Consider a spacecraft orbiting Saturn. Its high$-$gain $d=15\mathrm{.}0ft$ diameter antenna communicates with NASA's Deep Space Network $($DSN$)$ at the frequency $f=9\mathrm{.}5GHz($ X band$).$ Assume transmitting antenna efficiency ${}_t=0\mathrm{.}52$ and line losses at the transmitter $1\mathrm{.}80$ dB $.$ The receiving $D=70m$ diameter antenna at DSN has efficiency ${}_r=0\mathrm{.}57,$ line losses $1\mathrm{.}20$ dB $,$ and noise figure $F=1\mathrm{.}22.$

Assume the nominal atmospheric loss of $4\mathrm{.}0$ dB ; signal$-$to$-$noise ratio $\frac{E_b}{N_0}=9\mathrm{.}0dB$; and the distance between the spacecraft and the DSN station equal to $S=11\mathrm{.}0AU.$

a$)$ Calculate the power $P_t$ of the spacecraft transmitter at Saturn to support the communication bit rate $R=6000bi\frac{t}{sec}=6\mathrm{.}0$kbi$\frac{t}{sec}$

b$)$ Estimate the transmitting antenna $3-dB$ beamwidth $($in deg$)($the beamwidth will determine the requirements to spacecraft pointing for communications with the DSN site$).$