The Ultimate Cheat Sheet for Astrophysics Students, Cheat Sheet of Astrophysics

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A Web of Worlds presents

The Ultimate Cheat Sheet for Astrophysics Students

Lachlan Marnoch

www.webofworlds.net

v 1.

July 28, 2018

Contents

• 1 Physics
  • 1.1 Motion
    • 1.1.1 Velocity
    • 1.1.2 Acceleration
    • 1.1.3 Newton’s Laws
    • 1.1.4 Momentum
    • 1.1.5 Centripetal Force
    • 1.1.6 Kinetic Energy
    • 1.1.7 Projectile Motion
    • 1.1.8 Rotation
    • 1.1.9 Euler-Lagrange and the Hamiltonian
  • 1.2 Oscillations
    • 1.2.1 Springs
  • 1.3 Materials
    • 1.3.1 Density
  • 1.4 Energy
    • 1.4.1 Work
  • 1.5 Forces
    • 1.5.1 Buoyancy (Archimedes’ Principle)
    • 1.5.2 Friction
  • 1.6 Waves
    • 1.6.1 Wavelength
    • 1.6.2 Angular Frequency
  • 1.7 Newtonian Gravity
    • 1.7.1 Force of Gravity
    • 1.7.2 Gravitational Potential (potential energy per unit mass)
    • 1.7.3 Gravitational field
    • 1.7.4 Gravitational Potential Energy
    • 1.7.5 Kepler’s Third Law
  • 1.8 Electromagnetism
    • 1.8.1 Notation
    • 1.8.2 Maxwell’s Equations
    • 1.8.3 Lorentz Force
    • 1.8.4 Electric Field
    • 1.8.5 Dipole moment
    • 1.8.6 Electric potential
    • 1.8.7 Electric potential difference
    • 1.8.8 Electric potential energy
    • 1.8.9 Charge densities
    • 1.8.10 Current densities
    • 1.8.11 Circuits
    • 1.8.12 Capacitors
    • 1.8.13 Magnetic fields
    • 1.8.14 Inductors
    • 1.8.15 Materials
  • 1.9 Special Relativity
    • 1.9.1 Interval
    • 1.9.2 Four-vectors
    • 1.9.3 Frames of Reference
    • 1.9.4 Proper Velocity
  • 1.10 General Relativity
    • 1.10.1 Metrics
    • 1.10.2 Rindler coordinates
    • 1.10.3 Einstein notation
    • 1.10.4 Christoffel symbols
    • 1.10.5 Covariant derivatives
    • 1.10.6 Riemann curvature tensor
    • 1.10.7 Ricci curvature tensor
    • 1.10.8 Einstein’s equations
  • 1.11 Thermodynamics
    • 1.11.1 Ideal Gases
    • 1.11.2 Microstates
    • 1.11.3 Entropy
    • 1.11.4 Black bodies
  • 1.12 Quantum Mechanics
    • 1.12.1 The Uncertainty Principle
    • 1.12.2 Bras and Kets
    • 1.12.3 Rules for an Inner Product
    • 1.12.4 The Born Rule
    • 1.12.5 Expectation
    • 1.12.6 Variance
    • 1.12.7 Standard Deviation
    • 1.12.8 Trace
    • 1.12.9 Partial Trace
    • 1.12.10 The Schr¨odinger Equation
    • 1.12.11 Heisenberg equation of motion
    • 1.12.12 Operators
• 2 Astrophysics & Astronomy
  • 2.1 Astrometry
    • 2.1.1 Redshift
    • 2.1.2 Apparent magnitude
    • 2.1.3 Absolute magnitude
    • 2.1.4 Relative magnitudes
    • 2.1.5 Flux-magnitude relationship
    • 2.1.6 Color
