetic field is basically frozen-in onlarge scales, while the turbulent dynamo keeps amplifying the field on smallerand smaller scales as the collapse proceeds.(A color version of this figure is available in the online journal.)

high-resolution spectra obtained inside the Jeans volume (Fig-ure 4), by shifting them to the correct position with respect tothe fixed frame of reference. However, the spectra at late timesduring the collapse, obtained with this method, do not containany information on scales far outside the Jeans volume. Thus,in the second step, we add spectral information on large scales,by gradually extracting larger boxes, centered on the core at afixed grid resolution. Using this method, we can test the spec-tral energy scale-by-scale. We call this method “scale-by-scaleextraction.”

The result of this two-step approach is shown in Figure 5,where we plot the spectra of magnetic energy in the fixed frameof reference, i.e., as a function of the initial (at ! = 0) Jeanswavenumber, kJ0, of the collapsing system. The first thing tonote is that the spectra on large scales obtained with the scale-by-scale extraction method connect reasonably well with thespectra computed inside the Jeans volume and re-normalized tomatch the fixed frame of reference. Some deviation is seenonly at k/kJ0 ! 300 for the spectrum at ! = 12, whichcan be taken as a measure of the uncertainty in the spectraobtained with our scale-by-scale extraction method. Given thetotal range of scales that we aim to probe here, the difference inthe spectra obtained with our two-step approach is acceptable.We also tested whether extending the scale-by-scale extractionto smaller scales (inside the Jeans volume) matches the high-resolution spectra of step one. We found that they do within anuncertainty of about 25%, i.e., the slopes and peak positions arereproduced reasonably well with the scale-by-scale extractionmethod. However, the high-resolution spectra inside the Jeansvolume are more accurate, and we thus prefer the two-stepapproach described above.

Three main results can be extracted from Figure 5. First, themagnetic field always grows fastest on the smallest availablescales in the simulation, with the peak located at around 30 gridcells. Second, as the collapse proceeds, the field is amplified onsmall scales inside the core, but is essentially frozen-in on largescales. Third, the large-scale spectra follow a power law closeto k3.0. This particular exponent for the power law is consistentwith the radial dependence of the field (see Figures 1 and 2) and

depends strongly on resolution. The magnetic energy on scalesoutside the core is

B2rms "

!P (B) dk " k4.0 . (5)

From this, it follows that Brms " k2.0 " r#2.0 outside the Jeansvolume, which is consistent with the power-law behavior of theradial profile of Brms in Figure 1. It should be re-emphasized,however, that the exact exponent of this power law is essentiallymeaningless, as it depends on the numerical Jeans resolution.Higher resolution will lead to a steeper increase of P (B) towardsmall scales (see Figure 2). In reality, the magnetic field willgrow much more strongly due to dynamo action than what wecan resolve in the present calculation with 128 cells per Jeanslength, and thus, our amplification rate is a lower limit (discussedfurther in Section 4).

3.6. Probability Distribution Function of the Gas Density

The probability distribution function (PDF) of the gas densityis a useful measure of the turbulence in any turbulent systemexhibiting density fluctuations. Moreover, the PDF is an es-sential ingredient for models of star formation (e.g., Padoan& Nordlund 2002; Krumholz & McKee 2005; Hennebelle &Chabrier 2008, 2009) and the gas distribution in galaxies (e.g.,Tassis 2007; Krumholz et al. 2009). The density PDF has beenstudied in some detail in non-self-gravitating, turbulent systems(e.g., Vazquez-Semadeni 1994; Padoan et al. 1997; Passot &Vazquez-Semadeni 1998; Lemaster & Stone 2008; Federrathet al. 2008b; Price & Federrath 2010) and in turbulent sys-tems including self-gravity (e.g., Klessen 2000; Federrath et al.2008a; Kainulainen et al. 2009; Cho & Kim 2011; Kritsuk et al.2011). However, it has not been analyzed yet in a collapsingsystem, in which turbulence is replenished by the gravitationalcollapse of a dense core.

