PH 508: Spacecraft design and operations

Dr Mark Price ([email protected]) ,
Prof Mark Burchell (convener), Prof Richard Holdaway (CCLRC),
Dr Vicky Fitzgerald.
Spring 2011.
Dr. Mark Price
◦ Room 103C
◦ E-mail: [email protected]
Lectures notes will eventually be available on
Moodle, but can be downloaded now from:
30 hours of lectures
◦ Low Earth Orbit (4 lectures and 1 workshop – MCP)
◦ Spacecraft Systems (11 lectures and 1 workshop –
◦ Project Management (5 lectures and 1 workshop –
◦ Orbital Mechanics (10 lectures and 1 workshop –
4 workshops (weeks TBD)
2 class tests (weeks 18 & 24)
To provide a basic understanding of the
major subsystems of a spacecraft system.
understanding of spacecraft trajectory and
orbits, including interplanetary orbits, launch
phase and attitude control.
To provide an awareness of the basic ideas of
how space is a (multi-billion dollar!) business
opportunity and some of the management
tools required in business.
Low Earth Orbit (4 lectures)
◦ The vacuum, radiation environment and thermal
environment that a spacecraft encounters in Low
Earth Orbit (LEO) and that environment’s effect on
the spacecraft materials (electronics, superstructure
Spacecraft Systems (11 lectures)
◦ An introduction to spacecraft and their
environment. Covers Spacecraft structures and
materials, thermal control, power systems, attitude
control systems, the rocket equation and
Project management (Dr. Vicky Fitzgerald)
Orbital mechanics (Prof. Richard Holdaway)
◦ Explains the evolving framework in which world-wide public and
private sector space activities are conceived, funded and
◦ Introduces the basics of business planning and management
applicable to any project!
◦ Using celestial mechanics (Newton’s laws) to the real world
application of satellite/spacecraft missions.
◦ Basic equations of motion are outlined in order to give an
understanding of the causes and effects of orbit perturbations.
◦ Descriptions are given of different types of orbit and methods are
outlined for the determination and prediction of satellite and
planetary orbits
◦ Assessment of mission analysis problems such as orbital choice,
ground station usage, satellite station-keeping and orbital
An understanding of the way in which space
missions are configured both from the point-ofview of the constituent subsystems, mission
profile (i.e., the project aims) including the
influence of the space environment.
Appreciate the constraints and trade-offs which
led to one mission configuration over another.
Appreciate space activities from a commercial
viewpoint and be familiar with basic management
tools for planning work (e.g., Gant charts, Pert
charts etc.)
Make (valid) approximations and solve problems
using a mathematical approach.
Spacecraft Systems Engineering by Fortescue,
Stark and Swinerd (3rd edition).
[*NB: No joke, this is pretty much ESSENTIAL ]
Library has 6 copies (Classmark: TL875,
location: Level 2 West. Also has some copies
in core text collection (1 week loan?).
Library has many copies (>15) of the second
edition (same classmark and location)
Amazon price: £37.95
Please contact me ([email protected]) if
you really can’t source/afford a copy!
Orbital Motion by A. Roy. (3rd edition) Library
classmark: QB 355, 6 copies. Amazon price: £53.19
Space vehicle design by Griffin and French [Classmark:
TL 875]
Space mission analysis and design by Wertz & Larson
[Classmark TL 790]
Satellite Technology and its applications by Chetty
[Classmark TL796]
Spacecraft Attitude determination and control by Wertz
[Classmark TL3260]
Rocket and Spacecraft propulsion by Turner [Classmark:
TL 872]
Basic elements
of a space mission
F&S, Fig. 1.3, Page 7
Crude overview: [Read: chapter 2, F&S]
Ground phase (vehicle construction)
Pre-launch phase (payload and rocket integration)
Launch phase
Space operations phase
Other (planetary, asteroid belt, cometary
environments, de-orbital/end of life phase)
Can be sub-divided further into:
Manufacture stage
Assembly stage
Test and checkout stage
Handling stage
Transportation stage
Storage (prior to rocket/payload integration)
Manufacture and construction stage
Could be argued that this phase is the same as for
any other industrial product. Incorrect!
Spacecraft manufacture is very expensive, and very
few spacecraft are actually made.
Spacecraft (even the lowly comms satellite) are very
complicated with many subassemblies and built-in
redundant systems.
