A Web of Worlds presents The Ultimate Cheat Sheet for Astrophysics Students

Lachlan Marnoch www.webofworlds.net

v 1.0 July 28, 2018 Contents

1 Physics 5 1.1 Motion...... 5 1.1.1 Velocity...... 5 1.1.2 Acceleration...... 5 1.1.3 Newton’s Laws...... 5 1.1.4 Momentum...... 5 1.1.5 Centripetal Force...... 5 1.1.6 Kinetic Energy...... 5 1.1.7 Projectile Motion...... 6 1.1.8 Rotation...... 6 1.1.9 Euler-Lagrange and the Hamiltonian...... 7 1.2 Oscillations...... 7 1.2.1 Springs...... 7 1.3 Materials...... 8 1.3.1 Density...... 8 1.4 Energy...... 8 1.4.1 Work...... 8 1.5 Forces...... 8 1.5.1 Buoyancy (Archimedes’ Principle)...... 8 1.5.2 Friction...... 8 1.6 Waves...... 8 1.6.1 Wavelength...... 8 1.6.2 Angular Frequency...... 8 1.7 Newtonian Gravity...... 8 1.7.1 Force of Gravity...... 8 1.7.2 Gravitational Potential (potential energy per unit mass)...... 8 1.7.3 Gravitational field...... 8 1.7.4 Gravitational Potential Energy...... 8 1.7.5 Kepler’s Third Law...... 9 1.8 Electromagnetism...... 9 1.8.1 Notation...... 9 1.8.2 Maxwell’s Equations...... 9 1.8.3 Lorentz Force...... 9 1.8.4 Electric Field...... 9 1.8.5 Dipole moment...... 10 1.8.6 Electric potential...... 10 1.8.7 Electric potential difference...... 10 1.8.8 Electric potential energy...... 10 1.8.9 Charge densities...... 10 1.8.10 Current densities...... 10 1.8.11 Circuits...... 11 1.8.12 Capacitors...... 11 1.8.13 Magnetic fields...... 12 1.8.14 Inductors...... 12 1.8.15 Materials...... 12 1.9 Special Relativity...... 13

1 1.9.1 Interval...... 13 1.9.2 Four-vectors...... 14 1.9.3 Frames of Reference...... 15 1.9.4 Proper Velocity...... 15 1.10 General Relativity...... 15 1.10.1 Metrics...... 15 1.10.2 Rindler coordinates...... 16 1.10.3 Einstein notation...... 16 1.10.4 Christoffel symbols...... 16 1.10.5 Covariant derivatives...... 17 1.10.6 Riemann curvature tensor...... 17 1.10.7 Ricci curvature tensor...... 17 1.10.8 Einstein’s equations...... 17 1.11 Thermodynamics...... 17 1.11.1 Ideal Gases...... 17 1.11.2 Microstates...... 17 1.11.3 Entropy...... 18 1.11.4 Black bodies...... 18 1.12 Quantum Mechanics...... 18 1.12.1 The Uncertainty Principle...... 18 1.12.2 Bras and Kets...... 18 1.12.3 Rules for an Inner Product...... 18 1.12.4 The Born Rule...... 18 1.12.5 Expectation...... 18 1.12.6 Variance...... 19 1.12.7 Standard Deviation...... 19 1.12.8 Trace...... 19 1.12.9 Partial Trace...... 19 1.12.10 The Schr¨odingerEquation...... 19 1.12.11 Heisenberg equation of motion...... 19 1.12.12 Operators...... 19

2 Astrophysics & Astronomy 22 2.1 Astrometry...... 22 2.1.1 Redshift...... 22 2.1.2 Apparent magnitude...... 22 2.1.3 Absolute magnitude...... 22 2.1.4 Relative magnitudes...... 22 2.1.5 Flux-magnitude relationship...... 22 2.1.6 Color...... 22 2.1.7 Metallicity...... 22 2.2 Stars...... 22 2.2.1 Stellar Structure Equations...... 22 2.2.2 Timescales...... 23 2.2.3 Gravitational potential energy...... 23 2.2.4 Eddington Limit (hydrostatic equilibrium)...... 23 2.2.5 Mass-Luminosity Relationship...... 23 2.3 Galaxies...... 23 2.3.1 Hubble Eliptical Galaxy Classification...... 23 2.3.2 S´ersicProfile...... 23 2.3.3 Density of stars in the Milky Way Galaxy...... 24 2.4 Black Holes...... 24 2.4.1 Schwarszchild Radius...... 24 2.5 Instrumentation...... 24 2.5.1 Lensmaker’s equation...... 24 2.5.2 Focal ratio / Focal number...... 24 2.5.3 Field of view...... 24 2.5.4 Resolution Limits...... 24

2 2.5.5 Nyquist sampling...... 24 2.5.6 Plate scale...... 24 2.5.7 Fitting error...... 24 2.5.8 Adaptive optics error...... 25 2.5.9 Signal-to-noise ratio...... 25 2.5.10 Atmospheric Extinction...... 25 2.5.11 Rocket science...... 25

3 Mathematics 26 3.1 Notation...... 26 3.2 Algebra...... 26 3.2.1 Factorisation...... 26 3.2.2 Absolute Value...... 26 3.2.3 Quadratics...... 26 3.2.4 Logarithms...... 27 3.2.5 Vectors...... 27 3.2.6 Factorials...... 27 3.2.7 Inner product definition...... 28 3.2.8 Complex Numbers...... 28 3.2.9 Power Series...... 28 3.2.10 Matrix Operations...... 28 3.2.11 Matrix Types...... 29 3.2.12 Change of Basis Unitary...... 32 3.2.13 Commutator...... 32 3.2.14 Anticommutator...... 32 3.2.15 Cauchy-Schwarz Inequality...... 32 3.3 Geometry...... 32 3.3.1 Pythagorean theorem...... 32 3.3.2 Properties of shapes...... 32 3.3.3 Properties of solids...... 32 3.3.4 Circular formulae...... 33 3.3.5 Useful Functions...... 33 3.3.6 Coordinates...... 34 3.3.7 Hyperbolic Functions...... 34 3.4 Trigonometry...... 35 3.4.1 Identities...... 35 3.5 Calculus...... 39 3.5.1 Limits...... 39 3.5.2 Properties...... 39 3.5.3 Differentiation...... 39 3.5.4 Partial Differentiation...... 41 3.5.5 The Differential...... 41 3.5.6 Line Element...... 41 3.5.7 Integration...... 41 3.5.8 Vector Calculus...... 43 3.5.9 Dirac Delta Function...... 44 3.5.10 Approximations...... 44

4 Statistics 45 4.1 Variance...... 45 4.2 Standard Deviation...... 45

A Values 46 A.1 Physics...... 46 A.1.1 Physical Constants...... 46 A.1.2 Useful Quantities...... 47 A.2 Astronomy...... 47 A.2.1 Useful Quantities...... 47 A.3 Mathematics...... 48

3 B Units of Measurement 49 B.1 Natural Units...... 49 B.2 SI System...... 49 B.2.1 Base Units...... 49 B.2.2 Derived Units...... 50 B.3 CGS (centimetres-grams-seconds)...... 51 B.4 Astronomy units...... 52 B.4.1 Astronomical system...... 52 B.4.2 Equatorial Coordinate System...... 53 B.5 United States customary units (aka Imperial Units)...... 53 B.5.1 Length...... 53 B.6 Degrees of Angle...... 53 B.7 Miscellaneous Units...... 53 B.7.1 Pressure...... 53 B.8 Prefixes...... 54

C Mathematical Stuff 55 C.1 Trigonometric Values...... 55 C.1.1 Pythagorean Triples...... 55

D Boring stuff 56 D.1 Version History...... 56 D.2 Licensing...... 56 D.3 Contact...... 56 D.4 Credits...... 56

4 Chapter 1

Physics

1.1 Motion 1.1.1 Velocity ∆~x d~x • ~v = = = ~x˙ ∆t dt 1.1.2 Acceleration ∆~v d~v d2~x • ~a = = = = ~x¨ ∆t dt dt2 1.1.3 Newton’s Laws Newton’s First Law • When viewed in an inertial reference frame, an object either remains at rest or continues to move at a constant velocity, unless acted upon by a net force.

Newton’s Second Law d~p • F~ = m~a = net dt

Newton’s Third Law

• F~A = −F~B • When one body exerts a force on a second body, the second body simultaneously exerts a force equal in magnitude and opposite in direction on the first body.

