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Physics Notes 2010 Core Module 1: Space Contextual Outline Scientists have drawn on advances in areas such as aeronautics, material science, robotics, electronics, medicine and energy production to develop viable spacecraft. Perhaps the most dangerous parts of any space mission are the launch, re-entry and landing. A huge force is required to propel the rocket a sufficient distance from the Earth so that it is able to either escape the Earth’s gravitational pull or maintain an orbit. Following a successful mission, re-entry through the Earth’s atmosphere provides further challenges to scientists if astronauts are to return to Earth safely. Rapid advances in technologies over the past fifty years have allowed the exploration of not only the Moon, but the Solar System and, to an increasing extent, the Universe. Space exploration is becoming more viable. Information from research undertaken in space programs has impacted on society through the development of devices such as personal computers, advanced medical equipment and communication satellites, and has enabled the accurate mapping of natural resources. Space research and exploration increases our understanding of the Earth’s own environment, the Solar System and the Universe. This module increases students’ understanding of the history, nature and practice of physics and the implications of physics for society and the environment. SI Units

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8(06#%'0<'=6>,%+#2"' (0/"' (0/' SI Prefixes

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Gravity is a force of attraction that exists between any two masses o Usually a very small, if not negligible force o When one or both of masses is as large as a planet the force becomes significant o Weight is the force on an object due to a gravitational field – vector quantity

Gravitational field – vector field within which a mass will experience a force o Direction of field line at any point is towards centre of the Earth o Field vector: a single vector that describes the strength and direction of a uniform vector field o For a gravitational field, the field vector is g – same as direction of associated force o Net force applied to a mass will cause it to accelerate according to Newton’s 2nd Law – g also represents

acceleration due to gravity

!"8 9:;7-'(#,+-,#-#1+-(*%#'(#*0-6',-,'/(-7#%(%0*<#'.#0%7-,%4#,/#)/0=#4/(%# Gravitational potential energy (Ep): the energy of a mass due to its position within a gravitational field o Equal to the work done to move the object to the point o Work done is the measure of how much energy is required to displace an

object a specified distance o Ground is our defined zero level – Ep = 0

Ep = work done to move to the point = Fs = (mg) x h = mgh!

o If an object is moved against the gravitational field, positive work is done – acquires energy o If an object is moved with the gravitational field, negative work is done – loses energy

#

#

#

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W = force exerted (N) on mass m, m = mass (kg), g = field vector or acceleration due to gravity (N kg-1 or m s-2)

W = mg

Ep = mgh Ep = gravitational potential energy (J) m = mass (kg), g = acceleration due to gravity (m s-2), h = height (m)

3

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On a large scale, Ep defined as the work done to move an object from infinity (or some point very far away)

to a point within a gravitational field o Ep = 0 at an infinite distance from the planet o Strange side effect as a consequence

Because gravitation is a force of attraction work must be done on an object to move it from a point, x, to infinity – gains energy Ep

Put another way, the work that would be done in moving an object from a very large distance away from a planet to its current distance = Ep

Negative sign indicates work done by the system (not on the system) in moving an object from a very large distance away to its present distance

Negative work represents Ep lost by the system as objects are brought together (converted into other forms of energy, most probably KE)

Since Ep reduced below the zero level it is appropriate that it should appear as a negative value Missing energy lends stability to a system – e.g. Earth would need to get energy back from somewhere if

it were to separate from the Sun – can be thought of as binding energy since the lack of this energy binds a system together

Ep = -G

m1m2 r

Ep = gravitational potential energy (J) G = 6.72 x 10-11 m1 = mass of planet (kg), m2 = mass of object (kg), r = distance separating centres of masses (m)

4

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Variations in the value of g o Geographical location – minor variations in value of g around Earth’s surface occur because:

Lithosphere varies in thickness & structure due to factors such as tectonic plate boundaries & dense mineral deposits – places where thick/dense will experience greater gravitational force

Earth not a perfect sphere – flattened at poles therefore g will be greater at poles – closer to centre Spin of Earth creates a centrifuge effect that reduces the effective value of g - Effect greatest at equator, no effect at poles

o Altitude – formula for g shows value of g varies with altitude above Earth’s surface As altitude increases, g decreases dropping to zero only when altitude

Length of Pendulum vs. T^2

0

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0.00 0.20 0.40 0.60 0.80 1.00 1.20

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A projectile is any object launched into the air then left to complete their unpowered flight o Subjected to 1 force – force of gravity, & 1 acceleration – g (9.8 ms-2 downwards near Earth’s surface) o To simplify analysis of projectile motion air resistance is ignored

Air resistance depends on many factors such as shape, surface area & texture of the projectile as well as velocity through the air – acts as a retarding force in both the horizontal & vertical directions

Path of projectile distorted away from a perfect parabola The trajectory of a projectile is the path that it follows during its flight o In absence of air resistance the trajectory of a projectile will be in shape of a parabola o The motion of a projectile can be regarded as two separate & independent motions superimposed upon

each other – first noted by Galileo Frame of reference can be used – y-axis for vertical motion & x-axis for horizontal motion The two motions are perpendicular, & therefore independent thus we can treat them separately Adding the vectors will produce the resulting direction and magnitude of motion at any one time

o Vertical motion – subject to acceleration due to gravity, ay = 9.8ms-2 down Symmetrical – will rise up, slow to a halt, then fall back to Earth Will speed up until, when back at its starting point, it is going as fast as it was when thrown Time taken to fall from peak height to starting point exactly equals time taken to rise to peak height

o Horizontal motion – once under way experiences no acceleration (ignoring air resistance) ax = 0 ms-2

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Body

Centre of Orbit

Mass (kg)

(3 sig. figs.)

Radius (km)

(3 sig. figs.)