Operational constraints mean that each spacecraft (or
rocket) is only used once (slight exception is the
space shuttle) and cannot be ‘test driven’. It has to
work, and it has to work first time!
[Q: Approximate cost of comms./science satellites?]
Manufacture and construction stage (continued)
The end user environment (see later in course)
imposes unusual constraints in terms of mass,
volume, power, allowable materials, reliability,
technology etc. over conventional manufacturing.
All this pushes up the cost, construction time and
complexity of the end product (spacecraft or
Need to impose the highest standards of quality
control to guarantee the manufactured end
Test and checkout stage
 A time-consuming and therefore expensive
process. Remember: it has to work, and it has to
work first time!
◦ Example: A chip fails during a PCB test. What do you do?
◦ Answer: Replace ALL chips from the same manufacturer
on all PCBs.
◦ Example: A solar panel generates insufficient power for
the end user requirement. What do you do?
◦ Answer: Rebuild panel, delaying the mission by 6
◦ HST mirror!
Handling stage
To be ‘space qualified’ all components and
assemblies have to handled in a clean
Human operatives have to wear paper
coveralls, gloves, hairnets, facemasks.
All handling of sub-assemblies is
documented meticulously (do not drop a
spanner onto the spacecraft, or one of the
flight detectors on the floor!)
Transportation stage:
Manufacture and checkout facility may be a long
distance (>1000 km) from the launch site (end
user requirement, depends on the required orbit)
Vibration and shocks are monitored during
transport by gauges.
Transported in a sealed environment to prevent
ingress of dust, moisture and temperature
Just moving a spacecraft is expensive!
Missions get delayed, sometimes for years. Estimated cost
of HST storage was $500M - $1Bn!
‘Ground Phase’ duration can last years. Time to build a
typical Spacecraft ~3 years, longer for scientific payloads.
Recent attempts have been made to try and ‘production
line’ the process particularly for telecommunication
satellites due to commercial pressures.
‘Ground Phase’ environment: ‘one-g’, standard
temperature and pressure, in a clean room.
[Aside: clean rooms are quantified by the number of
particles per cubic foot (US). A class ‘1000’ being dirtier
than a class ‘100’. Spacecraft and scientific payload
assembly is normally done in a class ‘100’ or ‘1000’]
Spacecraft is delivered to the launch site for
integration with rocket.
The whole assembly is then moved to the
This period can last days, but delays can (and
do) occur.
Environment: ‘one-g’ gravity, temperature
and pressure that of the launch site, cool dry
air can be flowed over the spacecraft to give a
certain degree of environmental control.
Duration: typically ~1000 seconds, from
ground to orbit.
Environment: Vibration/noise, acceleration
shock (‘g-force’), mechanical shock
(vibration), thermal changes (ground – space),
pressure change (1 atmosphere – vacuum).
Vibration/noise caused by:
◦ Burning of fuel
◦ Operation of pumps, compressors, turbines etc.
◦ Aerodynamic buffeting
Vibration/noise environment
Two modes of transport to payload
1. Vibration through payload mounting structure
2. Acoustically through the atmosphere
At its worst (peak intensity) at launch (from
reflections off ground, main firing of rocket)
and during transonic flight through the
atmosphere. Uneven air flow over the rocket
causes buffeting/shearing forces.
Dr Mark Price ([email protected]) ,
Prof Mark Burchell (convener), Prof Richard Holdaway (CCLRC),
Dr Vicky Fitzgerald.
Spring 2011.
Typical Ariane 4 & 5 noise spectrum
Definition of noise intensity, I
I  20 log 10 
 2  10 
I has units of decibels, ‘dB’.
P is the overpressure (i.e., a ‘relative’
pressure) referenced to 20 millipascals (at
Vibration is given by spectral density, units of g2/
Hz (g = acceleration)
Ariane 4 vibration
Spectrum (F&S, Fig 2.2).
Actual shape is rocket
and payload dependent.
Both vibration and noise can damage a
Designers/engineers do ‘destruction’ tests on
engineering/flight spare components to make
sure the flight model Spacecraft will survive.
The fairing (the detachable cover over the
Spacecraft) vibrates. Due to weight
constraints, the fairing has to be lightweight
and thus is prone to vibration.
Acceleration (‘g force’ – normalised to Earth standard
Acceleration is not constant. Has several peaks
and troughs during the launch phase.