1.1.4 Momentum • ~p = γm~v ≈ m~v

• ∆~p = F~ ∆t ∆~p d~p • F~ = = ∆t dt 1.1.5 Centripetal Force mv2 • F = c r 1.1.6 Kinetic Energy 1 2 • K = 2 mv

5 1.1.7 Projectile Motion 2 2 • vy = uy + 2ay∆y

• x = uxt F • ∆y = u ∆t + 1 a ∆t2 = u t + 1 y ∆t2 y 2 y y 2 m 1.1.8 Rotation Angular Velocity dθ • ω = = θ˙ dt v • ω = r • ~v = ~r × ~ω

Angular Acceleration dω d2θ • α = = =ω ˙ = θ¨ dt dt2

Moment of Inertia Point Mass • I = mr2 Several Point Masses • I = P mr2 Continuous mass • I = r2 dm ´ Parallel axis theorem

2 • I = Icom + md Thin disc rotating about centre MR2 • I = 2 Thin hoop rotating about centre • I = MR2 Thin rod rotating about centre ML2 • I = 12 Thin rod rotating about end ML2 • I = 3

Rotational Kinetic Energy

1 2 • Krot = 2 Iω

Total Kinetic Energy

1 2 2 • Ktot = Ktrans + Krot = 2 (mrcom + Icom)ω

6 Angular Momentum • L~ = I~ω = ~r × ~p

Torque dL • ~τ = I~α = = ~r × F~ dt 1.1.9 Euler-Lagrange and the Hamiltonian Lagrangian P • ` = T − V = a(q)q ˙lq˙m lm

• = K(q ˙l) − U(ql)

Generalised coordinates & momenta ∂L • pk ≡ ∂q˙k

Euler-Lagrange Equation d ∂` ∂` • − = 0 dt ∂x˙ ∂x

Action

tB • S[x(t)] = `(x ˙(t), x(t)) dt t´A

Hamiltonian P •H = plq˙ − L l ∂H • P˙ = − ∂Q ∂H • Q˙ = ∂P • P˙ = −ω2Q

• Q˙ = P

1.2 Oscillations 1.2.1 Springs Force of a Spring

• F~ = −ks~x

Potential Energy of a Spring 1 • U = k x2 s 2 s

Angular Frequency of a Spring rk • ω = s m

7 1.3 Materials 1.3.1 Density m dm • ρ = = V dV 1.4 Energy 1.4.1 Work b • W = F~ · d~l ≈ F~ · ~s ´a 1.5 Forces 1.5.1 Buoyancy (Archimedes’ Principle)

• Fbuoy = mdisplacedg = ρdVdg

1.5.2 Friction

• FK ≈ µK F⊥

• FS ≤ µSF⊥

1.6 Waves

• a sin(ωt − kx + φ)

2π • k = λ

1.6.1 Wavelength • v = fλ

1.6.2 Angular Frequency 2π • ω = = 2πf T 1.7 Newtonian Gravity 1.7.1 Force of Gravity GmM • F~ = rˆ = −m∇~ Φ(~r) ≈ −mgyˆ = m~g G r2

1.7.2 Gravitational Potential (potential energy per unit mass) GM(~r ) Gµ(r~0) • Φ(~r) = − P i = − d3r~0 |~r − ~r | ~0 i i ´ |~r − r | 1.7.3 Gravitational field GM • ~g(~r) = = −∇Φ(~r) r2 1.7.4 Gravitational Potential Energy GmM • U = − ≈ mgh G r

8 1.7.5 Kepler’s Third Law T 2 4π2 • = = constant r3 G(m + M)

1.8 Electromagnetism 1.8.1 Notation • ~r = ~r − r~0

1.8.2 Maxwell’s Equations

Integral form Differential form

1 ρ Gauss’s Law E~ · d~a = ρ dV ∇~ · E~ = vS ε0 tV ε0

P Q = enc ε0

Gauss’s Law for Magnetism B~ · d~a = 0 ∇~ · B~ = 0 vS

d ∂B~ Maxwell-Faraday equation E~ · d~l = − B~ · d~a ∇~ × E~ = − dt s ∂t ¸b S d ∂E~ Amp´ere’scircuital law B~ · d~l = µ J~ · d~a + µ ε E~ · d~a ∇~ × B~ = µ (J~ + ε ) 0 s 0 0 dt s 0 0 ∂t ¸b S S d = µ (I + ε E~ · d~a) 0 enc 0 dt S´ 1.8.3 Lorentz Force On a point charge • F~ = q(E~ + ~v × B~ )

On a current • dF~ = I d~l × B~ ´ • F~ = IL~ × B~

1.8.4 Electric Field ρ(r~0) ~ ˆ • E = 2 r dτ V´ r

From a single point charge

~ 1 q • E = 2 rˆ 4πε0 r

9 From a dipole

~ 2p •| Eaxis| ≈ 3 4πε0r ~ p •| E⊥| ≈ 3 4πε0r

1.8.5 Dipole moment • ~p = qd~

1.8.6 Electric potential 1 Q • V = 4πε0 r −ρ •∇ 2V = ε0

In a single-point charge field 1 q • ∆(~r) = 4πε0 r

1.8.7 Electric potential difference

~a • ∆(~r) = − E~ · d~l ~´b

In a single-point charge field 1 1 1 • ∆(~r) = Q( − ) 4πε0 b a

1.8.8 Electric potential energy 1 qQ • UE = q∆V = 4πε0 r

Energy stored in an electrostatic field distribution

1 2 • UE = 2 = 0E × volume

1.8.9 Charge densities Surface dq Q • σ = = da A

Line dq Q • λ = = dl L 1.8.10 Current densities Volume dI~ I • J~ = = = σ(E~ + ~v × B) = |q|nu(E~ + ~v × B) d~a⊥ A⊥ • ∇~ · J~ = 0

10 Surface dI~ I • K~ = = = σ~v ~ dl⊥ l

1.8.11 Circuits Electron drift velocity

• v¯ = uE~net

Current per unit charge

• i = nAcsv¯ = nAcsuEnet

Current dq • I = ei = enA uE = cs net dt

Electrical Power • P = IV = I2R

Voltage (Electric potential difference) • V = ∆V = IR = −ε

Electromotive Force (EMF) from a Non-Coulomb force F d •  = NC e

Resistance Lρ L • R = = A σA

• Rseries = R1 + R2 + ... + Rn 1 1 1 1 • = + + ... + Rparallel R1 R2 Rn

1.8.12 Capacitors Capacitance Q εA kε A • C = = = 0 V d d

Energy stored in a capacitor CV 2 • W = 2

Electric field in a capacitor Q • E = ε0A

Potential difference across a capacitor dQ • ∆V = − Aε0

11 1.8.13 Magnetic fields µ I~ × ˆ • B~ (~ ) = 0 r dl r 4π 2 ´ r µ q~v × ˆ µ I d~l × ˆ • dB~ = 0 r = 0 r 4π r 2 4π r 2 Magnetic field due to a wire µ 2I • B~ = 0 φˆ 4π r

Magnetic vector potential µ J~(r~0) • A~(~ ) = 0 dτ r 4π ´ r • ∇~ × A~ = B~

• ∇~ × (∇~ × A~) = −µ0J~

• ∇~ · A~ = 0

1.8.14 Inductors • ε = −LI

Energy stored in an inductor LI2 • W = 2 1.8.15 Materials Macroscopic Maxwell’s Equations (Materials)

Integral form Differential form

P Gauss’s Laws P~ · d~a = − QB ∇~ · P~ = −ρB vS P D~ · d~a = Qf ∇~ · D~ = ρf vS

Gauss’s Law for Magnetism B~ · d~a = 0 ∇~ · B~ = 0 vS d ∂B~ Maxwell-Faraday equation E~ · d~l = − B~ · d~a ∇~ × E~ = − dt s ∂t ¸b S ∂ ∂D~ Amp´ere’scircuital law H~ · d~l = I + D~ · d~a ∇~ × H~ = J~ + f,enc ∂t s f ∂t ¸b S

Dielectric constant ε • k = = εr ε0

• ε = kε0 = εrε

Susceptibility

• χe = 1 − εr

12 Polarisability

• P~ = ε0χeE~ = n~p

Bound Charge Surface

• σB = ~p · nˆ Volume

• ρB = −∇~ · P~ Total

• QB = σB + ρB = ~p · nˆ − ∇~ · P~

Electric displacement

• D~ = εE~ = kε0E~ = ε0E~ + P~

Magnetic field B~ • H~ = − M~ µ0

Magnetic dipole • ~m = I~a

Bound current

• J~B = ∇~ × M~

• K~ B = M~ × nˆ

1.9 Special Relativity 1.9.1 Interval • ∆s2 = −c2∆t2 + ∆x2 + ∆y2 + ∆z2 • ds2 = −c2 dt2 + dx2 + dy2 + dz2

• ∆s2 < 0 is a timelike interval. Events separated by this interval can be causally related. • ∆s2 = 0 is a lightlike interval. Events separated by this interval can be causally related, but only by a lightspeed signal. • ∆s2 > 0 is a spacelike interval. Events separated by this interval CANNOT be causally related.

