Added 6th and 7th week notes: GNSS and EP

Author: mariomerinomartinez

README.md

@@ -13,6 +13,8 @@ Each chapter in the notes corresponds to one week of the course:
 3. Space Environment
 4. Space Systems I
 5. Space Systems II
+6. Global Navigation Satellite Systems
+7. Fundamentals of electric propulsion
 
 # Important information
 
@@ -39,4 +41,4 @@ NOTES.
 
 # Release notes
 
-Last updated: 20170326
+Last updated: 20170403

chapters/chapter6.tex

@@ -0,0 +1,134 @@
+%----------------------------------------------------------------------
+\section[GNSS]{Global Navigation Satellite Systems}
+%----------------------------------------------------------------------
+ 
+%----------------------------------------------------------------------
+\paragraph{Operation principles of a GNSS position fix}
+%----------------------------------------------------------------------
+
+The basic ideas behind the operation of a GNSS can be summarized as follows:
+%
+\begin{enumerate}
+
+\item The satellites of a GNSS constellation each have a very precise on
+board clock that is synchronized with all other clocks in the constellation.
+
+\item The position vector of each satellite in the constellation is known (or
+can be made known) to any user of the system at all times.
+
+\item Each satellite broadcasts a time-stamped, identifiable signal in all
+directions. Each signal encodes, among other pieces of information, the exact
+time at which the signal was broadcast. 
+
+\item Any number of users can receive these signals simultaneously: they are
+all passive (i.e. receiving) users.
+
+\item A user receiving a satellite signal can compare the broadcast time with
+his/her own clock: the travel time for the signal $\Delta t = t_{RX} - t_{TX}$
+is a direct measure of the distance (or range) $d$ between the user and the
+emitting satellite, since the speed of light $c$  is a physical constant: $d =
+c\Delta t$.
+
+\item Assuming the instantaneous position of the satellites is known to the 
+user, using 3 such distance measurements to 3 different satellites allows, in 
+principle, to determine the 3 unknowns $x,y,z$ of the position of the user.
+
+\item However, for this to work, the user would need to have his/her clock
+perfectly synchronized with the constellation clocks. As this would be 
+impractical, a 4th distance measurement with a 4th satellite is needed to 
+determine the 4 unknowns of the problem: $x,y,z$ and $t$, the time of the user.
+
+\end{enumerate}
+%
+Obtaining a position (and time) fix for a user requires therefore solving a 
+system of 4 non-linear equations, one for each pseudo-range measurement:
+%
+\begin{align}
+c^2(t_{RX} - t_{TX,1})^2 &= 
+(x_{RX} - x_{TX,1})^2 + (y_{RX} - y_{TX,1})^2 + (z_{RX} - z_{TX,1})^2,
+\\
+c^2(t_{RX} - t_{TX,2})^2 &= 
+(x_{RX} - x_{TX,2})^2 + (y_{RX} - y_{TX,2})^2 + (z_{RX} - z_{TX,2})^2,
+\\
+c^2(t_{RX} - t_{TX,3})^2 &= 
+(x_{RX} - x_{TX,3})^2 + (y_{RX} - y_{TX,3})^2 + (z_{RX} - z_{TX,3})^2,
+\\
+c^2(t_{RX} - t_{TX,4})^2 &= 
+(x_{RX} - x_{TX,4})^2 + (y_{RX} - y_{TX,4})^2 + (z_{RX} - z_{TX,4})^2,
+\end{align}
+%
+where $x_{TX,i},y_{TX,i},z_{TX,i}$ is the position of the $i$-th transmitting
+satellite, $t_{TX,i}$ the time at which the signal from the $i$-th satellite
+was sent, and $x_{RX},y_{RX},z_{RX},t_{RX}$ are the unknowns (position and 
+time of the receiving user).
+
+%----------------------------------------------------------------------
+\paragraph{Operation principles of a GNSS velocity fix}
+%----------------------------------------------------------------------
+
+Once the procedure to determine the position is clear, it would be possible to
+compute the velocity of a user by observing how his/her position changes in
+time. In other words, by taking the derivative of the position vector:
+%
+\begin{equation}
+\bm v = \frac{\dd \bm r}{\dd t} \simeq \frac{\bm r(t_2)-\bm r(t_1)}{t_2-t_1}. 
+\end{equation}
+%
+
+There is, however, another, more accurate way to determine the user's velocity:
+using the Doppler shift of the signal frequency. This is done according to the
+following basic ideas:
+%
+\begin{enumerate}
+
+\item The exact frequency at which the satellite signals are emitted is known 
+to the users of the system.
+
+\item The velocity vector of each satellite in the constellation is known (or
+can be made known) to any user of the system at all times.
+
+\item Whenever the distance between a satellite and a user changes in time 
+(either because the satellite is moving or the user is moving),
+the perceived signal frequency by the user changes slightly: the frequency 
+will increase if the two of them are coming closer together, and it will 
+decrease if they are moving apart. This physical phenomenon is known as 
+\emph{Doppler shift}.
+
+\item Any user listening to the satellite signals can determine the Doppler 
+shift of any of them, $\Delta f = f_{RX} - f_{TX}$.
+
+\item As the user knows the velocity of each satellite, combining 3 such 
+measurements with 3 different satellites 
+it is possible to determine the user's velocity (3 unknowns: $v_x,v_y,v_z$).
+
+\item However, the cheap user's clock may have a drift with time. This may
+lead the user to wrong measurements of the Doppler shifts. Thus, a 4th 
+measurement with a 4th satellite is needed to determine the drift rate of the 
+user clock as an extra unknown.
+
+\end{enumerate} 
+%
+Obtaining a velocity (and clock rate) fix for a user requires therefore
+solving the following system of 4 non-linear equations:
+%
+\begin{align}
+\dot{t}_{RX}f_{RX,1}-f_{TX,1} = \frac{f_{TX,1}}{c} (\bm v_{RX}-\bm v_{TX,1}) 
+\cdot \frac{\bm r_{RX}-\bm r_{TX,1}}{\left|\bm r_{RX}-\bm r_{TX,1}\right|}
+\\
+\dot{t}_{RX}f_{RX,2}-f_{TX,2} = \frac{f_{TX,2}}{c} (\bm v_{RX}-\bm v_{TX,2}) 
+\cdot \frac{\bm r_{RX}-\bm r_{TX,2}}{\left|\bm r_{RX}-\bm r_{TX,2}\right|}
+\\
+\dot{t}_{RX}f_{RX,3}-f_{TX,3} = \frac{f_{TX,3}}{c} (\bm v_{RX}-\bm v_{TX,3}) 
+\cdot \frac{\bm r_{RX}-\bm r_{TX,3}}{\left|\bm r_{RX}-\bm r_{TX,3}\right|}
+\\
+\dot{t}_{RX}f_{RX,4}-f_{TX,4} = \frac{f_{TX,4}}{c} (\bm v_{RX}-\bm v_{TX,4}) 
+\cdot \frac{\bm r_{RX}-\bm r_{TX,4}}{\left|\bm r_{RX}-\bm r_{TX,4}\right|}
+\end{align}
+%
+where $\bm v_{TX,i}$ is the velocity of the $i$-th transmitting
+satellite, $f_{TX,i}, f_{RX,i}$ the frequencies at which the signal from the 
+$i$-th satellite was respectively broadcast and received,
+and $\bm v_{RX}, \dot{t}_{RX}$ are the unknowns (position and 
+clock drift rate of the receiving user). Note that the vector
+$(\bm r_{RX}-\bm r_{TX,i})/\left|\bm r_{RX}-\bm r_{TX,i}\right|$ is a unit 
+vector that points in the direction from the $i$-th satellite to the user.