These peaks and troughs happen as various
stages ignite and drop-off and during jettisoning
of the payload fairing.
Early ‘brute force’ rockets had high values of ‘g’.
Early Mercury rockets peaked at ~6.7g.
Space shuttle has a ~3g sustained peak. A
sustained thrust reduces the peak g experienced.
Note: starts at
1g (ground) ends
at 0g (free-fall).
Ariane 4 acceleration
profile (F&S – Fig. 2.3)
If –gz (ie., ‘up’) is > 5g blood supply to brain
stops (heart is pumping against 5g). Leads to
Can survive front-back motions at higher g as
limitation is the squashing/tearing of soft tissue
in the lungs.
During an uncontrolled descent, some Russian
cosmonauts survived >18g! (Soyuz 18 + others?)
For very short time periods (<msec) accelerations
can be very much higher (~hundreds of g) as
fairings detach etc.
Heat experienced by payload is not due to the
radiative/conductive heat from the rocket
motor. Payload is well insulated from this
The payload fairing is forcing its way through
the atmosphere and gets hot (think inverse
heat shield).
Atmospheric frictional heating occurs and
this heat can be conducted to the payload or
radiatively from the fairing’s inner surface.
Effect decreases with altitude due to
decreasing atmospheric density.
Total energy input, E, into payload can be
calculated via:
E 
 F dx
Where dx is each interval of distance travelled
F is the drag force experienced travelling dx
The drag force, F, is defined as:
F 
C D Av
CD is the drag coefficient - a function of
atmospheric density, ρ. Typical values are
between 0.5 – 2. Also a function of altitude
A is the cross-sectional area of the spacecraft
in line of flight
v is the velocity.
Emery equation difficult to solve exactly as
some terms are altitude dependent.
Main points:
◦ Energy input is large
◦ Dominated at low altitude by increasing velocity
(acceleration through thick atmosphere).
◦ Decreases at high altitude at ρ decreases.
But this is energy input. The temperature is a
function of the heat capacity of the fairing
and the way it dissipates heat and how that
heats the payload.
Fairing temperature:
◦ Atmospheric frictional heating
◦ Specific heat capacity
◦ Radiative, conductive and convective heat loss
Payload temperature determined by:
◦ Radiative and conductive thermal pathways from fairing.
◦ Then direct radiative input after fairing is jettisoned
from space environment. Jettison occurs at an altitude of
~100 km.
◦ Heat input into Ariane V is typically 500 W m-2 with a
peak of 1135 W m-2.
Atmospheric pressure/density
Affects heating of fairing and thus payload
Affects the noise/vibration environment
Affects the final velocity achieved
Thus is very important!
As the atmospheric pressure drops, the
pressure in the payload bay drops. The
depends on the venting through the fairing.
Need good venting paths to avoid sudden
pressure drops and large pressure
differentials across the payload (‘pop!’).
◦ For Ariane rockets pressure venting is ~10 mBar s-1.
◦ For the Space Shuttle, the venting pressure rate is
How does the atmospheric pressure change
with density?
However, the absolute pressure varies not just with altitude,
but with the Sun! Due to fluctuating energy input from the
Sun, weather, time-of-day, solar cycle.
Int. Ref.
Different species
concentration as
function of altitude
(US Standard
Things to ‘take home’
◦ The design and implementation of a space mission
is a complicated and expensive task.
◦ Each separate phase has to implement the highest
possible level of quality control. It has to work, and
it has to work first time!
◦ Many different things to consider when designing a
mission: power requirements, weight, thermal
control, mechanical robustness, system
redundancy, etc.
[Will crop up again in PH608, and probably PH711]
Can be broadly categorised into:
◦ Near Earth Environment
◦ Deep Space
◦ Other ‘local’ environment (planetary orbits, asteroid
belt, cometary etc.).
As ‘Near Earth’ is local space we’ll start with
the general case: deep space.
Deep Space: gravity
“Zero gravity” . Not true. You are always
subject to gravity, but in a freefall state (or
coasting) you appear to have ‘Zero gee’.
‘Zero gee’ is now going out of fashion to be
replaced with the more correct term: “microgravity (μg)”.
Spacecraft vibrations can shake the structure
giving rise to a μg environment.