Gamma Factor 1 • γ = r v 1 − ( )2 c dt • γ = dτ

13 Mass-energy

2 • Erest = mc 1 • E = γmc2 = mc2 r v 1 − ( )2 c

Relativistic kinetic energy • K = γmc2 − mc2

Length contraction l r v • l = 0 = l 1 − ( )2 v γ 0 c

Time dilation t • t = γt = 0 v 0 r v 1 − ( )2 c

Mass dilation m • m = γm = 0 v 0 r v 1 − ( )2 c

Relative Velocity 0 0 ∆x ux − vx • ux = = v u ∆t 1 − x x c2

Relativistic Momentum m~v • ~p = γ~v = p1 − (v/c)2

1.9.2 Four-vectors Four-space ct x • s = x =    y  z

Four-velocity  c  ds vx • u = = γ   dτ vy vz

Four-momentum E/c  c   px  vx • p =   = γm   = mu  py  vy pz vz

14 1.9.3 Frames of Reference Condition for an inertial frame d2x d2y d2z • = = = 0 dt2 dt2 dt2

Galilean Transformations • x0 = x + vt • y0 = y

• z0 = z • All assuming x is along the axis of motion and x = x’ when t = 0.

Lorentz Boosts vx • t0 = γ(t − ) c2 • x0 = γ(x − vt)

• y0 = y • z0 = z • (x is along the axis of motion) ct0  γ −vγ 0 0 ct x0  −vγ γ 0 0 x •   =      y0   0 0 1 0  y  z0 0 0 0 1 z

General Lorentz transformation b00  γ −vγ 0 0 b0 b01 −vγ γ 0 0 b1 •   =     b02  0 0 1 0 b2 b03 0 0 0 1 b3 • Motion along the x-axis.

Proper Time r tB 1 tB v2(t) • τ = dt = 1 − dt γ c2 t´A t´A 1.9.4 Proper Velocity ds • u = dτ 1.10 General Relativity 1.10.1 Metrics Minkowski −1 0 0 0  0 1 0 0 • η =    0 0 1 0 0 0 0 1

• ds2 = −c2 dt2 + dx2 + dy2 + dz2

15 Schwarzschild  2GM  −(1 − ) 0 0 0  c2r   2GM  • g =  0 (1 − )−1 0 0   c2r   2   0 0 r 0  0 0 0 r2 sin2 θ 2GM 2GM • ds2 = −(1 − )c2 dt2 + (1 − )−1 dr2 + r2 dθ2 + r2 sin2 θ dφ2) c2r c2r 1.10.2 Rindler coordinates Line element gx0 • ds2 = −(1 + )2(c dt0)2 + dx0 c2 1.10.3 Einstein notation • Contravariant: eα

• Covariant: eα

γ • tαβ = gβγ tα

β βγ • tα = g tαγ ∂x0α ∂xδ • t0α = tγ β ∂xγ ∂x0β δ ∂xγ ∂x0β • t0 β = t δ α ∂x0α ∂xδ γ

Metrics

2 α β • ds = gαβ dx dx 1 • gαβ = gαβ ( 1 α = β • δα = β 1 α 6= β

α γ α • δγ a = a αγ α • g gγβ = δβ

Four-vector product

α β α • a · b = gαβa b = aβb

1.10.4 Christoffel symbols ∂g ∂g ∂g • Γα = 1 gαδ( δβ + δγ − βγ ) βγ 2 ∂xγ ∂xβ ∂xδ ∂g ∂g ∂g • Γ = 1 ( δβ + δγ − βγ ) αβγ 2 ∂xγ ∂xβ ∂xδ d2xµ dxα dxβ • + Γµ = 0 dτ αβ dτ dτ

16 1.10.5 Covariant derivatives ∂tα •∇ tα = β + Γα tδ − Γδ tα γ β ∂xγ γδ β γβ δ ∂tαβ •∇ tαβ = + Γα tδβ + Γβ tαδ γ ∂xγ γδ γδ ∂t •∇ t = αβ − Γδ t − Γδ tαδ γ αβ ∂xγ γα δβ γβ ∂t β •∇ t β = α − Γδ t β + Γβ t δ γ α ∂xγ γα δ γδ α 1.10.6 Riemann curvature tensor ∂Γα ∂Γα • Rα = βδ − βγ + Γα Γ − Γα Γ βγδ ∂xγ ∂xδ γ βδ δ βγ ∂2g ∂2g ∂2g ∂2g • R = 1 ( αδ − αγ − βδ ) + βγ αβγδ 2 ∂xβ∂xγ ∂xβ∂xδ ∂xα∂xγ ∂xα∂xδ

• Rαβγδ = −Rβαγδ

• Rαβγδ = −Rβαδγ

• Rαβγδ = Rδγαβ

• Rαβγδ + Rαδβγ + Rαγδβ=0

1.10.7 Ricci curvature tensor γ • Rαβ = R αγβ

α • R = R α

1.10.8 Einstein’s equations 8πG • R − 1 g R + Λg = T µν 2 µν µν c4 µν 1.11 Thermodynamics 1.11.1 Ideal Gases Ideal Gas Law

• pV = NkBT

Heat / Thermal Energy • Q = mc∆T

Heat Capacity dQ • C = dT

Specific Heat Capacity C • c = m 1.11.2 Microstates (q + N − 1) • Ω = q!(N − 1)

17 1.11.3 Entropy

• S = kB ln Ω

1.11.4 Black bodies Energy of a photon • E = hf

Wien’s Displacement Law b 1 • λ = = (2.8977729 × 10−3) max T T

Stefan-Boltzmann Law • I = σT 4

Spectrum 2hc2 1 • B (T ) = λ λ5 hc exp( − 1) λkBT 2hν 1 • B (T ) = ν c2 hν exp( ) − 1 kBT 1.12 Quantum Mechanics 1.12.1 The Uncertainty Principle ¯h • ∆x∆p ≥ 2 ¯h • ∆E∆t ≥ 2 1.12.2 Bras and Kets •| ψi = hψ|†

1.12.3 Rules for an Inner Product •h ψ|φi ≡ (|ψi, |φi) • Symmetric: hψ|φi = hφ|ψi∗ • Linear in second component

• Anti-linear in first component

1.12.4 The Born Rule • P = |hψ|ψi|2

1.12.5 Expectation •h Ai = A|Ψ(x, t)|2 dx ´ •h Ai = hψ|A|ψi

18 1.12.6 Variance • var(A) = hψ|A2|ψi − hψ|A|ψi2

1.12.7 Standard Deviation q • δA = pvar(A) = hψ|A2|ψi − hψ|A|ψi2

1.12.8 Trace P • Tr(A) = hxj|A|xji j

1.12.9 Partial Trace

• T rB(|aiha| ⊗ |bihb|) ≡ |aiha|Tr(|bihb|)

• Tr(kAB) = T rA(T rB(kAB)) = T rB(T rA(kAB))

• ρB = T rA(ρAB) • The partial trace is linear

1.12.10 The Schr¨odingerEquation ∂ • i¯h Ψ(r, t) = Hˆ Ψ(r, t) ∂t ¯h2 ∂2Ψ(x, t) ∂Ψ(x, t) •− + V (x)Ψ(x, t) = i¯h 2m ∂x2 ∂t ¯h2 ∂2ψ(x) •− + V (x)ψ(x, t) = Eψ(x) 2m ∂x2 ∂ • Hˆ |Ψ(t)i = i¯h |Ψ(t)i ∂t 1.12.11 Heisenberg equation of motion d i • Aˆ(t) = [H,ˆ Aˆ(t)] dt ¯h 1.12.12 Operators

• ajk = hj|A|ki

Diagonalizable Operator P • A = λj|λjihλj| j

Normal Operator P • A = |λjihλj| j

Eigenstate Operators

n • (|λkihλk|) = |λkihλk|

Identity P • I = |xjihxj| j

19 Projector • P = |ψihψ|

Density operator P • ρ ≡ Pj|ψjihψj| j

• Hermitian: ρ† = ρ • Normalised: Tr(ρ) = 1 • Positive Semi-Definite: hψ|ρ|ψi ≥ 0, ∀ |ψi ∈ H • purity = Tr(ρ2) 1 – ≤ Tr(ρ2) ≤ 1 d – Pure: Tr(ρ2) = 1 1 – Maximally mixed: Tr(ρ2) = d

• ρA = T rB(ρAB) •h Ai = Tr(ρA)

Pauli Operators . 0 1 • σ = X = |0ih1| + |1ih0| = x 1 0

. 0 −i • σ = Y = i|1ih0| − i|0ih1| = y i 0

. 1 0  • σ = Z = |0ih0| − |1ih1| = z 0 −1

. 1 0 • I = |0ih0| + |1ih1| = (Sometimes included) 0 1

• T rx = Tr(Y ) = Tr(Z) = 0

• With respect to Hilbert-Schmidt Inner Product: √ ||X|| = ||Y || = ||Z|| = ||I|| = 2

Properties • Unitary • Hermitian

• λ = ±1

Photon Annihilation and Creation Operators √ • aˆ|ni = n|n − 1i √ • aˆ†|ni = n + 1|n + 1i • aˆ|αi = α|αi