Radius (m)

(3sig. figs.)

g on Surface (m/s/s)

(3 sig. figs.) = Gm/r2

Weight of a 100kg person on surface

(N) = g x 100

Jupiter Sun 1.90E+27 6.99E+04 6.99E+07 25.9 2590 Saturn Sun 5.68E+26 5.82E+04 5.82E+07 11.2 1110

Neptune Sun 1.02E+26 2.46E+04 2.46E+07 11.2 1120 Uranus Sun 8.68E+25 2.54E+04 2.54E+07 9.00 900 Earth Sun 5.97E+24 6.38E+03 6.38E+06 9.80 980 Venus Sun 4.87E+24 6.05E+03 6.05E+06 8.87 887

6

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Acceleration equations – whenever an object changes its velocity it has accelerated

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Initial velocity can be resolved into vertical & horizontal components using

trigonometry Velocity of projectile at other times during motion can be found by combining

vertical & horizontal velocities in vector addition At the end of an object’s trajectory it has the same initial velocity & same angle

to the horizontal, although it is now directed below the horizontal Maximum height: vy y Trip time: vy = 0, calculate t to rise to peak, double this to find trip time Range: find trip time, calculate range using x = uxt (no acceleration)

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Galileo was the first to conduct a mathematical analysis of projectile motion and document such work Instead of analysing the motion as a whole, he divided the motion into two components: o Motion in x & y planes are independent of each other because they are perpendicular o Vertical: realising that the vertical motion is affected by the downwards force of gravity – a = 9.8 ms-2 o Horizontal: whilst the horizontal component experiences no acceleration

Also realised that the motion of a projectile is parabolic in nature, due to a uniform acceleration and a constant horizontal motion (ignoring air resistance)

Led him to consider reference frames – what all measurements are referred to Galileo also postulated that all objects accelerated towards Earth at same rate – g same for all objects o Heavier objects don’t fall faster than lighter ones (only true if air resistance is ignored) o Discovered by rolling balls down highly polished inclines instead of dropping them, to reduce the air

resistance. o Enabled him to slow down the motion enough to make more accurate observations of the motion

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Escape velocity: the initial velocity required by a projectile to rise vertically & just escape the gravitational field of a planet

o Only depends upon universal gravitation constant, mass & radius of the planet o Escape velocity is achieved when the kinetic energy (Ek) is equal to gravitational potential energy (Ep)

At a distance of infinity both Ek and Ep = 0, similarly due to the principal of conservation of momentum Ek and Ep must also equal to zero at the planet’s surface

o Earth’s escape velocity is 11 200 ms-1

or about 40 000 kmh-1 – same for all objects o Greater mass of planet = greater escape velocity, greater radius = smaller escape velocity o Does not apply to a rocket which continues its thrust well after launch

If you throw a stone directly up will rise to a certain height & fall back to Earth o If thrown faster will rise higher o If thrown fast enough it should rise & continue to rise, slowing down but never falling back to Earth,

finally coming to rest only when it has completely escaped the Earth’s gravitational field o The initial velocity required to achieve this is known as escape velocity

8"P Q5,7'(%#R%),/(I.#1/(1%;,#/&#%.1-;%#6%7/1',<# Newton wrote that it should be possible to launch a projectile fast enough so that it

achieves an orbit around the Earth o A stone thrown from a tall tower will cover a considerable range before striking

the ground o If thrown faster it will travel further before stopping, if thrown even faster it will

have an even greater range o If thrown fast enough then, as stone falls, Earth’s surface curves away so that

stone never actually lands but orbits the Earth o Only a thought experiment – could not test o Showed that at any altitude a specific velocity is required for an object to achieve

a stable circular orbit

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8

o If specific velocity exceeded slightly, object will follow an elliptical orbit around Earth o If exceeded further, object will follow parabolic or hyperbolic path away from earth o This is how space probes depart Earth

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The sensation of weight that you feel derives from your apparent weight (equal to the sum of the contact

forces resisting your true weight (W = mg)) o Includes normal reaction force of the floor on your body or the thrust of a rocket engine

‘g force’ used to express a person’s apparent weight as a multiple of his/her normal true weight (when standing on surface of the Earth)

o In a rocket the astronaut’s body exerts a downward weight force on the floor & the floor meets this with an upward reaction force equal to m x g

o In addition the floor exerts an upward accelerating force equal to m x a o Astronaut feels an apparent weight = mg + ma

Astronauts are placed under high accelerations in order to reach an orbit or escape velocity. The measure of “G’s” is more sensible and applicable when considering such high accelerations.

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A moving platform offers a boost to the velocity of a projectile launched from it if launched in the direction of the platform

o Earth orbits sun & rotates once on its axis per day hence the Earth itself is a moving platform with 2 different motions which can be exploited in a rocket launch to gain a boost in velocity

o Less fuel necessary to achieve target velocity, ability to carry greater payloads o Rotational motion can be exploited to achieve velocity needed for a stable orbit – launching at equator

toward the east in direction of Earth’s rotation Rotational velocity of the launch site relative to the Sun will add to the orbital velocity of the rocket

relative to the Earth to produce a higher orbital velocity achieved by the rocket relative to the Sun o Orbital motion around the Sun can be exploited for deeper space missions by planning the launch for a

time of year when the direction of the Earth’s orbital velocity corresponds to the desired heading Rocket allowed to orbit until direction of orbital velocity corresponds with Earth’s then its engines are

fired to push it out of orbit & further into space Orbital velocity relative to Sun adds to rocket’s orbital velocity relative to Earth to produce a higher

velocity relative to the Sun o Challenges:

The earth’s rotational and orbital motions pose a limitation to the direction and location of launches. - Since the equator is furthest from the centre of the earth, it has the greatest rotational velocity and

launch locations are generally needed to be located near to the equator. - In terms of the earth’s orbital motion, in considering interplanetary travel, our orbit in relation to

other planets must be considered, limiting the time of launch to achievable trajectories between planets.

weighttruenormalweightapparent

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! Rockets – powered projectiles – receive thrust from their engine/s for a significant portion of their upwards

flight & become more conventional projectiles only after their engines are exhausted o A rocket engine is different from most other engines – carries both its fuel & oxygen supply o Modern rockets use either solid or liquid propellants