Deep space: μg
◦ Small forces -> light structures can be employed
◦ Cheap to launch (but objects still have mass and
◦ Low self-damping -> vibration prone (rigidity more
important than strength, determines limiting mass).
◦ Difficult to test material behaviour on the ground
◦ Fluid flow problems in μg environment (‘bubbles’)
◦ Need active pumping/circulation system (no gravity
◦ Humans? Sleeping, eating, respiration etc. all affected by
μg environment.
Deep space: pressure (or lack of!)
At Geostationary Earth Orbit (GEO, altitude
~36,000 km) pressure ~10-15 Pa!
At altitudes >120 km don’t really use
pressure units, but a ‘number density’
(number per cubic metre, m-3). The molecules
of a gas are too separated to interact, so can
treat as separate species.
Different species
concentration as
function of altitude
(US Standard
[F&S, Pg. 22]
At an altitude of ~400 km the number density
is approximately 1012 – 1014 atoms per m3
depending on the species and solar activity.
Typical number density at low altitude inside
the atmosphere ~1024 m-3.
The result of such a low pressure is
outgassing. Solids give off materials
contained within them when the ambient
pressure ~ vapour pressure (10-11 – 10-4 Pa)
Outgassing examples:
◦ Metals → adsorbed/absorbed gases and water on
◦ Polymers →volatile components (normally organic)
which are part of their matrix material.
◦ Composite →absorbed water.
This process starts immediately the pressure
drops and can last months until all trapped
molecules are released.
Outgassing consequences:
◦ Polymers: loss of organic components leads to chemical changes
and thus to mechanical and electrical changes (they become
brittle, and their electrical conductivity changes).
◦ Composites: loss of water leads to mechanical shrinkage and thus
Outgassed material can condense onto nearby surfaces –
normally the coldest. This could be a telescope mirror and
lead to frosting/degrading of the surface.
The camera on NASA’s Stardust mission to visit a comet
acquired a layer of ‘goo’ (technical term) during launch
and lead to blurred images.
Solutions: Careful material selection before construction.
Vacuum bake out preflight, then coat the surface to seal it.
Sublimation: (solid → gas phase)
◦ Metals: happens at very low rate but does occur (see Table
next slide).
◦ Plastics/polymers: also occurs at low rate
◦ Lubricants: very high rate
A potential problem if a thin film is involved, or if
exposure is very long.
Sublimation of metals can lead to short/open circuits
in PCBs
Sublimation of polymers & lubricants can lead to
recondensing of material somewhere else, and
probably somewhere unwanted!
Solution: careful material selection. Use low volatility
oils and solid lubricants (graphite powder).
Note: very low rate for metals, but non-zero!
[F&S, Table 2.8, Pg. 40]
Whisker growth:
 Exacerbated by vacuum.
 Fibre-like monocrystalline filaments grow under
vacuum conditions.
 Some metals more prone than others. E.g., Mo,
W, Zn, Cd, Al, Sn, Ag.
 These conducting whiskers can lead to electrical
shorting of PCB and ICs and arcing within the
spacecraft (non-desirable!).
 Example: in the 1990s Hughes built satellites
which are now starting to fail. Supposed reason
for failure is they used tin (Sn) coatings on relays
and whisker growth now probably occurring.
Material strength & fatigue
Generally enhanced in vacuum by
approximately an order of magnitude
Possible reasons:
◦ Retardation of crack propagation?
◦ Material sublimation
◦ Thermal ‘hotspots’ (for non-conductive polymers,
no convective cooling).
Electromagnetic radiation environment:
Dominated by the Sun
Yellow dwarf star: class G2
Spectral peak ~440 nm (‘yellow’).
Spectrum approximated by a ~5800 K blackbody
Has enhanced UV and X-ray output over a
blackbody spectrum.
◦ Energy output ~3.85 x 1026 Watts (equivalent to the
energy output of burning all the Earth’s fossil fuels
for 50 milliseconds!).
◦ Radiation intensity falls off as 1/r2
Slight aside: the Sun’s radiation density at the
◦ Assume Earth – Sun distance = 149.6 million km =
1.49 x 1011 metres.
◦ Surface area of sphere at that distance (4πr2) =
2.81 x 1023 m2.
◦ Solar output = 3.85 x 1026 Watts
◦ Radiation density at earth’s surface = 3.85 x 1026/
2.81 x 1023 = 1369 Watts m-2 (ignoring
attenuation from atmosphere etc.) – generally
referred to as the ‘solar constant’.