•h α|aˆ† = α∗hα|

20 Atomic Energy Level Operators (for a two-level approximation)

• σˆ+ = |eihg|

• σˆ− = |gihe|

• σˆz = |eihe| − |gihg|

• σˆ+|gi = |ei

• σˆ−|ei = |gi

• σˆ+|ei = 0

• σˆ−|gi = 0

21 Chapter 2

Astrophysics & Astronomy

2.1 Astrometry 2.1.1 Redshift λ − λ v • obs emit ≈ λemit c λ • 1 + z = obs λemit

2.1.2 Apparent magnitude F • m − m0 = −2.5 log10( ) F0

2.1.3 Absolute magnitude d • M = m − 5 log ( ) 10 10 2.1.4 Relative magnitudes

(mb − ma) I • a = 100 5 Ib

2.1.5 Flux-magnitude relationship −0.4m • F = F0 × 10

2.1.6 Color F •− 2.5 log( f1 ) Ff2

2.1.7 Metallicity

(F e/H)∗ • Z = log10( ) = log10(F e/H)∗ − log10(F e/H) (F e/H) 2.2 Stars 2.2.1 Stellar Structure Equations Hydrostatic Equilibrium dP −GM ρ(r) • = r dr r2

22 Mass Conservation M • r = 4πr2ρ r

Energy Equation dL • r = 4πr2ρε dr

Radiative Transport dT 3 κρ¯ L • | = r dr rad 4ac T 3 4πr2 2.2.2 Timescales Thermal / Kelvin-Helmholtz Timescale 2 |U∗| GM∗ • τKH = = L∗ R∗L∗

• τKH ≈ 50millionyears

Nuclear Timescale / Main Sequence Lifespan

−3 9 M −3 • τN ≈ τ M ≈ 10 ( ) M

2.2.3 Gravitational potential energy −GM 2 • U ≈ ∗ R 2.2.4 Eddington Limit (hydrostatic equilibrium)

4πGMmpc 4 M • Ledd = ≈ 3.2 × 10 ( )[L ] σT M

−5 L • Medd = 3.1 ∗ 10 ( )[M ] L

Eddington Rate (mass loss)

˙ Ledd −8 M • Medd = 2 ≈ 2.4 × 10 ( )[M /yr] ηc M

2.2.5 Mass-Luminosity Relationship  2.3, 0.23 M < 0.43M  L M 4, 1 0.43M < M < 2M • ≈ b( )a ; a, b = L M 3.5, 1.5 2M < M < 20M  1, 32000 M > 55M

2.3 Galaxies 2.3.1 Hubble Eliptical Galaxy Classification a − b • 10 × ( ) a 2.3.2 S´ersicProfile

R 1 • I(R) = I0 exp{−b[( ) n − 1]} Re

23 2.3.3 Density of stars in the Milky Way Galaxy

−z/z0 −R/h • ρ(R, z) = ρ0e e

2.4 Black Holes 2.4.1 Schwarszchild Radius 2GM • r = S c2 2.5 Instrumentation 2.5.1 Lensmaker’s equation 1 1 1 (n − 1)d • = (n − 1)[ − + ] f R1 R2 nR1R2 1 1 ≈ (n − 1)[ − ] (Thin lens approximation) R1 R2

2.5.2 Focal ratio / Focal number f • N = D 2.5.3 Field of view w w • FOV = D = D DT N fsys

2.5.4 Resolution Limits Diffraction limit (Rayleigh criterion) λ • ε = 1.22 d D

Seeing limit (Rayleigh criterion) λ • εs = 0.98 r0

Total Resolution limit

p 2 2 • ε εd + εs

2.5.5 Nyquist sampling 2p λ • = (When diffraction limited) fsys DT 2p • N = λ 2.5.6 Plate scale 1 206265 • [rad/m] = [arcsec/m] f f

2.5.7 Fitting error

2 dsub 5 dsub 5 • σfit = af ( ) 3 = 0.26( ) 3 r0 r0

24 2.5.8 Adaptive optics error

2 dsub 5 θ 5 τ 5 λ 2 • σtotal = 0.3( ) 3 + ( ) 3 + 28.4( ) 3 + CWFS( ) r0 θ0 τ0 F τdsub

2.5.9 Signal-to-noise ratio F t • SNR = p 2 F t + (Bsnpt) + (Dnpt) + (R np)

2.5.10 Atmospheric Extinction

• mλ = mλ,z − aλ(sec z)

2.5.11 Rocket science Tsiolkovsky rockey equation

m0 • ∆v = ve ln( ) mf

25 Chapter 3

Mathematics

3.1 Notation

a • [f(x)]b = f(a) − f(b)

3.2 Algebra 3.2.1 Factorisation • (a + b)2 = a2 + b2 + 2ab2 • (a − b)2 = a2 + b2 − 2ab2 • a2 − b2 = (a + b)(a − b) • (a + b)(a + c) = a2 + (b + c)a + bc • (a + b)3 = a3 + 3ab2 + 3a2b + b3 • (a − b)3 = a3 + 3ab2 − 3a2b − b3 • a3 + b3 = (a + b)(a2 − ab + b2) • a3 − b3 = (a − b)(a2 + ab + b2) • a2n − b2n = (an − bn)(an + bn)

3.2.2 Absolute Value •| ab| = |a||b| a |a| •| | = b |b| •| a + b| ≤ |a| + |b|

3.2.3 Quadratics Quadratic Formula For ax2 + bx + c = 0, a 6= 0: √ −b ± b2 − 4ac • x = 2a • b2 − 4ac > 0 - two real unequal solutions. • b2 − 4ac = 0 - repeated real solution. • b2 − 4ac < 0 - two complex solutions.

26 3.2.4 Logarithms y • y = logb(x); x = b

• logb(xy) = logb x + logb(y) x • log ( ) = log x − log (y) b y b b

p • logb(x ) = p logb x x • logb(b ) = x

logd(a) • logb(a) = logd(b)

√ p • logb( x) = log p q • p logb x + q logb(y) = log(x y )

• blogb x = x

• logb(b) = 1

• logb(1) = 0

3.2.5 Vectors Dot Product ~ • ~a · b = a1b1 + a2b2 + a3b3 + ... + anbn = ab cos θ

Cross Product

ˆi ˆj kˆ ~ ˆ ˆ ˆ • ~a × b = a1 a2 a3 = (a2b3 − a3b2)i + (a3b1 − a1b3)j + (a1b2 − a2b1)k = ab sin θnˆ

b1 b2 b3

• ~a ×~b = −~b × ~a • ~a × (~b + ~c) = ~a ×~b + ~a × ~c

• ~a ×~b = 0

3.2.6 Factorials • n! = n(n − 1)(n − 2)...(2)(1) • (n + 1)! = (n + 1)n!

Stirling’s approximation • n! ≈ n ln(n) − n + O(ln(n))

The Factorial Integral ∞ • xne−x dx ´0

27 3.2.7 Inner product definition 1. Linear in first variable: (αa + βb, c) = α(a, c) + β(b, c) 2. Positive-definite: (a, a) ≥ 0, (a, a) = 0 ⇐⇒ a = 0

3. Conjugate symmetrical: (a, b) = (b, a)∗ (a, b) = (b, a), b, a ∈ R

3.2.8 Complex Numbers • z = a + ib = <(z) + =(z)i

Euler’s Formula • eiθ = cos θ + i sin θ • reiθ = |z|eiθ = r(cos θ + i sin θ)

De Moivre’s Formula • (cos x + i sin x)n = cos(nx) + i sin(nx)

Complex Modulus √ • r = |z| = |a + ib| = a2 + b2 = p<2(z) + =2(z)

Complex Conjugate • z¯ = (a +¯ ib) = a − ib • (a +¯ ib)(a + ib) = |a + ib|2

Complex Argument a =(z) • θ = arg(z) = arctan( ) = arctan( ) b <(z)

3.2.9 Power Series ∞ P n • f(x) = fnx n=0

Notable Series ∞ xn x2 x3 x4 x5 • ex = P = 1 + x + + + + + ... n=0 n! 2 6 24 120 ∞ (−1)n x3 x5 • sin x = P x2n+1 = x − + − ... n=0 (2n + 1)! 3! 5! ∞ (−1)n x2 x4 • cos x = P x2n = 1 − + − ... n=0 (2n)! 2! 4!