Both expel hot gases through a nozzle upon combustion Law of Conservation of Momentum – during any interaction in a closed system the total momentum of

the system remains unchanged o During a launch the momentum of the gases shooting out the rear of the rocket must be equal to the

forward momentum of the rocket itself o In any one-second time interval:

While mass of gases during any given second is less than mass of the rocket their velocity is much greater – momenta are equal but opposite

Also

This is Newton’s 3rd Law of Motion – for every force there is an equal but opposite force Rocket is forcing a large volume of gas behind it & gases in turn force rocket forward – although the

two forces are equal & opposite the rocket experiences just one of them – the forward thrust o A rocket will accelerate according to Newton’s 2nd Law

o A rocket is subject to the following forces:

Its weight force directed downward Its thrust (delivered by engines) directed upward

F = ma mF

a

10

Reaction force of ground on rocket (equal to difference between weight & thrust while rocket on ground) directed upward

No net force until the thrust exceeds the weight of the rocket Air resistance directed downward against motion of rocket once it has left ground – significant as speed

builds but at relatively low speeds of early lift-off can be ignored When in orbit no thrust from engines, rocket is continually falling & only experiences weight force

o A rocket’s acceleration will not be constant because fuel constitutes up to 90% the mass of a typical rocket As fuel is burnt the mass of rocket decreases although thrust remains essentially constant Additionally, g reduces slightly with increasing altitude Air resistance decreases as air thins at high altitudes Results in rocket’s rate of acceleration increasing as flight progresses & velocity increases

logarithmically Acceleration equations above can only apply at an instant in time

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Prior to lift-off rocket is stationary so only weight & reaction force felt by astronaut – equal in magnitude but opposite in direction

o Apparent weight equals true weight & occupant experiences a g force of one When thrust of rocket exceeds weight there is a net force upwards – rocket accelerates upwards & the floor

exerts an upwards force on the occupant of (mg + ma) o Occupant experiences a g force of ((g + a)/9.8) which is a value > 1

Mass of rocket decreases as fuel is consumed hence acceleration & subsequent g force steadily increase, reaching maximum values just before the rocket has exhausted its fuel

o Increasing acceleration dangerous if it becomes too high as forces may become too large to withstand After a multi-stage rocket has jettisoned its spent stage there is only the downward acceleration due to

gravity before the next stage is ignited – has become a projectile in freefall o a = g so g force = 0 – astronauts experience a 0 apparent weight – weightlessness

2nd-stage rocket fires & quickly develops necessary thrust to exceed effective weight at its altitude, then starts to accelerate again

o g force greater than 1 & builds to maximum value before fuel supply is exhausted If third stage process is repeated

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Uniform circular motion: circular motion with a uniform orbital speed o Centripetal force: the force that acts to maintain circular motion and is directed towards the centre of the

circle

o Newton’s 2nd Law states that wherever there is a net force acting on an object

there is an associated acceleration Because it is on the horizontal plane gravity is not a factor Since centripetal force is the only force acting on the motion we can say that:

Fc = mass x centripetal acceleration

Fc = r

mv2

Fc = centripetal force (N) m = mass of object in motion (kg) v = instantaneous or orbital velocity of the mass (ms-1)

rv

ac

2

11

o Centripetal acceleration is always present in uniform circular motion. It is associated with centripetal force & is also directed towards the centre of the circle

Even though magnitude of velocity remains unchanged the direction is constantly changing – hence the velocity of an object in circular motion is constantly changing and it is accelerating

An astronaut in orbit around Earth still experiences an acceleration due to gravity of about 8.8 ms-2 – still has significant true weight

This acceleration due to gravity acts as the centripetal acceleration of the orbital motion & the astronaut’s weight forms the centripetal force

Acceleration = –g therefore no resisting forces acting on astronaut so no apparent weight 8"B#Z/?;-0%#]5-7',-,'6%7<#7/)#9-0,+#-(4#*%/D.,-,'/(-0<#/02',.##

Low Earth Orbit Geostationary Orbit Altitude > 250km to avoid drag

< 1000km – Van Allen belts start to appear Regions of high radiation trapped by

the Earth’s magnetic field & pose significant risk to live space travellers & electronic equipment

Approx. 35 800km Upper limits of Van Allen belts Near edge of magnetosphere Radius of orbit = 42168 km Can be calculated using Kepler’s Law of

Periods Period (T)

90 mins – 5 hours Approx. 24 hours – precisely matches that of Earth ‘geosynchronous’

Orbital velocity

At 250km approx 27900 km h-1

Affected by orbital decay Permanently over a fixed point on the Earth’s surface

If over Equator would appear stationary in the sky from Earth – always located in same direction

If not over Equator will appear from Earth to trace a ‘figure of eight’ every 24 hours

Not affected by orbital decay Uses Spying (cover whole surface of the Earth at

least once per day), space shuttles, Hubble telescope, long range weather forecasting satellites and land surveys

Scientific, communication satellites, weather satellites

Transfer orbit used to manoeuvre a satellite from one orbit to another – elliptical o Satellites headed for geostationary orbit placed into low Earth orbit then boosted up using a transfer orbit o Has a higher specific orbital energy that lies between that of the lower & higher circular orbits o In order to move a satellite into a different orbit its energy must be changed – Ek rapidly altered

12

Rockets fired to change satellite’s velocity which will increase or decrease the Ek (and therefore the total energy) to alter the orbit as desired

However, as soon as the satellite begins to change altitude, transformations between the Ep & Ek occur so speed is constantly changing

o Simplest & most fuel-efficient path is a Hohmann transfer orbit To move to a higher orbit a boost increases the satellite’s velocity, stretching the circular low Earth orbit

into a transfer ellipse When satellite reaches apogee (slowest point on the ellipse & furthest point from mass) it will be at

correct altitude but will be moving too slowly – rockets fired again to increase velocity to that required for the new higher, stable & circular orbit

