This thesis is an anthology of three theoretical problems in astrophysics with applicationsto planetary atmospheres, their immediate external environment, andtheir birthplaces.The first problem concerns the linear and nonlinear stability of equilibriummotions. In particular, we investigate the stability of a charged particle in a circularorbit subject to axisymmetric gravitational and electromagnetic forces. Weextend previous work on this problem by including a toroidal magnetic field. Insource free regions, we prove that the toroidal field has no effect on either the equilibriumorbit or its stability. However, in regions with charge or current sources,we find that the toroidal field can potentially alter the stability of the orbit althoughit plays no role in determining the equilibrium orbital coordinates. Weshow that the toroidal field enters the system as a gyroscopic force and can stabilizeotherwise unstable particle orbits for a range of physical parameters. We alsodemonstrate that gyroscopic stability so attained is only temporary and that theslightest dissipative forces can render the system unstable again, albeit at a slowerrate. Our results may apply to dust grains orbiting within a rotating planetarymagnetosphere.The second problem looks at how magnetic diffusion alters the character ofthe magnetorotational instability. The magnetorotational instability (MRI) is generallyregarded as the foremost contender for driving magnetohydrodynamic turbulencein differentially rotating astrophysical disks thereby facilitating accretion.In disks that are poorly ionized, the three non-ideal effects of ohmic, Hall andambipolar diffusion take hold. We conduct a systematic analysis of the non-idealMRI in the shearing sheet framework and elucidate the character of the eigenmodes.We derive expressions for the kinetic and magnetic stresses and uncovera new characteristic scale when the net magnetic field and angular velocity areanti-parallel. This scale may possibly signal a change in the nature of the ensuingturbulence provided dissipative effects are small. Non-ideal effects pervade disksaround young stars and our results may be relevant to the dynamical evolution ofcertain parts of such disks.The third problem deals with modeling irradiated atmospheres. Using theprinciples and methods of radiative transfer, we derive a plane-parallel equilibriumanalytical model of an atmosphere that receives radiant energy from aboveand below. Constructing analytical atmospheric models is a challenging exerciseand obtaining one with a frequency dependent opacity function is one of the mainpoints of difficulty. By using the picket-fence technique for modeling spectral lines,we are able to derive exact analytical solutions and thereby obtain thermal profilesof an atmosphere that receives strong collimated high frequency radiation fromabove in addition to thermally emitted radiant energy from below. Our modelalso includes the effects of coherent scattering in the lines and the continuum. Anobvious application of our analysis is to modeling planetary atmospheres, in particular,those outside of our own solar system that are presently being discoveredin the thousands.