3.2.10 Matrix Operations Determinant

a b • = ad − bc c d

28 Transpose

T • ajk = akj • (AB)T = BT AT

• Linear: (A + B)T = AT + BT • (rA)T = rAT

Hermitian Adjoint • A† ≡ (A∗)T = (AT )∗ • (AB)† = B†A†

• (A + B)† = A† + B†

Trace n P • Tr(A) = ajj j • Cyclic: Tr(AB) = Tr(AB) • Linear: Tr(A + B) = Tr(A) + Tr(B) Tr(aB) = aTr(B), a ∈ C

• Tr(SAS−1) = Tr(A)

Hilbert-Schmidt Inner Product • (A, B) ≡ Tr(A†B)

Rank-Nullity Theorem • rk(A) = dim(ker(A)) + dim(im(A))

3.2.11 Matrix Types Real Matrix

• ajk ∈ R

Square Matrix • m = n

Symmetric Matrix • A = AT

• ajk = akj • Square

Normal Matrix • A†A = AA† • Square • Diagonalisable

29 Diagonal Matrix

• ajk = 0, j 6= k

• λj = ajj • Square

ed11 0 ··· 0 • eD = 0 ed11 ··· 0

Diagonalisable Matrix • A = PDP −1

• Square

Identity Matrix • IA = A

• ijj = 1

• ijk = 0, j 6= k • Real • Square • Diagonal

• Symmetric • Hermitian

Hermitian Matrix • H = H†

∗ • hjk = hkj

• hjj ∈ R • λ ∈ R • Square

• Normal • All real, square matrices are Hermitian

Anti-Hermitian Matrix • H = −H†

∗ • Hjk = −Hkj • Square

Orthogonal Matrix • AT = A−1 • AAT = I

T • (AA )jk = δjk

30 Positive Semidefinite • A ≥ 0 • Aˆ† = A,ˆ Aˆ ≥ 0

• B = Aˆ†Aˆ is positive semidefinite for any linear operator Aˆ • Positive semidefinite matrices are Hermitian

Projector • Pˆ2 = Pˆ • λ = 1 or 0

• P1P2 7→ H1 ∩ H2 • Projectors are Hermitian

Real Matrix • A = A∗

∗ • Ajk = Ajk

Imaginary Matrix • A = −A∗

∗ • Ajk = −Ajk

Symmetric Matrix • A = AT

• Ajk = Akj • Square

Antisymmetric Matrix • A = −AT

• ajk = akj

• ajj = 0 • Square

Unitary Matrix • U †U = UU † = I

• U † = U −1

† • (U U)jk = δjk • Square • Normal

• Hermitian

31 3.2.12 Change of Basis Unitary • Basis a is (?)

† • (V )b = [U ]a(V )a

• [U]a = [(b0)a(b1)a...(bn)a]

3.2.13 Commutator • [A, B] = AB − BA • [A, A] = 0

• [A + B,C] = [A, C] + [B,C] • [A, BC] = [A, B]C + B[A, C]

3.2.14 Anticommutator •{ A, B} = AB + BA

3.2.15 Cauchy-Schwarz Inequality • |h~u,~vi|2 ≤ h~u,~ui · h~v,~vi

3.3 Geometry 3.3.1 Pythagorean theorem • a2 + b2 = c2 √ • a = b2 + c2

Higher dimensions • r = px2 + y2 + z2 √ p 2 2 2 • r = x1 + x2 + ··· + xn = ~r · ~r

Distance between two points In two dimensions

p 2 2 • d12 = (x2 − x1) + (y2 − y1) In higher dimensions

p 2 2 2 • dab = (b1 − a1) + (b2 − a2) + ··· + (bn − an)

3.3.2 Properties of shapes

Area Circumference Circle πR2 2πR Square L2 4L

3.3.3 Properties of solids

Surface Area Volume

2 4 3 Sphere 4πR 3 πR

32 3.3.4 Circular formulae Arc length • l = Rθ

Area of a sector R2θ • A = 2

Area of a segment R2 • A = (θ − sin θ) 2 3.3.5 Useful Functions Parabola • f(x) = a(x − h)2 + k • Vertex at (h, k) • Up-concave if a > 0; down-concave if a < 0

• f(x) = ax2 + bx + c b b • Vertex at (− , f(− )) 2a 2a • Up-concave if a > 0; down-concave if a < 0

Hyperbola x − h y − k • ( )2 − ( )2 = 1 a b • Centre at (h, k) b • Asymptotes through centre, slope ± a

Circle • (x − h)2 + (y − k)2 = R2

• Centre at (h, k)

Ellipse x − h y − k • 1 = ( )2 + ( )2 a b • Centre at (h, k) • Vertices a units right/left from the centre and vertices b units up/down from the center.

Sphere • R2 = (x − h)2 + (y − k)2 + (z − l)2

• Centre at (h, k, l):

33 Ball • R2 < (x − h)2 + (y − k)2 + (z − l)2 • Centre at (h, k, l):

3.3.6 Coordinates Transformations to Cartesian coordinates

Cartesian x = x y = y z = z dV = dx dy dz

Polar (2D) x = r cos φ y = r sin φ N/A dA = r drdφ

Cylindrical x = r cos φ y = r sin φ z = z dV = r dr dθ dz

Spherical x = r sin θ cos φ y = r sin θ sin φ z = r cos θ dV = r2 sin θ dr dθ dφ

Transformations from Cartesian coordinates

Cartesian x = x y = y z = z

y Polar (2D) r = px2 + y2 φ0 = arctan | | (φ depends on quadrant) N/A x y Cylindrical r = px2 + y2 φ0 = arctan | | (φ depends on quadrant) z = z x y z Spherical r = px2 + y2 + z2 φ0 = arctan | | (φ depends on quadrant) θ = arccos( ) x px2 + y2 + z2

3.3.7 Hyperbolic Functions Hyberbolic Sine ex − e−x e2x − 1 1 − e−2x • sinh x = = = 2 2ex 2e−x • sinh x = −i sin(ix)

Hyperbolic Cosine ex + e−x e2x + 1 1 + e−2x • cosh x = = = 2 2ex 2e−x • cosh x = cos(ix)

Hyperbolic Tangent sinh x ex − e−x e2x − 1 1 − e−2x • tanh x = = = = cosh x ex + e−x e2x + 1 1 + e−2x • tanh x = −i tan(ix)

Hyperbolic Cotangent 1 cosh x ex + e−x e2x + 1 1 + e−2x • coth x = = = = = tanh x sinh x ex − e−x e2x − 1 1 − e−2x • coth x = i cot(ix)

34 Hyperbolic Secant 1 2 2ex 2e−x • sechx = = = = cosh x ex + e−x e2x + 1 1 + e−2x • sechx = sec(ix)

Hyperbolic Cosecant 1 2 2ex 2e−x • cschx = = = = sinh x ex − e−x e2x − 1 1 − e−2x • cschx = i csc(ix)

Identities • cosh2 x − sinh2 x = 1

1 • sin θ cos θ = 2 sin(2θ)

3.4 Trigonometry

Definitions • SOH CAH TOA opposite • sin θ = hypotenuse adjacent • cos θ = hypotenuse opposite sin θ • tan θ = = adjacent cos θ adjacent cos θ • cot θ = = opposite sin θ hypotenuse • sec θ = adjacent hypotenuse • csc θ = opposite

3.4.1 Identities Pythagorean Identities • cos2 θ + sin2 θ = 1 • tan2 θ + 1 sec2 θ

• 1 + cot2 θ = csc2 θ

Reciprocals 1 • sin θ = csc θ 1 • cos θ = sec θ 1 • tan θ = cot θ 1 • cot θ = tan θ

35 1 • sec θ = cos θ 1 • csc θ = sin θ

As complex exponentials eix − e−ix • sin θ = 2i eix + e−ix • cos θ = 2 eix − e−ix • tan θ = i(eix + e−ix) i(eix + e−ix) • cot θ = eix − e−ix 2 • sec θ = eix + e−ix 2i • csc θ = eix − e−ix

Symmetries • sin(−θ) = − sin θ • cos(−θ) = cos θ • tan(−θ) = − tan θ • csc(−θ) = − csc θ • sec(−θ) = sec θ • cot(−θ) = − cot θ

π • sin( − θ) = cos θ 2 π • cos( − θ) = sin θ 2 π • tan( − θ) = cot θ 2 π • csc( − θ) = sec θ 2 π • sec( − θ) = csc θ 2 π • cot( − θ) = tan θ 2

• sin(π − θ) = sin θ • cos(π − θ) = − cos θ • tan(π − θ) = − tan θ • csc(π − θ) = csc θ • sec(π − θ) = − sec θ • cot(π − θ) = − cot θ

36 Angle sum and difference formulae • sin(α ± β) = sin α cos β ± cos α sin β • cos(α ± β) = cos α cos β ∓ sin α sin β tan α ± tan β • tan(α ± β) = 1 ∓ tan α tan β

Half-angle formulae θ 1 − cos θ • sin2( ) = 2 2 θ 1 + cos θ • cos2( ) = 2 2 θ 1 − cos θ • tan2( ) = 2 1 + cos θ θ tan θ sin θ 1 − cos θ • tan( ) = = = = csc θ − cot θ 2 1 + sec θ 1 + cos θ sin θ

Double-angle formulae • cos(2θ) = 2 cos2 θ − 1 = 1 − 2 sin2 θ = cos2 θ − sin2 θ • sin(2θ) = 2 sin θ cos θ 2 tan θ • tan(2θ) = 1 − tan2 θ

Sum to Product α + β • sin α + sin β = 2 sin( ) cos( α−β ) 2 2 α + β • sin α − sin β = 2 cos( ) sin( α−β ) 2 2 α + β • cos α + cos β = 2 cos( ) cos( α−β ) 2 2 α + β • cos α − cos β = −2 sin( ) sin( α−β ) 2 2

