Most-likely DCF Estimates of Magnetic Field Strength

<jats:p>The Davis–Chandrasekhar–Fermi (DCF) method is widely used to evaluate magnetic fields in star-forming regions. Yet it remains unclear how well DCF equations estimate the mean plane-of-the-sky field strength in a map region. To address this question, five DCF equations are applied to an idealized cloud map. Its polarization angles have a normal distribution with dispersion <jats:italic>σ</jats:italic> <jats:sub> <jats:italic>θ</jats:italic> </jats:sub>, and its density and velocity dispersion have negligible variation. Each DCF equation specifies a global field strength <jats:italic>B</jats:italic> <jats:sub>DCF</jats:sub> and a distribution of local DCF estimates. The “most-likely” DCF field strength <jats:italic>B</jats:italic> <jats:sub>ml</jats:sub> is the distribution mode, for which a correction factor <jats:italic>β</jats:italic> <jats:sub>ml</jats:sub> ≡ <jats:italic>B</jats:italic> <jats:sub>ml</jats:sub>/<jats:italic>B</jats:italic> <jats:sub>DCF</jats:sub> is calculated analytically. For each equation, <jats:italic>β</jats:italic> <jats:sub>ml</jats:sub> < 1, indicating that <jats:italic>B</jats:italic> <jats:sub>DCF</jats:sub> is a biased estimator of <jats:italic>B</jats:italic> <jats:sub>ml</jats:sub>. The values of <jats:italic>β</jats:italic> <jats:sub>ml</jats:sub> are <jats:italic>β</jats:italic> <jats:sub>ml</jats:sub> ≈ 0.7 when <jats:inline-formula> <jats:tex-math> <? CDATA ${B}_{\mathrm{DCF}}\propto {\sigma }_{\theta }^{-1}$? > </jats:tex-math> <mml:math xmlns:mml="http://www.w3. org/1998/Math/MathML" overflow="scroll"> <mml:msub> <mml:mrow> <mml:mi>B</mml:mi> </mml:mrow> <mml:mrow> <mml:mi>DCF</mml:mi> </mml:mrow> </mml:msub> <mml:mo>∝</mml:mo> <mml:msubsup> <mml:mrow> <mml:mi>σ</mml:mi> </mml:mrow> <mml:mrow> <mml:mi>θ</mml:mi> </mml:mrow> <mml:mrow> <mml:mo>−</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msubsup> </mml:math> <jats:inline-graphic xmlns:xlink="http://www.w3. org/1999/xlink" xlink:href="apjad1596ieqn1.gif" xlink:type="simple" /> </jats:inline-formula> due to turbulent excitation of Alfvénic MHD waves, and <jats:italic>β</jats:italic> <jats:sub>ml</jats:sub> ≈ 0.9 when <jats:inline-formula> <jats:tex-math> <? CDATA ${B}_{\mathrm{DCF}}\propto {\sigma }_{\theta }^{-1/2}$? > </jats:tex-math> <mml:math xmlns:mml="http://www.w3. org/1998/Math/MathML" overflow="scroll"> <mml:msub> <mml:mrow> <mml:mi>B</mml:mi> </mml:mrow> <mml:mrow> <mml:mi>DCF</mml:mi> </mml:mrow> </mml:msub> <mml:mo>∝</mml:mo> <mml:msubsup> <mml:mrow> <mml:mi>σ</mml:mi> </mml:mrow> <mml:mrow> <mml:mi>θ</mml:mi> </mml:mrow> <mml:mrow> <mml:mo>−</mml:mo> <mml:mn>1</mml:mn> <mml:mrow> <mml:mo stretchy="true">/</mml:mo> </mml:mrow> <mml:mn>2</mml:mn> </mml:mrow> </mml:msubsup> </mml:math> <jats:inline-graphic xmlns:xlink="http://www.w3. org/1999/xlink" xlink:href="apjad1596ieqn2.gif" xlink:type="simple" /> </jats:inline-formula> due to non-Alfvénic MHD waves. These statistical correction factors <jats:italic>β</jats:italic> <jats:sub>ml</jats:sub> have partial agreement with correction factors <jats:inline-formula> <jats:tex-math> <? CDATA ${\beta }_{\mathrm{sim}}$? > </jats:tex-math> <mml:math xmlns:mml="http://www.w3. org/1998/Math/MathML" overflow="scroll"> <mml:msub> <mml:mrow> <mml:mi>β</mml:mi> </mml:mrow> <mml:mrow> <mml:mi>sim</mml:mi> </mml:mrow> </mml:msub> </mml:math> <jats:inline-graphic xmlns:xlink="http://www.w3. org/1999/xlink" xlink:href="apjad1596ieqn3.gif" xlink:type="simple" /> </jats:inline-formula> obtained from MHD simulations. The relative importance of the statistical correction is estimated by assuming that each simulation correction has both a statistical and a physical component. Then the standard, structure function, and original DCF equations appear most accurate because they require the least physical correction. Their relative physical correction factors are 0.1, 0.3, and 0.4 on a scale from 0 to 1. In contrast, the large-angle and parallel-<jats:italic>δ</jats:italic> <jats:italic>B</jats:italic> equations have physical correction factors 0.6 and 0.7. These results may be useful in selecting DCF equations, within model limitations. </jats:p>