Magnetized Protoplanetary Disks around Young Stars in the Presence of Realistic Cooling

Abstract In this paper, we study the effects of magnetic fields and ohmic resistivity on the structure of protoplanetary disks (PPDs) in the presence of realistic radiative cooling. We use the self-similar method to solve the magnetohydrodynamic equations in spherical coordinates ( r , θ , ϕ ). Our solutions indicate that a strong magnetic field reduces the outflow region while increasing the mass accretion rate, which in turn leads to a thicker disk. Our model applied to the FU Orionis PPD indicates that in the inner regions of the disk ( r = 10 au), <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mover accent="true"> <mml:mrow> <mml:mi>M</mml:mi> </mml:mrow> <mml:mrow> <mml:mo>̇</mml:mo> </mml:mrow> </mml:mover> </mml:math> is the order of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mover accent="true"> <mml:mi>M</mml:mi> <mml:mo>̇</mml:mo> </mml:mover> <mml:mo>≈</mml:mo> <mml:mn>5</mml:mn> <mml:mo>×</mml:mo> <mml:mn>1</mml:mn> <mml:msup> <mml:mn>0</mml:mn> <mml:mrow> <mml:mo>−</mml:mo> <mml:mn>4</mml:mn> </mml:mrow> </mml:msup> <mml:msub> <mml:mi>M</mml:mi> <mml:mo>⊙</mml:mo> </mml:msub> <mml:mspace width="0.25em"/> <mml:msup> <mml:mi>yr</mml:mi> <mml:mrow> <mml:mo>−</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msup> </mml:math> and the mass outflow rate is <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:msub> <mml:mover accent="true"> <mml:mi>M</mml:mi> <mml:mo>̇</mml:mo> </mml:mover> <mml:mi>w</mml:mi> </mml:msub> <mml:mo>=</mml:mo> <mml:mn>0.1</mml:mn> <mml:msub> <mml:mover accent="true"> <mml:mi>M</mml:mi> <mml:mo>̇</mml:mo> </mml:mover> <mml:mi>acc</mml:mi> </mml:msub> </mml:math> , which is consistent with the observed range of accretion rates. Additionally, we calculated the cooling timescale in our model to assess one possible fragmentation condition necessary for planet formation. Our results indicate that, although the gravitational instability condition exists in the midplane of the disk, the cooling timescale is not short enough ( τ cool ≈ 1000Ω −1 ), and one possible mechanism for planet formation is not met.