Product to Sum 1 • sin α sin β = 2 [cos(α − β) − cos(α + β)] 1 • cos α cos β = 2 [cos(α − β) + cos(α + β)] 1 • sin α cos β = 2 [sin(α + β) + cos(α − β)] 1 • cos α sin β = 2 [sin(α + β) − sin(α − β)]

Law of Sines sin α sin β sin γ • = = a b c

Law of Cosines • a2 = b2 + c2 − 2bc cos α

• b2 = a2 + c2 − 2ac cos β • c2 = a2 + b2 − 2ab cos γ

37 Law of Tangents 1 a − b tan( 2 [α − β]) • = 1 a + b tan( 2 [α + β]) 1 b − c tan( 2 [β − γ]) • = 1 b + c tan( 2 [β + γ]) 1 a − c tan( 2 [α − γ]) • = 1 a + c tan( 2 [α + γ])

Mollweide’s Formula 1 a + b cos( 2 [α − β]) • = 1 c sin( 2 γ)

Small-angle approximations • sin θ ≈ θ θ2 • cos θ ≈ 1 − 2 • tan θ ≈ θ

Other identities 1 • sin θ cos θ = 2 sin(2θ) 2 1 • cos θ = 2 (cos(2θ) + 1)

Averages • sin¯ x = ¯cos x = 0

¯2 ¯2 1 • sin x = cos x = 2

Table of Identities

In terms of... sin θ cos θ tan θ sec θ cot θ csc θ √ √ tan θ sec2 θ − 1 1 1 sin θ = sin θ ± 1 − cos2 θ ±√ ± ±√ 1 + tan2 θ sec θ 1 + cot2 θ csc √ p 1 1 cot θ csc2 −1 cos θ = ± 1 − sin2 θ cos θ ±√ ±√ ± 1 + tan2 θ sec θ 1 + cot2 θ csc θ √ 2 √ sin θ 1 − cos θ 2 1 1 tan θ = ±p ± tan θ ± sec θ − 1 ±√ 1 − sin2 θ cos θ cot θ csc2 θ − 1 √ √ 2 1 1 2 1 + cot θ csc θ sec θ = ±p ± 1 + tan θ sec θ ± ±√ 1 − sin2 θ cos θ cot θ csc2 −1 p 1 − sin2 θ cos θ 1 1 √ cot θ = ± ± √ ±√ cot θ ± csc2 θ − 1 sin θ ± 1 − cos2 θ tan θ sec2 θ − 1 √ 1 1 1 + tan2 θ sec θ √ csc θ = ±√ ± ±√ ± 1 + cot2 θ csc sin θ 1 − cos2 θ tan θ sec2 θ − 1

38 3.5 Calculus 3.5.1 Limits 3.5.2 Properties • lim cf(x) = c lim f(x) x→a x→a • lim [f(x) ± g(x)] = lim f(x) ± lim g(x) x→a x→a x→a • lim [f(x)g(x)] = lim f(x) lim g(x) x→a x→a x→a

f(x) lim f(x) • lim = x→a , lim g(x) 6= 0 x→a g(x) lim g(x) x→a x→a • lim [f(x)]n = [ lim f(x)]n x→a x→a

Useful Limits • lim ex = ∞ x→∞ • lim ex = 0 x→−∞ • lim ln(x) = ∞ x→∞ • lim ln(x) = −∞ x→0− • lim x log x = 0 x→0

L’Hˆopital’srule f(x) f 0(x) • lim = lim x→a g(x) x→a g0(x)

Squeeze principle • For g(x) ≤ f(x) ≤ h(x) and lim g(x) = lim h(x) = L: x→a x→a lim f(x) = L x→a

3.5.3 Differentiation First Principles d f(x + h) − f(x) • f(x) = lim dx h→0 h

Nature of derivatives

Derivative Function f 0x > 0 Increasing f 0x = 0 Stationary f 0x < 0 Decreasing Second Derivative Function Stationary Points [f 0x = 0] f 00x > 0 Concave up Local Minimum f 0x = 0 No information Inflection Point f 00x < 0 Concave down Local Maximum

39 Product Rule • (uv)0 = uv0 + vu0 d • f(x)gx = f(x)g0(x) + f 0(x)g(x) dx

Quotient Rule u vu0 − uv0 • ( )0 = v v2 d f(x) g(x)f 0(x) − f(x)g0(x) • = dx g(x) g(x)2

Chain Rule dy dy du • = dx du dx d • f(g(x)) = f 0(g(x))g0(x) dx

Useful Derivatives d • xn = nxn−1 dx d • ax = ax ln(a) dx d • ex = ex dx d 1 • ln x = , x > 0 dx x d 1 • ln |x| = , x 6= 0 dx x d f 0x • ln(f(x)) = dx f(x) d 1 • log x = , x > 0 dx b x ln(b) d • sin x = cos x dx d • cos x = − sin x dx d • tan x = sec2 x dx d • sec x = sec x tan x dx d • csc x = − csc x cot x dx d • cot x = − csc2 x dx d 1 • sin−1 x =√ dx 1 − x2 d 1 • cos−1 x = √ dx − 1 − x2 d 1 • tan−1 x = √ dx − 1 + x2

40 3.5.4 Partial Differentiation First Principles ∂ f(x + h, y) − f(x, y) • f(x, y) = lim ∂x h→0 h

Jacobian Matrix   ∂f1 ∂f1 ∂f1  (~a) (~a) ··· (~a)  ∂x ∂x ∂x   1 2 n     ∂f2 ∂f2 ∂f2   (~a) (~a) ··· (~a)  ∂x ∂x ∂x  • Df~(~a) =  1 2 n     . . .. .   . . . .      ∂f ∂f ∂f   m (~a) m (~a) ··· m (~a) ∂x1 ∂x2 ∂xn

Definition of differentiability of a multivariable function ||f(~x) − f(~a) − Df(~a) · (~x − ~a)|| • lim = = 0 ~x→~a ||~x − ~a||

3.5.5 The Differential ∂F ∂F ∂F • dF = dx + dy + dz ∂x ∂y ∂z

3.5.6 Line Element • dS2 = dx2 + dy2 + dz2

3.5.7 Integration Properties • f(x) ± g(x) dx = ( f(x) dx ±g(x) dx

´a ´ ´ • f(x) dx = 0 ´a b a • f(x) dx = − f(x) dx ´a ´b b b •| f(x) dx| ≤ |f(x)| dx ´a ´a b a • If f(x) ≥ g(x) over [a, b], f(x) dx ≥ g(x) dx ´a ´b b • If f(x) ≥ 0 over [a, b], f(x) dx > 0 ´a b • If m ≤ f(x) ≤ M over [a, b], m(b − a) ≤ f(x) dx ≤ M(b − a) ´a

41 Integration by Parts • u0v = uv − uv0 ´ ´ • f 0(x)g(x) dx = f(x)g(x) − f(x)g0(x) dx ´ ´ b b 0 b 0 • f (x)g(x) dx = [f(x)g(x)]a − f(x)g (x) dx ´a ´a Integration by Substitution

b g(b) • u = g(x); dx = g0(x) dx; f(g(x))g0(x) dx = f(x) dx ´a g(´a)

Approximations Trapezoid rule

b ∆x • f(x) dx ≈ [f(x ) + 2f(x ) + 2f(x ) + ··· + 2f(x ) + f(x )] 2 0 1 2 n−1 n ´a Simpson’s rule

b ∆x • f(x) dx ≈ [f(x ) + 4f(x ) + 2f(x ) + ··· + 2f(x ) + 4f(x ) + f(x )] 3 0 1 2 n−2 n−1 n ´a Useful Indefinite Integrals • k dx = kx + C ´ 1 • log x dx = x(log x − log (e)) + C = x(log x − ) + C b b b b ln b ´ • ln x dx = x(ln x − 1) + C ´ • ex dx = ex + C ´ xn+1 • xn dx = + C, n 6= −1 n + 1 ´ 1 • dx = ln |x| + C x ´ 1 1 • dx = ln |ax + b| + C ax + b a ´ • sin x dx = − cos x + C ´ • cos x dx = sin x + C ´ • tan x dx = ln | sec x| + C ´ • sec x dx = ln | sec x + tan x| + C ´ • sec2 x dx = tan x + C ´ • csc2 x dx = − cot x + C ´ • sec x tan x dx = sec x + C ´ • csc x cot x dx = − csc x + C ´ 1 1 x • dx = tan−1( ) + C a2 + x2 a a ´ 1 x • √ dx = sin−1( ) + C 2 2 a ´ a + x