    • 2.1.7 Metallicity
  • 2.2 Stars
    • 2.2.1 Stellar Structure Equations
    • 2.2.2 Timescales
    • 2.2.3 Gravitational potential energy
    • 2.2.4 Eddington Limit (hydrostatic equilibrium)
    • 2.2.5 Mass-Luminosity Relationship
  • 2.3 Galaxies
    • 2.3.1 Hubble Eliptical Galaxy Classification
    • 2.3.2 S´ersic Profile
    • 2.3.3 Density of stars in the Milky Way Galaxy
  • 2.4 Black Holes
    • 2.4.1 Schwarszchild Radius
  • 2.5 Instrumentation
    • 2.5.1 Lensmaker’s equation
    • 2.5.2 Focal ratio / Focal number
    • 2.5.3 Field of view
    • 2.5.4 Resolution Limits
    • 2.5.5 Nyquist sampling
    • 2.5.6 Plate scale
    • 2.5.7 Fitting error
    • 2.5.8 Adaptive optics error
    • 2.5.9 Signal-to-noise ratio
    • 2.5.10 Atmospheric Extinction
    • 2.5.11 Rocket science
• 3 Mathematics
  • 3.1 Notation
  • 3.2 Algebra
    • 3.2.1 Factorisation
    • 3.2.2 Absolute Value
    • 3.2.3 Quadratics
    • 3.2.4 Logarithms
    • 3.2.5 Vectors
    • 3.2.6 Factorials
    • 3.2.7 Inner product definition
    • 3.2.8 Complex Numbers
    • 3.2.9 Power Series
    • 3.2.10 Matrix Operations
    • 3.2.11 Matrix Types
    • 3.2.12 Change of Basis Unitary
    • 3.2.13 Commutator
    • 3.2.14 Anticommutator
    • 3.2.15 Cauchy-Schwarz Inequality
  • 3.3 Geometry
    • 3.3.1 Pythagorean theorem
    • 3.3.2 Properties of shapes
    • 3.3.3 Properties of solids
    • 3.3.4 Circular formulae
    • 3.3.5 Useful Functions
    • 3.3.6 Coordinates
    • 3.3.7 Hyperbolic Functions
  • 3.4 Trigonometry
    • 3.4.1 Identities
  • 3.5 Calculus
    • 3.5.1 Limits
    • 3.5.2 Properties
    • 3.5.3 Differentiation
    • 3.5.4 Partial Differentiation
    • 3.5.5 The Differential
    • 3.5.6 Line Element
    • 3.5.7 Integration
    • 3.5.8 Vector Calculus
    • 3.5.9 Dirac Delta Function
    • 3.5.10 Approximations
• 4 Statistics
  • 4.1 Variance
  • 4.2 Standard Deviation
• A Values
  • A.1 Physics
    • A.1.1 Physical Constants
    • A.1.2 Useful Quantities
  • A.2 Astronomy
    • A.2.1 Useful Quantities
  • A.3 Mathematics
• B Units of Measurement
  • B.1 Natural Units
  • B.2 SI System
    • B.2.1 Base Units
    • B.2.2 Derived Units
  • B.3 CGS (centimetres-grams-seconds)
  • B.4 Astronomy units
    • B.4.1 Astronomical system
    • B.4.2 Equatorial Coordinate System
  • B.5 United States customary units (aka Imperial Units)
    • B.5.1 Length
  • B.6 Degrees of Angle
  • B.7 Miscellaneous Units
    • B.7.1 Pressure
  • B.8 Prefixes
• C Mathematical Stuff
  • C.1 Trigonometric Values
    • C.1.1 Pythagorean Triples
• D Boring stuff
  • D.1 Version History
  • D.2 Licensing
  • D.3 Contact
  • D.4 Credits