To move v rocket boosts - Boosts slow satellite, changing its orbit, first from higher circular orbit into a transfer ellipse that

reaches down to lower altitudes - Then from perigee (closet point to mass) of the transfer ellipse changed into a circular low Earth orbit

Elliptical orbits – theory assumes orbits are circular – usually slightly elliptical o Ellipses can be round or elongated – degree of stretch known as eccentricity

Circle is an ellipse with an eccentricity of zero o Most satellites are placed into near-circular orbits o Satellites move quickly when closest to Earth & slow down as they move further away

Once a launched rocket has achieved a sufficient altitude it can be accelerated into the desired orbit o Necessary speed dependent upon mass of Earth & geometry of orbit o If speed is not reached the spacecraft will follow a shortened elliptical orbit that dips back down toward

the atmosphere, possibly causing re-entry o If speed is exceeded the spacecraft will follow an elongated elliptical orbit taking it away from Earth

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Kepler’s Law of Periods o Period (T): the time taken to complete one orbit

for planet 1 = for planet 2 o Can be used to compare any two bodies orbiting the same object – e.g. two planets orbiting the sun o Alternative expression of this law is

for any satellites orbiting a common central mass o Law of Universal Gravitation can be used to derive an expression for the constant k, law becomes:

Orbital velocity – the instantaneous direction & speed of an object in circular motion along its path o The velocity required to maintain an object in orbit o For uniform circular motion magnitude is constant & inversely proportional to the period of the orbit

2

3

Tr

2

3

Tr

kTr

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r = radius of the orbit of any given satellite (m) T = the period of that satellite’s orbit (s) k = a constant

22

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4GM

Tr G = the universal gravitation constant

= 6.676 x 10-11 N m2 kg-2

M = mass of the central body (kg)

Tr

v2

periodcircletheofncecircumfere

veloctiyorbital

13

o If expression substituted in to Kepler’s Law of Periods a formula specific to a satellite emerges o Value of satellite’s orbit depends on mass of planet being orbited & radius of orbit o For a satellite orbiting a planet the radius of orbit = radius of planet + altitude of orbit o For objects orbiting same body, altitude is the only variable that determines the orbital velocity required

for a specific orbit o Orbital velocity decreases as radius of orbit increases o Orbital velocity is independent of the mass of the satellite

Orbital energy – any satellite travelling in a stable circular orbit at a given radius has a characteristic total mechanical energy E

o Sum of its kinetic energy Ek (due to its orbital velocity) & its gravitational potential energy Ep (due to its height)

From chapter 1:

For a stable circular orbit Ek is half that of Ep but positive in value Therefore

A lower orbit produces a more negative value of E – less energy A higher orbit corresponds to more energy Specific orbital energy (mechanical energy per kilogram)

All of this theory assumes these orbits are circular

rGM

v

v = orbital velocity (ms-1) G = universal gravitation constant = 6.676 x 10-11 N m2

kg-2 M = mass of the central body (kg) r = radius of the orbit (m)

14

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All satellites in low Earth orbit are subject to some degree of atmospheric drag that will eventually decay their orbit & limit their lifetimes

o Atmosphere very thin 1000km above Earth but sufficient friction to cause a gradual loss of energy to heat

Loss of orbital energy = loss of altitude, spirals back towards Earth o Accelerating process

The two forces acting on a low-orbiting satellite are its weight & atmospheric drag

As satellite descends drag increases - Air density increases as altitude decreases - As satellite loses altitude some of Ep transformed into Ek & it speeds

up – drag force proportional to v2 so as speed increases the drag increases more sharply

At an altitude of 80km atmospheric drag increases sufficiently to start slowing descending satellite – braking effect rapidly builds

At around 60km atmospheric drag increases sharply, leading to huge heat & g force – all but largest satellites are vaporised

o Satellites can be lifted periodically back to intended orbital altitude by small rocket boosters o Actual service life can be unpredictable as atmosphere itself can change – e.g. an increase in solar

radiation can cause atmosphere to expand & rise up to meet a satellite, increasing drag #

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De-orbiting – the process of deliberately leaving a stable Earth orbit & re-entering the atmosphere to return to the surface of the Earth

o Different to orbital decay – deliberate, impulsive (rapid) orbital manoeuvre resulting in an elliptical path down to the atmosphere – not spiral

o Retroburn – spacecraft’s rockets pointed ahead & short burn executed to quickly reduce v & thus Ek

Required transfer ellipse must be calculated in advance to determine when & for how long retroburn must occur to

v o Glide phase – spacecraft’s orbit changes & carries it down

to meet atmosphere at ‘re-entry interface’ around 120 km altitude

No further manoeuvring required but spacecraft is speeding up again as Ep Ek as it falls

o Re-entry into & through atmosphere – begins to slow down again Extreme heat – spacecraft has a velocity of tens of thousands of km/h – significant Ek o Additionally, altitude of spacecraft’s orbit means it has considerable gravitational potential energy o As spacecraft re-enters it experiences friction with molecules of the atmosphere

Enormous Ek converted into heat & spacecraft to reaches extreme temperatures o Best shape for re-entry is a blunt one – when it collides with upper atmosphere at re-entry speeds it

produces a shockwave of compressed air in front of itself (like bow wave of a boat) Most of heat is then generated in the compressing air & significantly less heat caused by friction of the

air against the object itself Nose of space-shuttle kept up during re-entry & enter backwards – flat underbelly Protective ‘heat shield’ still needed Capsules – ablation – nose cone covered with a ceramic material (e.g. fibreglass) which is vaporised or

‘ablated’ during re-entry heating – dissipates heat & carries it away Space shuttle – slower orbital speed allows it to use different approach – covered in insulating tiles made

of glass fibre approximately 90% air – excellent thermal insulator & conserves mass - Must be waterproofed between each flight as they are porous