chapters/chapter7.tex

@@ -0,0 +1,63 @@
+%----------------------------------------------------------------------
+\section[Electric Propulsion]{Fundamentals of Electric Propulsion}
+%----------------------------------------------------------------------
+ 
+%----------------------------------------------------------------------
+\paragraph{Key parameters of electric propulsion}
+%----------------------------------------------------------------------
+
+While chemical propulsion is \emph{energy limited}, as its performance depends
+on the amount of energy per unit mass stored in the chemical bonds of the
+propellant, electric propulsion is \emph{power limited}: the performance
+depends on how  much electric power is available on board.
+
+As with chemical propulsion, there are two basic parameters that characterize 
+an electric thruster:
+%
+\begin{enumerate}
+
+\item Specific impulse $I_{sp}$: this is a measure of the velocity at which
+the propellant is ejected from the thruster. A higher specific impulse allows
+fulfilling a propulsive mission (i.e. providing a given $\Delta V$ to the
+spacecraft) at a much lower expense of propellant. Electric propulsion devices
+have $I_{sp}^v$ in the order of $10$--$100$ km/s (about $1000$--$10000$ s if
+$I_{sp}$is expressed in seconds), whereas the best chemical rockets can only provide about $5$ km/s (about $500$ s).
+
+\item Thrust $F$: the force the propulsion system can generate to accelerate
+the spacecraft. Contrary to chemical propulsion, where $F$ can be very large,
+electric propulsion thrust levels are comparatively small: current thrusters
+provide $< 1$ N of force (typically, about $50$--$200$ mN).  Thrust is equal
+to the propellant mass flow rate $\dot m$ used in the thruster times the
+exhaust velocity, i.e., its specific impulse in velocity units: 
+%
+\begin{equation}
+F=\dot m I_{sp}^v.    
+\end{equation}
+
+
+\end{enumerate}
+%
+Electric propulsion provides tiny thrust levels, but it does so very 
+efficiently, using very little propellant. This enables space missions that 
+would be too ambitious to be carried out only with chemical propulsion, and to 
+lower the cost of existing missions. 
+Note, however, that electric propulsion cannot fully replace chemical 
+propulsion: large thrust levels are required to take off from the surface of a 
+planet, and for some quick propulsive maneuvers.
+
+Apart from $I_{sp}$ and $F$, there is another important figure of merit of an 
+electric thruster:
+%
+\begin{enumerate}[resume]
+
+\item Thrust efficiency $\eta_T$: it is a measure of how the input power $P$
+is  used for propulsion. The power contained in a jet of mass flow rate $\dot
+m$  and exhaust velocity $I_{sp}^v$ is $\dot m (I_{sp}^v)^2/2$. The thrust
+efficiency is defined as the ratio of the jet power over the input power:
+%
+\begin{equation}
+\eta_T = \frac{\dot m (I_{sp}^v)^2}{2P} = \frac{ F I_{sp}^v}{2P} =  
+\frac{ F^2}{2\dot m P} 
+\end{equation}
+
+\end{enumerate}

conquest-of-space-notes.pdf

@@ -29,6 +29,8 @@
 \include{chapters/chapter3}
 \include{chapters/chapter4}
 \include{chapters/chapter5}
+\include{chapters/chapter6}
+\include{chapters/chapter7}
 
 %----------------------------------------------------------------------
 \end{document}