1) Assuming the power output from the Sun is 3.85 x 1026 Watts what is
the radiation density at the surface of?
(1369 W m-2, 100% of Earth’s solar constant)
And what percentage is that density compared to the Earth’s solar
2) 130 decibels is defined as the pain threshold of hearing (at 1 kHz). What
pressure does this correspond to? Calculate the pressure for 120 dB
(permanent hearing damage), 85 dB (safe working limit) and 50 dB
(standard conversation level).
Show ALL working, and state where you obtained your mean Sun-planet
distance from please! (Wikipedia isn’t always correct!).
Electromagnetic radiation environment
So, if you put a Spacecraft into space its
temperature will stabilise as a function of:
Radiation incident on structure
Radiation reflected from structure
Radiative heat loss (its emissivity)
Its distance from the Sun
Solar activity levels
to a lot of physics,
Space physics and
Solar spectrum showing blackbody fit
and UV, X-ray excess [F&S, Fig. 2.7, Pg. 18]
Thermal considerations: acceptable
temperature values
◦ Spacecraft structure: -100 → +100° C (approx.)
◦ Electronics: -10 →+40° C
◦ Inner solar system will be too hot, so need reflective
surfaces, shades and passive cooling (radiators).
◦ Out solar system will be too cold, so need heaters
(power hungry) and insulation.
Ultra-violet radiation
◦ causes material degradation and embrittlement (particularly of
plastics and polymers).
◦ Also changes the resistivity of some materials (again, mostly
polymers and plastics).
◦ Leads to deterioration of plastics, paints, adhesives, epoxy resins
◦ Use protective coatings (UV absorbing paint) and metal films to
protect vulnerable materials.
X-rays and gamma-ray radiation cause similar problems but
have much lower flux (and also much harder to protect
The charged particle environment.
Solar wind – outward flow from Sun. Mainly protons
driven outward by radiation pressure.
At Earth, these protons have a velocity of 450 km s-1
(a kinetic temperature, Tk ~105)
At Earth, number density ~9 cm-3.
Solar flares give an enhanced flux and a higher
energy (30 MeV, factor of several thousand higher
than normal).
After a solar flare the Earth sees two events:
◦ The first approximately 20 minutes after the flare
◦ The second ~1 day later with an enhanced solar wind and a
velocity of ~1000 km s-1.
Cosmic rays (a misnomer, actually particles)
Originate outside the Solar System from
supernovae, neutron stars, black holes etc.
Mostly highly energetic (MeV – TeV) H+, He++,
Cn+, On+ and Fen+ ions.
◦ Organic materials suffer.
◦ Electronics suffer single event upsets (SEUs) and
(eventual) degradations. Especially semi-conductors.
◦ Degradation due to disruption of crystalline lattice.
◦ SEUs cause bit errors in CPU and subsequent software
malfunction and latch-up etc.
Cosmic rays (continued)
Solutions (partial)
◦ Shielding (normally Ti foil). Partially effective but can
produce secondary ‘showers’ of lower energy particles
with shorter stopping distances. May make things worse!
◦ Use ‘radiation hard’ components. Some technologies
more resilient than others. Generally a trade-off
between active area and disruption (large ICs have big
active volume, but one event may only cause one bit
error, a small SMT IC may suffer thousands of bit
◦ Software error detection and correction algorithms and
coincidence detection.
Now look at the differences between a ‘generic’ deep
space environment and the local space around the
Earth – the NEO environment.
Thermal effects:
 Albedo and “Earth Shine” (the Earth reradiates some
energy back into space). Radiation density
~ 200 W m-2.
 Eclipses (Earth passes between Spacecraft and Sun).
Spacecraft cools
Solar cells inoperative
35 minutes per orbit (~hours) for LEO
1.2 hours per orbit (1 day) for GEO
Well understood effect, and thus can be accounted for in
design and operation and is not a problem.
Meteoroids (small, natural) and debris (man-made)
The Earth enhances the number density and flux of
interplanetary particles via:
◦ Gravitational focussing (particles are gravitationally
◦ Atmospheric focussing (particles interact with upper
atmosphere, slow down, and get captured).
◦ Aero capture
Total enhancement over interplanetary space
environment is ~ a factor of 10.