42 Useful Definite Integrals ∞ x3 π4 • dx = ex − 1 15 ´0 ∞ n! • xne−ax dx = an+1 ´0 3.5.8 Vector Calculus Vector derivative   ∂f    ∂x      ∂f ∂f ∂f ~ ∂f  • grad(f) = ∇f =   = xˆ + yˆ + zˆ  ∂y  ∂x ∂y ∂z     ∂f  ∂z

b • (∇~ f) · d~l = f(a) − f(b) ´a The Laplacian ∂2f ∂2f ∂2f • δf = ∇2f = + + ∂x2 ∂y2 ∂z2

Divergence In Cartesian Coordinates ∂f ∂f ∂f • div(f) = ∇~ · f~ = x + y + z ∂x ∂y ∂z In Cylindrical Coordinates 1 ∂ 1 ∂f ∂f • ∇~ · f~ = (rf ) + φ + z r ∂r r r ∂φ ∂z In Spherical Coordinates 1 ∂ 1 ∂ 1 ∂f • ∇~ · f~ = (r2f ) + (f sin θ) + φ r ∂r r r sin θ ∂θ θ r sin θ ∂φ

Curl In Cartesian Coordinates

xˆ yˆ zˆ

∂ ∂ ∂ ∂f ∂f ∂f ∂f ∂f ∂f ~ ~ z y x z y x • curl(f) = ∇ × f = = ( − )ˆx + ( − )ˆy + ( − )ˆz ∂x ∂y ∂z ∂y ∂z ∂z ∂z ∂x ∂y

fx fy fz

In Cylindrical Coordinates 1 ∂f ∂f ∂f ∂f 1 ∂ ∂f • ∇~ × f~ = ( z − φ )ˆr + ( r − z )φˆ + ( (rf ) − r )ˆz r ∂φ ∂z ∂z ∂r r ∂r φ ∂φ In Spherical Coordinates 1 ∂ ∂f 1 1 ∂f ∂ 1 ∂ ∂f • ∇~ × f~ = ( (f sin θ) − θ )ˆr + ( r − (rf )) + ( (rf ) − r ) r sin θ ∂θ φ ∂φ r sin θ ∂φ ∂r φ r ∂r θ ∂θ

43 Vector Second Derivatives • ∇~ · (∇~ × ~v) = 0

• ∇~ (∇~ (∇~ · f~))

Vector Laplacian • ∇~ 2f~ = ∇~ (∇~ · f~) − ∇~ × (∇~ × f~)

Stokes’ Theorem • (∇~ × f~) · d~a = ~v · d~l s S B¸

Divergence Theorem • (∇~ · f~)dV = ~v · d~a t V S¸ 3.5.9 Dirac Delta Function ( 0 x 6= 0 • δ(x) = ∞ x = 0 ( 0 x 6= a • δ(x − a) = ∞ x = a

∞ • δ(x) dx = 1 −∞´ • f(x)δ(x) = f(0)δ(x)

3.5.10 Approximations • f(x + ∆x) ≈ f(x) + ∆xf 0(x)

44 Chapter 4

Statistics

4.1 Variance

• var(x) = h(x − hxi)2i = hx2i − hxi2

4.2 Standard Deviation

• σx = pvar(x) = phx2i − hxi2

45 Appendix A

Values

A.1 Physics A.1.1 Physical Constants c: Speed of light 1 • =√ = 299 792 458 m · s−1 ≈ 3 × 108 m · s−1 ε0µ0 G: Universal gravitational constant • = 6.67408(31) × 10−11 m3 · kg−1 · s−2 ≈ 6.67 × 10−11 m3 · kg−1 · s−2 g: Average acceleration due to gravity at sea level on Earth

• = 9.80665 m · s−2 ≈ 9.8 m · s−2 h: Planck constant • = 6.626 070 040 × 10−34J · s ≈ 6.626 × 10−34J · s ¯h: Reduced Planck constant h • = = 1.054 571 726 × 10−34J · s ≈ 1.055 × 10−34J · s 2π kB: Boltzmann constant • = 1.3 806 488 × 10−23 J · K−1 ≈ 1.38 × 10−23 J · K−1 ke: Coulomb’s constant 1 • = = 8.987 551 787 × 109 N · m · C−2 ≈ 9 × 109 N · m · C−2 4πε0

NA: Avogadro constant • = 6.022 140 857(74) × 1023 mol−1 ≈ 6.022 × 1023 mol−1

ε0: Vacuum permittivity

1 −12 −1 −6 −1 • = 2 = 8.854 187 817 × 10 F · m ≈ 8.85 × 10 F · m µ0c

µ0: Vacuum permeability

−7 −2 1 −6 −2 • = 4π × 10 N · A = 2 ≈ 1.257 × 10 N · A ε0c

46 A.1.2 Useful Quantities

−3 Density of air (ρA): 1.2922 kg · m 3 −3 Density of water (ρw): 10 kg · m −31 −31 Mass of an electron (me): 9.10 938 291 × 10 kg ≈ 9 × 10 kg −27 −27 Mass of a neutron (mn): 1.674 927 351 × 10 kg ≈ 1.675 × 10 kg −27 −27 Mass of a proton (mp): 1.672 621 777 × 10 kg ≈ 1.672 × 10 kg Speed of sound in air: 343.2 m · s−1 Specific heat capacity of water: 4.186 ×103 J · kg−1 · K−1

A.2 Astronomy A.2.1 Useful Quantities ◦ Surface Temperature of the Sun: (T ): = 5778 K = 5505 C

Planetary Properties Body Mass Average Radius Semi-major axis Eccentricity Orbital period Mercury 3.3011 × 1023 kg 2.4397 × 106 m 5.790905 × 1010 m 0.205630 0.240856 yr ' 0.055 M⊕ 0.3829 R⊕ 0.387098 AU −7 1.66 × 10 M 1.74 × 10−4 M 24 6 11 Venus 4.8675 × 10 kgX 6.0518 × 10 m 1.08208000 × 10 m 0.006772 0.615198 yr ♀ 0.815 M⊕ 0.9499 R⊕ 0.723332 AU −6 2.447 × 10 M 2.56 × 10−3 M 24 6 11 Earth ⊕ 5.97237 × 10 Xkg 6.3710 × 10 m 1.49598023 × 10 m 0.0167086 1.000017 yr 1 M⊕ 1 R⊕ 1.000001 AU −6 3.003 × 10 M 2.67 × 10−3 M 23 6 11 Mars 6.4171 × 10 kgX 3.3895 × 10 m 2.27939200 × 10 m 0.0934 1.88082 yr ♂ 0.107 M⊕ 0.53 R⊕ 1.523679 AU −7 3.226 × 10 M 3.38 × 10−4 M 27 7 11 Jupiter 1.8982 × 10 kgX 6.9911 × 10 m 7.4052 × 10 m 0.0489 11.862 yr X 317.8 M⊕ 10.97 R⊕ 5.2044 AU −4 9.55 × 10 M 1 M 26 7 12 Saturn 5.6834X × 10 kg 5.8232 × 10 m 1.43353 × 10 m 0.0565 29.4571 yr Y 95.159 M⊕ 9.14 R⊕ 9.5826 AU −4 2.86 × 10 M 0.299 M 25 7 12 Uranus 8.68 × 10X kg 2.5362 × 10 m 2.87504 × 10 m 0.046381 84.0205 yr Z 14.536 M⊕ 3.98 R⊕ 19.2184 AU −5 4.36 × 10 M 0.046 M 26 7 12 Neptune 1.0243 ×X10 kg 2.4622 × 10 m 4.5 × 10 m 0.009456 164.8 yr [ 17.147 M⊕ 3.86 R⊕ 30.11 AU −5 5.15 × 10 M 0.054 M X

47 A.3 Mathematics n 1 1 1 1 1 Euler’s number (e): P = + + + + ··· n=0 n! 1 1 1 · 2 1 · 2 · 3 = 2.71828182845904523536028747135266249775724709369995 ... ≈ 2.7182 ≈ 2.718 C C Pi (π): = d 2r = 3.14159265358979323846264338327950288419716939937510 ... ≈ 3.14159 ≈ 3.142

48 Appendix B

Units of Measurement

B.1 Natural Units

Handy when you’re dealing with small things.

Charge: elementary charge (e)

• The electric charge of a proton. • = 1.602 176 565 × 10−19 C ≈ 1.6 × 10−19 C Energy: electron volt (eV ) • The work done to move an electron across one volt of potential. • = e · V = 1.602 176 565 × 10−19 J ≈ 1.6 × 10−19 J

B.2 SI System

Universally acknowledged as the best system of units.

B.2.1 Base Units Amount of Substance: mole (mol) • The amount of substance of a system which contains as many elementary entities as there are atoms in 0.012 kg of carbon-12. • = 6.022 140 857 × 1023 Electric Current: ampere (A) • The constant current which, if maintained in two straight parallel conductors of infinite length, of negligible circular cross-section, and placed 1 m apart in vacuum, would produce between these conductors a force equal to 210−7N · m−1 of length. • = C · s−1 Force: newton (N) • = 0.224809 lbf

Length: metre (m) 1 • The distance traveled by light in vacuum in s 299 792 458 • = 3.2808 ft Luminous intensity: candela (cd)

49 • The luminous intensity, in a given direction, of a source that emits monochromatic radiation of frequency 5.4 × 1014 Hz and that has a radiant intensity in that direction 1 of watt per steradian. 683 Mass: kilogram (kg)

• = 2.205lb Thermodynamic Temperature: kelvin (K)

1 • 273.16 of the thermodynamic temperature of the triple point of water. • = T [◦C] + 273.15

Time: second (s) • The duration of 9 192 631 770 periods of rotation corresponding to the two hyperfine levels of the ground-state of the caesium-133 atom.