15

Decelerating g forces – greater angles of re-entry = greater rates of deceleration o Heat built-up faster as Ek is converted & higher g forces experienced by occupants of spacecraft o Research suggested g forces on astronauts be restricted to 3 g if possible, & 8 g was maximum safe load

although symptoms such as chest pain & loss of consciousness could be experienced at this level o Research conducted into ways to increase a human’s tolerance to g forces:

Traverse application of g load is easiest to cope with – blood not forced away from brain – astronauts should be lying down at take-off, not standing or sitting vertically

‘Eyeballs-in’ application of g loads easier to tolerate than ‘eyeballs-out’ – astronauts should lift-off forwards (facing up) but re-enter backward (facing up)

Supporting body in as many places as possible – contoured couch, built of fibreglass & moulded to suit the body of a specific astronaut settled on – using this can be subjected to loads of up to 20 g

Ionisation blackout – as heat builds up around spacecraft during re-entry, atoms in air around become ionised, forming a layer around the spacecraft

o Radio signals cannot penetrate, preventing communication between ground & spacecraft o Length of blackout depends upon re-entry profile o Apollo capsule blackout – 3-4 minutes whereas space shuttle suffers a longer period of 16 minutes

Reaching the surface – capsule reaches ‘entry interface’ at 120 km at an angle between 5.2° & 7.2° o Then descends over a range of 2800-4600 km, continually slowing down & at some point suffering

ionisation blackout o In last part of descent parachutes released to slow it to about 33 km h-1 o Splashes down into ocean , recovered by naval vessel o Early Soviet capsules descended over land, ejection seats used & descended to ground by parachute o Space shuttle has wings so pilot can control attitude of space shuttle & direct its descent to reduce g forces

During maximum deceleration, nose held up at 40° to slow progress & present the underbelly as a protected, blunt surface

After this, flown in a series of sharp-banking S-turns to control descent

When 500m above ground speed brakes applied (special flaps to increase drag) to reduce landing angle, landing gear deployed & shuttle touches down on runway

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When de-orbiting spacecraft’s orbit altered into a transfer ellipse that intersects the atmosphere at a desired angle

o Shallow angle selected to minimise extreme heating & g forces that can destroy a spacecraft o If angle is too shallow spacecraft may skip off atmosphere instead of penetrating it o For Apollo missions optimum angle was between 5.2° and 7.2° to the horizon

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Konstantin Tsiolkovsky (1857-1935) Played an important role in the development of the Soviet and Russian space programs Built first wind tunnel for his research Pioneer of the astronautic theory (engineering dealing with machines designed to work beyond earth’s

surface) Determined earth’s escape velocity: approximately 8 km/s o Determined this could be achieved by a multi-stage rocket fuelled by liquid hydrogen and liquid oxygen

Suggested propulsion by reaction (thrust) derived Tsiolkovsky Rocket Equation

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16

o The mathematical relation between the changing mass of a rocket as it burns fuel, the velocity of the exhaust gases, and the rocket’s change in velocity

o Maximum velocity a rocket can achieve – determined by specific impulse of propellants & empty mass of rocket fuel

o No rocket powered by conventional propellants can achieve relativistic velocities – mass ratios become too large

o Equation made it possible to find the exact amount of fuel required for the escape velocity of a rocket, using the mass of the spacecraft

His manuscripts/articles written between 1903 and 1914 are widely recognized as the first scientifically possible proposals to use rockets for space exploration.

Designed theoretical multi-stage rockets to back-up his research, created numerous scaled models First suggestion of liquid oxygen and liquid hydrogen as fuel: widely regarded today as the most efficient

rocket propellant Published over 500 articles on space travel and related subjects (although included are his science fiction

novels) Proposed several other innovations: metal blimp, aeroplane, spaceship, space elevator, spinning space

stations for artificial gravity, mining asteroids for minerals, space suits o Addressed problem of eating, drinking and sleeping in weightless environments o Biological systems to provide food and oxygen for space colonies.

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E6(783(*-.#2(*':%3+(&:(83.=(%-;3%832(,'(;2#<&%'( Vector: any quantity that has both magnitude & direction, e.g. force A vector field can be said to exist in any space within which a force vector can act, e.g. magnetic fields o Has strength & direction which corresponds to the vector that acts within it

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A gravitational field is a field within which any mass will experience a gravitational force o Surrounds any mass, vector field – has both strength and direction o Weak force so it takes a massive object to create a significant gravitational field o A large object will have a strong gravitational field and attract other masses toward it o If mass has little or no tangential velocity, will be dragged into the massive object and towards the

centre of the gravitational field. o If mass has some degree of tangential velocity, will be pulled into orbit or have trajectory through space

altered by the massive object with the force acting on the object pulling it towards the massive object o On a small scale (e.g. interior of a room) field lines – or lines of force – appear parallel & point down o On a large scale can see field lines have a radial pattern & point towards the centre of a mass o The closer to a mass, the closer the field lines indicating the field, & its force, are stronger o Gravitational field of a planet or star extends some distance from it o Any large object near another large object will have a gravitational field of its own & the 2 fields will

combine to form a more complex field

17

There is a point between the 2 where the strength of the field is zero – gravitational attraction between the two bodies are precisely equal but opposite in direction

Will be closer to the smaller body #

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Newton’s Law of Universal Gravitation: o Describes the force of attraction between any two masses o Every mass in universe attracted to every other mass in universe o Always an attractive force & is exerted equally on both masses o Body with larger mass will be less affected – a = F/m o Depends only upon the value of the two masses & their separation distance o The force is inversely proportional to the square of the distance – double the distance = ! the force o Formula for acceleration due to gravity can be derived by substituting F = ma (a = g)

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Gravity holds the solar system together, tying the planets to the Sun & the satellites to their planets, keeping them on an orbital path & determining the speed of their rotations