Man-made debris – now a major concern
Created by:
Dead satellites
Old upper stages of rockets
Fragments from exploded rockets/stages
Flecks of paint
Aluminium oxide spheres (microns in size) from solid
rocket burns. 1 burn generates 1020 such spheres.
When any of this debris hits something get a
cloud of smaller ejected debris. The process is
self propagating! (‘going viral’!).
[F&S, Fig. 2.21, Pg. 35
Probably out of date!]
Impact damage on Endeavour (STS-118)
and Challenger window (STS-7)
 Gravity gradient varies 10-3 – 10-11 g.
Depends on spacecraft size, configuration,
 Accelerations (μg) due to attitude control,
internal movement (mechanisms, pumps,
 Difference from “zero gee” is important for
some applications, experiments involving
crystal growth etc., but not for the majority.
Partial vacuum
 At orbital altitudes (few hundred km) vacuum is hard,
but not total.
 This causes a drag force, slows the Spacecraft down
and its orbit drops. Therefore need a altitude control
 The ISS needs regular reboosts during its life to
maintain its orbit.
 Also get low density plasmas and subsequent arcing
between points at different electrical potential.
 Solutions:
◦ Good design (as always!)
◦ A ‘wake shield’. Fly Spacecraft behind a big shield. The
vacuum behind the shield is improved.
Atomic oxygen (not an obvious problem!)
 Atomic O recombines on Spacecraft surfaces,
giving off a blue glow. This glow can interfere
with scientific observations (optical astronomy).
 Atomic O also erodes some materials (e.g.,
kapton thermal blankets) chemically and
physically (encounter velocity ~8 km s-1). Erosion
rates ~1 μm/day at altitude of 250 – 300 km.
 Alters the reflective/emissive properties of
materials thus degrading thermal control.
 Affects other materials, many plastics, silver etc.
Magnetic field
 Earth’s magnetic field influences charged
◦ Reflects some charged particles to give partial
protection in LEO
◦ Traps others (Van Allen belts – has a confinement
period of years). Regions of trapped particles may
be unavoidable
The South Atlantic Anomaly (SAA) is a
particular problem as hard to avoid (altitude
of a few hundred km).
Spacecraft charging
 Caused by exposure to plasma and/or ionising
 Particularly bad at geosynchronous altitudes
 Clouds of trapped low energy (~keV) electrons
formed by magnetic disturbances
 Interact with Spacecraft
Charging of dielectrics
Build-up of large potential differences
Electrical arcing
Physical damage
Disruption of electronics
Spacecraft charging (continued)
 Solution: provide conductive paths to prevent
large potential differences. E.g., coat glass
solar cells with indium oxide (indium oxide is
conducting, but transparent).
 Protection
◦ Shielding
◦ Careful orbit selection
◦ De-orbit old spacecraft upper stages or push into a
higher ‘graveyard’ orbit.
◦ Thermal. Changes with distance from Sun.
◦ Magnetic fields
 Strong: Jupiter (severe problems), Saturn, Uranus and
 Weak: Mercury, Venus and Mars
◦ No debris, but can get more natural dust (i.e., Saturn’s
rings) but usually tenuous.
Asteroid belt
◦ Low number density
◦ Dust (?)
◦ Charged gas/plasma near nucleus
◦ Random loss of material at high speed
◦ ‘Dusty’ environment
Summary – you should now have an understanding of
the various environments that a spacecraft could
Debris, etc.
Also an understanding of how these environments
affect the Spacecraft and its subsystems
◦ General case of deep space
◦ Specific case of NEO and the differences between the two.
◦ Other areas around Solar System bodies.
 The various phases of a space mission from
‘concept’ through to ‘end-of-life’ phase.
 An appreciation of some of the details of each of
these phases and how financial, engineering and
science constraints etc. affect mission design.
 How a spacecraft’s environment changes from
ground level, near earth orbit and deep space.
 How these environments (radiation, thermal, dust
etc.) feedback into the final mission design.
 Understand how to use the drag equation to
work out the force on a body as it travels
through the atmosphere
 Calculate the solar constant for Earth and
(other bodies) making justifiable
 Derive the escape velocity of a body.
Dr. Mark Price
◦ Room 103C
◦ E-mail: [email protected]
Lectures notes will eventually be available on
Moodle, but can be downloaded now from:

similar documents