1.1.7 Projectile Motion

• v^2 y = u^2 y + 2ay ∆y
• x = uxt
• ∆y = uy ∆t + 12 ay ∆t^2 = uy t + (^12)

Fy m

∆t^2

1.1.8 Rotation

Angular Velocity

• ω =

dθ dt = θ˙

• ω = v r
• ~v = ~r × ~ω

Angular Acceleration

• α = dω dt

d^2 θ dt^2

= ˙ω = θ¨

Moment of Inertia

Point Mass

• I = mr^2

Several Point Masses

• I =

mr^2

Continuous mass

• I =

r^2 dm

Parallel axis theorem

• I = Icom + md^2

Thin disc rotating about centre

• I =

M R^2

Thin hoop rotating about centre

• I = M R^2

Thin rod rotating about centre

• I =

M L^2

Thin rod rotating about end

• I =

M L^2

Rotational Kinetic Energy

• Krot = 12 Iω^2

Total Kinetic Energy

• Ktot = Ktrans + Krot = 12 (mr^2 com + Icom)ω^2

Angular Momentum

• L~ = I~ω = ~r × ~p

Torque

• ~τ = I~α = dL dt

= ~r × F~

1.1.9 Euler-Lagrange and the Hamiltonian

Lagrangian

• ` = T − V =

lm

a(q) ˙ql q˙m

• = K( ˙ql) − U (ql)

Generalised coordinates & momenta

• pk ≡

∂L

∂ q˙k

Euler-Lagrange Equation

d dt

∂`

∂ x˙

∂`

∂x

Action

• S[x(t)] =

t´B

tA

`( ˙x(t), x(t)) dt

Hamiltonian

• H =

l

pl q˙ − L

• P˙ = −

∂H

∂Q

• Q˙ =

∂H

∂P

• P˙ = −ω^2 Q
• Q˙ = P

1.2 Oscillations

1.2.1 Springs

Force of a Spring

• F~ = −ks~x

Potential Energy of a Spring

• Us =

ksx^2

Angular Frequency of a Spring

• ω =

ks m

1.7.5 Kepler’s Third Law

T 2

r^3

4 π^2 G(m + M )

= constant

1.8 Electromagnetism

1.8.1 Notation

• ~r = ~r − ~r′

1.8.2 Maxwell’s Equations

Integral form Differential form

Gauss’s Law

v S

E^ ~ · d~a = 1 ε 0

t V

ρ dV ∇ ·~ E~ =

ρ ε 0

Qenc ε 0

Gauss’s Law for Magnetism

v S

B^ ~ · d~a = 0 ∇ ·~ B~ = 0

Maxwell-Faraday equation

b

E^ ~ · d~l = − d dt

s S

B^ ~ · d~a ∇ ×~ E~ = − ∂

B~

∂t

Amp´ere’s circuital law

b

B^ ~ · d~l = μ 0

s S

J^ ~ · d~a + μ 0 ε 0 d dt

s S

E^ ~ · d~a ∇ ×~ B~ = μ 0 ( J~ + ε 0 ∂

E~

∂t

= μ 0 (Ienc + ε 0

d dt

S

E^ ~ · d~a)

1.8.3 Lorentz Force

On a point charge

• F~ = q( E~ + ~v × B~)

On a current

• d F~ = I

d~l × B~

• F~ = IL~ × B~

1.8.4 Electric Field

• E~ =

V

ρ(r~′) r^2

ˆr dτ

From a single point charge

• E~ =

4 πε 0

q r^2 ˆr

From a dipole

• | E~axis| ≈

2 p 4 πε 0 r^3

• | E~⊥| ≈

p 4 πε 0 r^3

1.8.5 Dipole moment

• ~p = q d~

1.8.6 Electric potential

• V =

4 πε 0

Q

r

• ∇^2 V =

−ρ ε 0

In a single-point charge field

• ∆(~r) =

4 πε 0

q r

1.8.7 Electric potential difference

• ∆(~r) = −

´^ ~a ~b

E^ ~ · d~l

In a single-point charge field

• ∆(~r) =

4 πε 0

Q(

b

a

1.8.8 Electric potential energy

• UE = q∆V =

4 πε 0

qQ r

Energy stored in an electrostatic field distribution

• UE = 12 =  0 E^2 × volume

1.8.9 Charge densities

Surface

• σ =

dq da

Q

A

Line

• λ =

dq dl

Q

L

1.8.10 Current densities

Volume

• J~ =

d~I d~a⊥

I

A⊥

= σ( E~ + ~v × B) = |q|nu( E~ + ~v × B)