Deriving orbital velocity – in order to launch a satellite the orbital velocity must be known o Gravity serves as the centripetal force that maintains the orbital motion of a satellite o By equating Newton’s Law of Universal Gravitation with the centripetal force of an orbiting satellite, we

are able understand & calculate the orbital velocity required for a satellite to remain in orbit at a particular height

Radius of orbit, r, is the sum of the radius of the central body & the altitude of the orbit From formula, v required for a particular orbit only depends on mass of central body & radius of orbit Vital in understanding the motion of satellites since it is required to derive orbital velocity The greater the radius of orbit, the slower the orbital velocity required to maintain the orbit

F = force of gravity between two masses (N) G = universal gravitation constant = 6.67 x 10-11 N m2

kg-2 m1, m2 = the two masses involved (kg) d = the distance between their centres of mass (m)

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18

Deriving the constant in Kepler’s Law of Periods – Isaac Newton was able to determine the constant in Kepler’s Law of Periods (r3/T2 = k) when devising Law of Universal Gravitation

o Important in understanding orbital motion o r/T substituted into equation

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The slingshot effect, or planetary swing-by or gravity-assist manoeuvre, is a strategy used with space probes to achieve a change in velocity with little expenditure of fuel

o A spacecraft deliberately passes close to a large mass (e.g. a planet) so that the mass’s gravity pulls the spacecraft towards it

o Causes spacecraft to accelerate & it heads around the planet & departs in a different direction

o Departure speed of spacecraft relative to planet is same as approach speed relative to planet but change in direction can result in a real change in velocity relative to the Sun

o In general, a spacecraft will approach a planet at an angle to the planet’s orbital path

o By swinging behind the planet an increase in speed can be achieved o By swinging in front of the planet’s path a decrease in speed is

achieved o Interaction behaves as an elastic collision as bodies don’t touch so

there are no energy loses vi = initial velocity of spacecraft Vi = initial velocity of planet m = mass of spacecraft K x m = mass of planet vf = final velocity of the spacecraft Vf = final velocity of the planet Applying conservation of momentum to the collision and conservation of kinetic energy to the collision then solving simultaneously

o Expression represents maximum velocity achievable from slingshot effect by a head-on rendezvous At other angles, a spacecraft achieves lower velocities Final velocity of planet is marginally less than its initial velocity Ek of system has been conserved – spacecraft gains kinetic energy & planet loses this amount

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The aether was the proposed medium for light & other EM waves, before it was realised that these waveforms don’t need a medium in order to travel

o Early 1800s physicists turned to other wave motions, e.g. water waves, to better understand light o All of these waveforms need a medium through which to travel so believed light requires a medium o Nobody could find such a medium but belief was very strong in existence of the ‘luminiferous aether’ o Properties:

Filled all of space Very low density Perfectly transparent Permeated all matter & yet was completely permeable to material objects Great elasticity to support & propagate the light waves

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If aether existed our earth, moving through space at 30 000 ms-1 as it orbits the sun, should be moving through the aether

o From our point of view we should experience a flow of aether past us – an ‘aether wind’ o The speed of light should vary due to the presence of the aether wind o However aether thought to be extremely tenuous so any aether wind would be hard to detect o Many experiments designed & performed to detect it but all failed – assumption that detection

mechanisms were not sensitive enough o Definitive experiment to detect aether wind performed by A.A. Michelson & E. W. Morley in 1887 o Raced 2 light rays over 2 courses of same length, one into the supposed aether wind & one across o Then swung the apparatus through 90 degrees to interpose (swap) the rays o Looked for a difference between the rays as they finished the race, from which could calculate the value of

the aether wind (like a current)

20

o Method of comparing the light rays involves a very sensitive effect ‘interference’ – apparatus called an ‘interferometer’

o When looking into the telescope a pattern of light & dark bands will be seen – if aether wind exists, so one light ray is faster than the other, when apparatus is rotated the interference pattern should shift

o No such shift was observed – experiment repeated many times at different times of day & year, with more sensitive equipment but no evidence of aether model ever found, ‘null result’

o Scientific community adapted theory to keep it alive, however no modification survived close scrutiny Suggested Earth carried aether with it so that there was no relative motion Lorentz suggested apparatus used contracted in direction of motion

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1905 Einstein showed aether was not necessary at all – not influenced by Michelson-Morley result o Michelson-Morley experiment helped scientists of 20th century to reject aether model & accept Einstein’s

relativity o Showed that light travelled at a constant speed in all directions and challenged the aether theory by

showing that there was no effect of the so called ‘aether wind’. o Important experiment in helping others to decide between competing theories o However, did not sway scientific belief at the time – aether supporters saw null result as indication that

model needed improvement #

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Frames of reference are objects or coordinate systems with respect to which we take measurements An inertial frame of reference is a non-accelerated environment o Only standing at rest or moving with uniform velocity o A non-inertial frame of reference experiences acceleration o No mechanical experiment or observation from within frame can detect if frame is moving with constant

velocity or at rest #

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The principle of relativity – Galileo proposed that all steady motion is relative & cannot be detected without reference to an outside point

o E.g. if travelling inside a car you cannot tell if you are moving at a steady velocity or standing still without looking out the window

o Applies only in inertial frames of reference o Within an inertial frame of reference you cannot perform any mechanical experiment or observation that

would reveal whether you are moving with uniform velocity or standing still o In late 1800s belief in aether posed a difficult problem for the principle of relativity because aether was

supposed to be stationary in space & light supposed to have a fixed velocity relative to aether Meant that if a scientist set up equipment to measure speed of light from back of a train carriage to the

front & observed that light was slower than it should be, train must be moving into the aether This optical experiment provides a way to violate the principle of relativity where no mechanical

experiment could

21

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Around 1900, Einstein puzzled over apparent violation of the principle of relativity posed by the aether model

o Conducted a thought experiment: if I were travelling in a train at the speed of light & I held up a mirror, would I be able to see my own reflection?