Figure 6 shows the time evolution of the PDFs of thelogarithmic density:

s $ ln"

"

%"&

#, (6)

where %"& denotes the mean density in the core, i.e., insidethe Jeans volume. The PDF at ! = 0 is purely a result of theinitial, radial density distribution, following a Bonnor–Ebertprofile. This profile exhibits a flat inner core for radii smallerthan the Jeans radius and can be approximated with a powerlaw of the form " " r## with # ! 2.2 at large radii (Ebert1955; Bonnor 1956). Using the derived relation between apower-law radial distribution and the corresponding densityPDF (see Appendix A), we can estimate the power-law exponentof the density PDF from the power-law exponent of the radialdistribution, #. Thus, the PDF of the logarithmic density,s, follows a power law for small logarithmic densities withexponent #3/# and falls off more steeply toward higherdensities, due to the flattening of the Bonnor–Ebert profile inthe center of the core (see Figure 1).

Both in the regime of turbulent decay (! ! 4) and in thecollapse regime (! " 4), the volume-weighted PDF develops alog-normal form:

ps(s) ds = 1$

2$% 2s,turb

exp

%

# (s # %s&)2

2% 2s,turb

&

ds , (7)

8

Federrath et al. (2011)

Introduction Importance of the initial conditions Impact of magnetic fields

Numerical Setup

constant values

mass M = 100 M�, size R = 0.1 pc

isothermal T = 20 K

supersonic turbulence M = 3.5

varying density profile

rescaled Bonnor-Ebert sphere

power-law profiles (ρ ∝ r−3/2)

varying turbulent random seed, mixed modesvarying B-fields (Bouchut, Klingenberg, Waagan 2007, 2010)

homogeneous magnetic field (B2)

tangled magnetic field (B1,2,3)

Introduction Importance of the initial conditions Impact of magnetic fields

Magnetic field data

Three different strengths: vrms/vA = q = 0.5, 1, 2

peak mode of field: kpeak = 30

power spectrum of tangled fields:

P(k) ∝{k3 for k ∈ 1..30

k−1.5 for k ∈ 30..∞(1)

With M = 3.45, c = 26800 cm s−1 and〈ρ〉 = 1.759× 10−18 g cm−3:

B1,2,3 =434.7µG

q1,2,3= 217.3, 434.7, 869.3 µG . (2)

Ekin/Epot ≈ 0.1

Emag/Epot ≈ 0.1

Introduction Importance of the initial conditions Impact of magnetic fields

Comparison hom. vs. tangled field (PL)

Power-law density field

homogeneous B field tangled B field

Introduction Importance of the initial conditions Impact of magnetic fields

Comparison tangled fields

Power-law density field

tangled field B1 tangled field B2 tangled field B3

Introduction Importance of the initial conditions Impact of magnetic fields

Bonnor-Ebert Sphere (hom. vs. tan.)

Bonnor-Ebert density field

homogeneous B field tangled B field

Introduction Importance of the initial conditions Impact of magnetic fields

Bonnor-Ebert Sphere (tangled)

Bonnor-Ebert density field

tangled field B1 tangled field B2 tangled field B3

Introduction Importance of the initial conditions Impact of magnetic fields

Bonnor-Ebert Sphere (hom. vs. tan.) Seed 2

Bonnor-Ebert density field

homogeneous B field tangled B field

Introduction Importance of the initial conditions Impact of magnetic fields

Bonnor-Ebert Sphere (tangled) Seed 2

Bonnor-Ebert density field

tangled field B1 tangled field B2 tangled field B3

Introduction Importance of the initial conditions Impact of magnetic fields

Spectrum of the magnetic field spectrum

Time evolution of the field

10−6

10−5

10−4

10−3

10−2

10−1

1 10 100

PB(k)

k

t = 3kyrt = 9kyrt = 16 kyrt = 22 kyrt = 28 kyr

fast transfer to large scales

comparison to observations ⇒ need small scale driver

Introduction Importance of the initial conditions Impact of magnetic fields

Conclusions

magnetic fields reduce total number of stars

tangled magnetic fields

produce more stars than homogeneous fieldslead to different cloud morphologyresult in same collapse time

structure of the field is more important than strength

strong coupling of magnetic fields and velocities

numerics: ⇒ accurately follow turbulence and magnetic field