• ∇ ·~ J~ = 0

1.8.13 Magnetic fields

• B~(~r) = μ 0 4 π

´ (^) I~ × ˆr r^2

dl

• d B~ =

μ 0 4 π

q~v × rˆ r^2

μ 0 4 π

I d~l × rˆ r^2

Magnetic field due to a wire

• B~ =

μ 0 4 π

2 I

r φˆ

Magnetic vector potential

• A~(~r) =

μ 0 4 π

´ (^) J~(r~′) r

dτ

• ∇ ×~ A~ = B~

• ∇ ×~ (∇ ×~ A~) = −μ 0 J~
• ∇ ·~ A~ = 0

1.8.14 Inductors

• ε = −LI

Energy stored in an inductor

• W =

LI^2

1.8.15 Materials

Macroscopic Maxwell’s Equations (Materials)

Integral form Differential form

Gauss’s Laws

v S

P^ ~ · d~a = − ∑^ QB ∇ ·~ P~ = −ρB

v S

D^ ~ · d~a = ∑^ Qf ∇ ·~ D~ = ρf

Gauss’s Law for Magnetism

v S

B^ ~ · d~a = 0 ∇ ·~ B~ = 0

Maxwell-Faraday equation

b

E^ ~ · d~l = − d dt

s S

B^ ~ · d~a ∇ ×~ E~ = − ∂

B~

∂t

Amp´ere’s circuital law

b

H^ ~ · d~l = If,enc + ∂ ∂t

s S

D^ ~ · d~a ∇ ×~ H~ = J~f + ∂

D~

∂t

Dielectric constant

• k =

ε ε 0

= εr

• ε = kε 0 = εr ε

Susceptibility

• χe = 1 − εr

Polarisability

• P~ = ε 0 χe E~ = n~p

Bound Charge

Surface

• σB = ~p · nˆ

Volume

• ρB = −∇ ·~ P~

Total

• QB = σB + ρB = ~p · ˆn − ∇ ·~ P~

Electric displacement

• D~ = ε E~ = kε 0 E~ = ε 0 E~ + P~

Magnetic field

• H~ =

B~

μ 0

− M~

Magnetic dipole

• m~ = I~a

Bound current

• J~B = ∇ ×~ M~
• K~B = M~ × ˆn

1.9 Special Relativity

1.9.1 Interval

• ∆s^2 = −c^2 ∆t^2 + ∆x^2 + ∆y^2 + ∆z^2
• ds^2 = −c^2 dt^2 + dx^2 + dy^2 + dz^2
• ∆s^2 < 0 is a timelike interval. Events separated by this interval can be causally related.
• ∆s^2 = 0 is a lightlike interval. Events separated by this interval can be causally related, but only by a lightspeed signal.
• ∆s^2 > 0 is a spacelike interval. Events separated by this interval CANNOT be causally related.

Gamma Factor

• γ =

v c

)^2

• γ =

dt dτ

1.9.3 Frames of Reference

Condition for an inertial frame

d^2 x dt^2

d^2 y dt^2

d^2 z dt^2

Galilean Transformations

• x′^ = x + vt
• y′^ = y
• z′^ = z
• All assuming x is along the axis of motion and x = x’ when t = 0.

Lorentz Boosts

• t′^ = γ(t −

vx c^2

• x′^ = γ(x − vt)
• y′^ = y
• z′^ = z
• (x is along the axis of motion)

ct′ x′ y′ z′

γ −vγ 0 0 −vγ γ 0 0 0 0 1 0 0 0 0 1

ct x y z

General Lorentz transformation

b′^0 b′^1 b′^2 b′^3

γ −vγ 0 0 −vγ γ 0 0 0 0 1 0 0 0 0 1

b^0 b^1 b^2 b^3

• Motion along the x-axis.