If aether model was right, light could go no faster than the train & could never catch up with the mirror to return as a reflection

Principle of relativity is violated because seeing one’s reflection disappear would be a way to detect motion

If principle of relativity were not violated reflection must be seen normally which means that it is moving away from the mirror holder at 3 x 108 ms-1

This would mean that an observer on the embankment next to the train would see that light travelling at twice its normal speed

Einstein decided principle of relativity must not be violated & reflection in mirror must always be seen – meant aether did not exist

Also decided that train rider & person on embankment must observe light travelling at normal speed Realised that if both observers saw the same speed of light (since v = d/t) then the distance & time

witnessed by both observers must be different o Published ideas in 1905 in the paper ‘On the Electrodynamics of Moving Bodies’ which presented:

1st postulate: the laws of physics are the same in all frames of reference – the principle of relativity always holds

2nd postulate: the speed of light in empty space always has the same value, c, which is independent of the motion of the observer – everyone observes same speed of light regardless of their motion

A statement: the luminiferous aether is superfluous – no longer needed to explain the behaviour of light o A constant speed of light was extremely revolutionary because of the resulting implications:

Proved aether model was superfluous Thought experiments, and subsequently physical experiments showed that as observed velocity

increases, time dilates, length contracts and mass increases. Essentially this theory states that nothing in the universe is constant except for the speed of light. Hence,

mass, length and time are dependent upon the relative motion between frames of reference. P"W#T4%(,'&<#,+-,#'&#1#'.#1/(.,-(,#,+%(#.;-1%#-(4#,'?%#2%1/?%#0%7-,'6%#

In Newtonian physics distance & velocity can be relative terms but time is an absolute & fundamental quantity

o Observer on embankment outside train would see light travelling twice as far & at twice normal speed o Einstein’s theory says both observer on embankment & train rider will measure the same value for the

velocity of light, called ‘c’ Could only be true if observer & rider observed different times as well as different distances (v = d/t)

o Einstein radically altered assumptions of Newtonian physics so that now speed of light is absolute & space & time are both relative quantities that depend upon the motion of the observer

Measured length of an object & time taken by an event depends entirely upon the velocity of the observer

o Further, since neither space nor time are absolute the theory of relativity has replaced them with the concept of a space-time continuum

Any event has four dimensions (3 space coordinates plus a time coordinate) that fully define its position within its frame of reference

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Metre first defined in 1793 when French government decreed it to be 1 x 10-7 times the length of the Earth’s quadrant passing through Paris

When discovered that this was incorrect, redefined as the distance between 2 marks on a bar In 1874 the Systeme Internationale of units was set up – distance between 2 lines scribed on a bar of

platinum-iridium alloy

22

Since redefined twice to improve accuracy as technology & science improves Current definition uses the constancy of the speed of light in a vacuum (299 792 458 ms-1) & the definition

of one second to achieve a highly accurate definition consistent with the idea of space-time o 1 metre = the length of path travelled by light in a vacuum during 1/299 792 458 of a second o The old meter standard would not hold with distance & time dilation posed by the absolute constancy of

the speed of light #

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To better understand how time is affected by relativity, Einstein analysed our perception of simultaneous events

o The relativity of simultaneity: Einstein argued that if an observer sees 2 events to be simultaneous then any other observer, in relative motion to the first generally will not judge them to be simultaneous

o When we state the time of an event we are making a judgement about simultaneous events E.g. if we say ‘school begins at 9 am’ we are really saying the ringing of a certain bell & the appearance

of ‘9 am’ on a certain clock are simultaneous events Simultaneous events in one frame of reference are not necessarily observed simultaneous in another

o Thought experiment – an operator of a lamp rides in middle of a train carriage Doors at either end are light-operated – at an instant in time when the operator is alongside an observer

on an embankment, the operator switches on the lamp which, in turn, opens the doors Operator of the lamp will see the 2 doors opening simultaneously – distance of each door from lamp is

same & light travels at same speed both forward & backward so doors receive light at the same time & open simultaneously

Observer on embankment sees differently – after lamp is turned on, but before light has reached doors, the train has moved so that the front door is now further away & back door is closer

Sees light travelling forward & backward at the same speed but the forward journey is now longer than the backward journey, so back door seen to open before front door – not simultaneous

Both observers judge the situation correctly from their different frames of reference & this is a direct consequence of the constancy of the speed of light

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As an object approaches the speed of light its mass approaches an infinite value – prevents any object exceeding c

o An applied force is required to create acceleration, acceleration leads to higher velocities which leads to increased mass, further acceleration will require even greater force

o As mass becomes infinite, an infinite force would be required to achieve any acceleration at all – sufficient force can never be supplied to accelerate beyond the speed of light

o If a force is applied to an object work is done on it – energy is given to the object o In the situation above the energy is kinetic energy as the object speeds up – but at near c the object doesn’t

speed up as would be expected o The applied force is giving energy to object but object does not acquire the expected Ek – acquires extra

mass o Einstein made inference – mass (or inertia) of the object contained the extra energy o Relativity results in a new definition of energy:

2mcEE k

E = total energy (J) Ek = kinetic energy m = mass (kg) c = speed of light (3 x 108 ms-1)

23

When an object is stationary, so Ek = 0, it still has some energy due to its mass – ‘mass energy’ or ‘rest energy’

Rest energy: the energy equivalent of a stationary object’s mass, measured within the its rest frame o This equation states clearly that there is an equivalence between mass & energy

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Rest frame: the frame of reference within which a measured event occurs or a measured object lies at rest The relativity of length – as a consequence of perceiving time differently, observers in differing frames of

reference also perceive length differently (lengths parallel to direction of motion) o Thought experiment – time traveller has arranged the light clock so it runs the length of the train with

lamp & sensor located on back wall & mirror on front wall As train passes observer on embankment, light clock emits a light pulse which travels to front wall then

returns to back wall where it is picked up by the sensor which then clicks Journey observed by both people but what is the length of journey that each perceives?