B.2.2 Derived Units Angle: radian (rad) • A full circle divided by 2π. 180 • = m · m−1 = ≈ 57.3◦ = 206265 arcsecs π Electric Charge: coulomb (C) • = A · s = 6.242 × 1018 e

Electrical capacitance: farad (F ) • = m−2 · kg−1 · s4 · A2 Electrical conductance: siemens (S) • = A · V −1 = kg−1 · m−2 · s3 · A2

Electrical inductance: henry (H) • = W b · A−1 = kg · m2 · s−2 · A−2 Electrical potential difference / Voltage: volt (V )

• = W · A−1 = kg · m2 · s−3 · A−1 Electrical resistance: ohm (Ω) • V · A−1 = kg · m2 · s−3 · A−2 Energy: joule (J)

• = N · m = kg · m2 · s−2 Force: newton (N) • = kg · m · s−2

Frequency: hertz (Hz) • = s−1 Illuminance: lux (lx) • = lm · m2 = m−2 · cd

Luminous flux: lumen (lm)

50 • = cd · sr = cd

Magnetic flux: weber (W b) • = V · s = kg · m2 · s−2 · A−1 Magnetic flux density: tesla (T ) • = kg · s−2 · A−1

Power: watt (W ) • = J · s = kg · m2 · s−3 Pressure: pascal (P a)

• = N · m−2 = kg · m−1 · s−2 Radioactivity: becquerel (Ω) • Decays per second • = s−1

Solid angle: steradian (sr) • = m2 · m−2 Temperature: degree Celcius (◦C) • T [C] = T [K] − 273.15

B.3 CGS (centimetres-grams-seconds)

Commonly used in astronomy, to everyone’s disappointment.

Acceleration: gal (Gal) • = cm · s−2 = 10−2 m · s−2 Energy: erg (erg) • = g · cm2 · s−2 = 10−7 J

Force: dyne (dyn) • = g · cm · s−2 = 10−5 N Length: centimetre (cm) • = 0.01 m

Mass: gram (g) • = 10−3 kg Power: erg per second (erg/s)

• = g · cm2 · s−2 = 10−7 W Pressure: barye (Ba) • = g · cm−1 · s−2 = 10−1 P a Time: second (s)

Velocity: centimetre per second (cm/s) • = 10−2 m · s−1

51 Viscosity (dynamic): poise (P )

• = g · cm−1 s−1 = 10−1 P a · s Viscosity (kinematic): stokes (St) • = g · cm2 s−1 = 10−4 m2 · s−1 Wavenumber: kayser (K)

• = cm−1 = 100 m−1

B.4 Astronomy units B.4.1 Astronomical system Distance: astronomical unit (AU)

• Roughly the distance from the Earth to the Sun. • = 1.4960 × 1011 m = 4.8481 × 10−6 pc = 1.5813 × 10−5 ly

Mass: solar mass (M )

30 30 • = 1.98855 × 10 kg ≈ 2 × 10 kg = 1048 M = 332 950 M Time: Day X • = 86 400s

Complimentary units

Distance: Solar radius (R ) • = 6.957 × 108 m = 695 700 km ≈ 7 × 108 m Distance: parsec (pc) • The distance at which the parallax of an object over the course of the Earth’s orbit is one arcsec. • = 3.0857 × 1016 m = 2.0626 × 105 AU = 3.26156 ly Distance: light year (ly) • The distance travelled by light in a vacuum in a year. • = 9.4607 × 1015 m = 6.3241 × 104 AU = 0.3066 pc

Mass: Earth mass (M ) • = 5.9722 × 1024kg ≈ 6 × 1027kg Mass: Jupiter mass (M ) • = 1.898 × 10X27kg ≈ 1.9 × 1027kg

Specific Flux: Jansky (Jy) • = 10−26 W · m−2 · Hz−1

52 B.4.2 Equatorial Coordinate System Right Ascension (α) 1 Hour (h): circle = 15◦ 24 1 h 1 Minute (m): = circle = 150 60 1440 1 m 1 h 1 Second (s): = = circle = 15” 60 3600 86400

Declination (δ) Declination is measured using normal degrees (see Degrees of Angle) from the equator.

B.5 United States customary units (aka Imperial Units) B.5.1 Length 127 Point (p): = mm 360 127 Pica (P/): = 12 p = mm 30 Inch (in or ”): = 6 P/ = 25.4 mm Foot (ft or 0): = 12 in = 0.3048 m

Yard (yd): = 3 ft = 0.9144 m Mile (Mi): = 5280 ft = 1760 yd = 1.609344 km

B.6 Degrees of Angle 1 π Degree (◦): circle = rad ≈ 0.0174532925199433 rad 360 180 1 ◦ 1 π Minute of arc (arcmin or 0): = circle = rad 60 21600 10800 1 1 ◦ 1 π Second of arc(arcsec or ”): arcmin = = circle = rad 60 3600 206265 648000 B.7 Miscellaneous Units B.7.1 Pressure Bar (bar): = 105 P a ≈ 0.9869 atm Atmosphere (atm): = 101325 P a = 1.01325 bar 1 101325 Torr (torr): = atm = P a ≈ 133.3224 P a 760 760

53 B.8 Prefixes atto (a) = ×10−18 femto (f) = ×10−15 pico (p) = ×10−12 nano (n) = ×10−9 micro (µ) = ×10−6 milli (m) = ×10−3 centi (c) = ×10−2 deca (da) = ×101 hecto (h) = ×102 kilo (k) = ×103 mega (M) = ×106 giga (G) = ×109 tera (T ) = ×1012 peta (P ) = ×1015 exa (E) = ×1018

54 Appendix C

Mathematical Stuff

C.1 Trigonometric Values C.1.1 Pythagorean Triples

π 6 S A √ 2 3 π 4 √ T C 2 1

π π π π 2 3 4 2 rad ◦ sin cos tan 1 1 0 0 0 1 0 √ √ π/6 30 1/2 3/2 1/ 3 √ √ π/4 45 1/ 2 1/ 2 1 √ √ π/3 60 3/2 1/2 3 π/2 90 1 0 ±∞ √ √ 2π/3 120 3/2 −1/2 − 3 √ √ 3π/4 135 1/ 2 −1/ 2 −1 √ √ 5π/6 150 1/2 − 3/2 −1/ 3 π 180 0 −1 0 √ √ 7π/6 210 −1/2 3/2 1/ 3 √ √ 5π/4 225 −1/ 2 −1/ 2 1 √ √ 4π/3 240 − 3/2 −1/2 3 3π/2 270 −1 0 ±∞ √ √ 5π/3 300 − 3/2 1/2 − 3 √ √ 7π/4 315 −1/ 2 1/ 2 −1 √ √ 11π/6 330 −1/2 3/2 −1/ 3 2π 360 0 1 0

55 Appendix D

Boring stuff

D.1 Version History v 0.1 2016: This project is begun in a trio of physical exercise books as The Little Book of Physics Formulae, The Little Book of Mathematics Formulae, and The Little Book of Astronomy Formulae v 0.6 2016: The process of transferring the formulae from paper to Latex is initiated, but aban- doned (or drifted away from). v 0.7 2018-03-20: The project is resurrected (probably because the author started MRes), up- loaded to Overleaf, and cleaned up. v 0.8 2018-07-12: All remaining formulae imported from the original books. v 0.9 2018-07-26: Formulae imported from undergrad formula sheets. v 1.0 2018-07-27: First public release, with some additions from 0.9.

D.2 Licensing

This work is licensed under a Creative Commons “Attribution- ShareAlike 4.0 International” license.

(Basically, as long as you credit me and share under a similar license, feel free to use this however you want)

D.3 Contact

Visit www.webofworlds.net for science fiction, science fact, geeky opinions, and maybe some Python code. Suggestions or corrections are welcome at [email protected]

D.4 Credits

Unit Circle: By Jim.belk [CC BY-SA 3.0 (https://creativecommons.org/licenses/by-sa/3.0) or GFDL (http://www.gnu.org/copyleft/fdl.html)], from Wikimedia Commons (https: //commons.wikimedia.org/wiki/File:Unit_circle_angles_color.svg)

Periodic Table: By Dmarcus100 [CC BY-SA 4.0 (https://creativecommons.org/licenses/ by-sa/4.0)], from Wikimedia Commons (https://commons.wikimedia.org/wiki/File: Periodic_Table_Of_Elements_Atomic_Mass_Black_And_White.jpg)

56 57