Proper Time

• τ =

t´B

tA

γ

dt =

t´B

tA

v^2 (t) c^2

dt

1.9.4 Proper Velocity

• u =

ds dτ

1.10 General Relativity

1.10.1 Metrics

Minkowski

• η =

• ds^2 = −c^2 dt^2 + dx^2 + dy^2 + dz^2

Schwarzschild

• g =

2 GM

c^2 r

2 GM

c^2 r

)−^1 0

0 0 r^2 0 0 0 r^2 sin^2 θ

• ds^2 = −(1 −

2 GM

c^2 r

)c^2 dt^2 + (1 −

2 GM

c^2 r

)−^1 dr^2 + r^2 dθ^2 + r^2 sin^2 θ dφ^2 )

1.10.2 Rindler coordinates

Line element

• ds^2 = −(1 +

gx′ c^2

)^2 (c dt′)^2 + dx′

1.10.3 Einstein notation

• Contravariant: eα
• Covariant: eα
• tαβ = gβγ tαγ
• tαβ^ = gβγ^ tαγ
• t′αβ =

∂x′α ∂xγ

∂xδ ∂x′β^

tγ δ

• t′ αβ^ =

∂xγ ∂x′α

∂x′β ∂xδ^

tγ δ

Metrics

• ds^2 = gαβ dxα^ dxβ
• gαβ^ =

gαβ

• δαβ =

1 α = β 1 α 6 = β

• δαγ aγ^ = aα
• gαγ^ gγβ = δβα

Four-vector product

• a · b = gαβ aαbβ^ = aβ bα

1.10.4 Christoffel symbols

• Γαβγ = 12 gαδ^ (

∂gδβ ∂xγ^

∂gδγ ∂xβ^

∂gβγ ∂xδ^

• Γαβγ = 12 (

∂gδβ ∂xγ^

∂gδγ ∂xβ^

∂gβγ ∂xδ^

d^2 xμ dτ

• Γμαβ

dxα dτ

dxβ dτ

1.11.3 Entropy

• S = kB ln Ω

1.11.4 Black bodies

Energy of a photon

• E = hf

Wien’s Displacement Law

• λmax =

b T

= (2. 8977729 × 10 −^3 )

T

Stefan-Boltzmann Law

• I = σT 4

Spectrum

• Bλ(T ) = 2 hc^2 λ^5

exp(

hc λkB T

• Bν (T ) =

2 hν c^2

exp( hν kB T

1.12 Quantum Mechanics

1.12.1 The Uncertainty Principle

• ∆x∆p ≥

¯h 2

• ∆E∆t ≥ ¯h 2

1.12.2 Bras and Kets

• |ψ〉 = 〈ψ|†

1.12.3 Rules for an Inner Product

• 〈ψ|φ〉 ≡ (|ψ〉, |φ〉)
• Symmetric: 〈ψ|φ〉 = 〈φ|ψ〉∗
• Linear in second component
• Anti-linear in first component

1.12.4 The Born Rule

• P = |〈ψ|ψ〉|^2

1.12.5 Expectation

• 〈A〉 =

A|Ψ(x, t)|^2 dx

• 〈A〉 = 〈ψ|A|ψ〉

1.12.6 Variance

• var(A) = 〈ψ|A^2 |ψ〉 − 〈ψ|A|ψ〉 2

1.12.7 Standard Deviation

• δA =

var(A) =

〈ψ|A^2 |ψ〉 − 〈ψ|A|ψ〉^2

1.12.8 Trace

• Tr(A) =

j

〈xj |A|xj 〉

1.12.9 Partial Trace

• T rB (|a〉〈a| ⊗ |b〉〈b|) ≡ |a〉〈a|Tr(|b〉〈b|)
• 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¨odinger Equation

• i¯h

∂t

Ψ(r, t) = HˆΨ(r, t)

¯h^2 2 m

∂^2 Ψ(x, t) ∂x^2

• V (x)Ψ(x, t) = i¯h

∂Ψ(x, t) ∂t

• −

¯h^2 2 m

∂^2 ψ(x) ∂x^2

• V (x)ψ(x, t) = Eψ(x)

• Hˆ|Ψ(t)〉 = i¯h

∂t

|Ψ(t)〉

1.12.11 Heisenberg equation of motion

d dt

Aˆ(t) = i ¯h

[ H,ˆ Aˆ(t)]

1.12.12 Operators

• ajk = 〈j|A|k〉

Diagonalizable Operator

• A =

j

λj |λj 〉〈λj |

Normal Operator

• A =

j

|λj 〉〈λj |

Eigenstate Operators

• (|λk〉〈λk|)n^ = |λk〉〈λk|

Identity

• I =

j

|xj 〉〈xj |