Train traveller:

Observer on embankment: because train is moving at the same time the forward leg of the light pulse’s journey is longer & the return leg is shorter

Since is always less than one, the length of the train as observed by the person on the

embankment is less than that observed by the person inside the train Train shortens & the faster it goes the shorter it gets

L0 = length of an object measured from its rest frame (m) Lv = length of an object measured from a different frame of

reference (m) v = relative speed of the two frames of reference (ms-1) c = speed of light (3 x 108 ms-1)

2mcEE = rest energy (J) m = mass (kg) c = speed of light (3 x 108 ms-1)

length of light journey = ct0 = 2L0

2

2

0 1cv

LLv

2

2

1cv

24

o Length contraction: the shortening of an object in the direction of its motion as observed from a reference frame in relative motion

The length of an object measured within its rest frame is called its proper length, L0, or rest length Measurements of this length, Lv, made from any other inertial reference frame in relative motion parallel

to that length are always less – moving objects shorten in the direction of their motion As velocity approaches the speed of light, the observed length approaches 0 If a spaceship were blasting past a planet at near light speed, people on planet would see a very short

spaceship of nearly 0 length but space travellers wouldn’t notice any change to length of ship Travellers would observe a wafer-thin planet in their windows, since from their inertial frame of

reference it is the planet in rapid motion Any observer sees length contracted in other frames of reference if there is relative motion

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The relativity of time – time is perceived differently by observers in relative motion to each other o Thought experiment – uses a ‘light clock’ – light pulse released by lamp & travels length of clock, then

reflected back to a sensor next to the lamp, when sensor receives pulse of light it ‘clicks’ Traveller seated in a speeding train, light clock arranged vertically with lamp at ceiling & mirror on

floor, observer watching from embankment outside train When light pulse is released how long does it take to travel down to mirror & return to ceiling, as seen

by both the train traveller & the observer? Train traveller in the rest frame: if L is the height of the carriage, for the total journey we can say that:

distance = 2L = ct0

Observer on embankment: from outside the train the observer sees the light travelling along a much longer journey & its length can be determined using Pythagoras’ theorem

t0 is time taken for the clock to go ‘click’ as observed by the train traveller while tv is the time taken as observed by the person on the embankment

The term is always less than one so that tv is always greater than t0

- Clock takes longer to click as observed by person on embankment or, put another way, the outside

observer hears the light clock clicking slower than does the train traveller Time is passing more slowly on the train as observed by the person outside the train

2

20

1cv

ttv

t0 = time taken in the rest frame of reference (s) = proper time tv = time taken as seen from the frame of reference in relative

motion to the rest frame (s) v = relative velocity (ms-1) c = speed of light (3 x 108 ms-1)

2

2

1cv

25

o Time dilation: the slowing down of events as observed from a reference frame in relative motion The time taken for an event to occur within its own rest frame is called the proper time t0 Measurements of this time, tv, made from any other inertial reference frame in relative motion to the first

are always greater – moving clocks run slow – degree of time dilation varies with velocity Has been experimentally verified by comparing atomic clocks that have been flown over long journeys

with clocks that have remained stationary for the same period Any observer sees time dilated in another frame of reference – no absolute frame of reference No inertial frame is to be preferred over another & relativistic effects are

reversible if viewed from a different frame #

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Mass dilation: the increase in the mass of an object as observed from a reference frame in relative motion

o Measurements of this mass mv, made from any other inertial reference frame in relative motion to the first, are always greater – degree of mass dilation varies with velocity – moving objects gain mass

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Designers of a new kind of spacecraft claim these craft could journey to Proxima Centauri, our closest neighbouring star, at a speed of 0.1c (10% the speed of light). If true, how long would it take? Distance to Proxima Centauri 13 km A speed of 0.1c = 1.08 x 108 km h-1

o However, this is the time taken as observed from Earth – space travellers will, according to relativity,

record a slightly different travel time

The spacecraft reaches its destination 73.25 days short of 40 years

2

20

1cv

mmv

t0 = mass measured in the rest frame of reference (kg) = rest mass tv = mass as seen from the frame of reference in relative motion

to the rest frame (kg) v = relative velocity (ms-1) c = speed of light (3 x 108 ms-1)

When distances & speeds are this large, calculation is simpler if distance is expressed in light-years & speed is expressed in terms of c:

Method 1: if time recorded on Earth, tv, is 40 years, the rest time, t0, lapsed on spacecraft, can be calculated using the time dilation equation

Method 2: the occupants of the spacecraft see the distance they have to cover contracted according to the length contraction equation:

26

o Big impact upon space travel when speeds become relativistic ( When speeds are less than this, effects are almost negligible

Astronauts in orbit around Earth won’t observe any noticeable effect but on Galatica in the course of

one Earth day - Just over 20 mins have passed on board - Lengths have contracted to just 1.4% of their original lengths - The 4 light-year trip to Proxima Centauri would be completed in just over 20 days according to the

ship’s clock If light takes 4 years to cover the distance how could this starship, travelling at nearly the speed of light

manage the journey in 20 days? - As observed from Earth the starship does take 4 years; however, the clocks on the starship, both

electronic & biological are running slow so that by their reckoning only 20 days pass Time dilation o Allows travel into the future at high speeds, but not back to the past o Astronauts in a relativistic spacecraft will age slower than people back on Earth

Comparatively live longer during space travel, people on Earth will pass away before they return (Twins paradox)

Length Contraction: o As a spacecraft speeds up the apparent distance to objects ahead decreases

Trips on a relativistic spacecraft will appear to cover less distance to observers in the spacecraft o Could possibly allow travel to distant stars etc.

Time dilation & length contraction could theoretically allow exceptionally long space journeys within reasonable periods of time, as judged by the travellers

Mass Dilation: o Energy costs of achieving these speeds would be prohibitive even if possible – acceleration is most energy

costly phase of a space mission o As the speed of a spacecraft c its mass , more thrust required

Accelerations beyond 0.9c to involve ever greater forces & energy input for only marginal increases Speeds that can be achieved limited

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