The $70-m$ Deep Space Network antennas are used for deep space communications with interplanetary spacecraft at a frequency of $8\mathrm{.}4$ GHz $($X$-$band$).$ The surfaces of the DSN antennas have rms deviations of the surface with respect to a perfect paraboloid of $=1\mathrm{.}0$ mm $.$ At $8\mathrm{.}4$ GHz the aperture efficiency is $55\%.$

a$)$ Calculate the gain of a $70-m$ DSN antenna at $8\mathrm{.}4$ GHz $.$

b$)$ Calculate the $3-dB$ beamwidth of a $70-m$ DSN antenna at $8\mathrm{.}4$ GHz $.$

c$)$ Recent interplanetary missions have started to use Ka$-$band $(32-GHz)$ for communications to allow for greater data throughput. Calculate the gain and $3-$dB beamwidth for Ka$-$band, still accounting for the $1-$ mm surface rms$.$ At $32$ GHz the aperture efficiency is $45\%.$

d$)$ How do the gain and beamwidth at Ka$-$band compare to X$-$band?

e$)$ If Ka$-$band has higher data throughput, why has it only been used recently? List two or more potential disadvantages of using Ka$-$band over X$-$band for deep space communications.