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Year 1 > Frontiers Equations > Flashcards

Frontiers Equations Flashcards

1
Q

Flux of a point source

A

F = L/4πr²

L is luminosity
r is radial distance from source

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2
Q

Arcminute and Arcsecond

A

1’ = (1/60)ᵒ
1’’ = (1/3600)ᵒ

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3
Q

Kepler’s Third Law

A

T² ∝ a³

T is the orbital period (years)
a is average distance from sun (AU)

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4
Q

Virial Theorem

A

2KE + PE = 0

True for bound orbits (systems in equilibrium)

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5
Q

Distance to a nearby star using parallax

A

θ = 1AU / D

D is distance to nearby star
θ is the parallax angle

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6
Q

Apparent magnitude of star

A

m = constant - 2.5log₁₀F

F is the flux

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7
Q

Absolute magnitude of star

A

M = m - 5log₁₀(D/10pc)

D is the distance to the star

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8
Q

Wein’s Displacement law

A

λₘₐₓ = 2.9X10⁻³ / T(K)

T(K) is temperature in kelvin

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9
Q

Stefan Boltzmann Law (The luminosity of a spherical blackbody)

A

L = 4π R² T⁴
L is luminosity
R is radius of star
T is temperature

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10
Q

class of elliptical galaxies

A

= 10(1- b/a)

b/a is the ratio between the semi minor and semi major axis

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11
Q

Number of Galaxies per unit volume in (L, L+dL)

A

= Φ(L)dL

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12
Q

Number of Galaxies per unit volume

A

N = ∫ Φ(L)dL

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13
Q

Total Galactic Luminosity

A

Lₜₒₜ = ∫ L Φ(L)dL

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14
Q

Schechter function

A

Φ(L) = KL⁻¹ e⁻ᴸ/ᴸ*

K is the normalisation constant
L* is the knee of the function

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15
Q

Galactic Velocity and M(r)

A

M(r) = rv²/G

M(r) is the tot mass enclosed by the galaxy in radius r
v is tangent rotational velocity

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16
Q

Vₗₒₛ and doppler

A

Vₗₒₛ / c = (λobs - λlab) / λlab

Vₗₒₛ is tangent rotational velocity of galaxy relative to line of sight
λobs is the observed wavelength
λlab is the actual wavelength

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17
Q

Vₗₒₛ and inclination angle

A

Vₗₒₛ = Vₜᵣᵤₑ sin(i)

Vₜᵣᵤₑ is the actual tangent rotational velocity
i is the inclination angle

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18
Q

Doppler broadening

A

Δ λ / λ = σ / c

σ is the RMS of random stellar velocities
Δ λ is the spectral line width

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19
Q

Approximate energy of a particle at the black hole event horizon

A

Δ E ≈ 1/2 Δmc²

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20
Q

Luminosity of a particle at the event horizon

A

L = ΔE / Δt = 1/2 Δmc² / Δt
Δm /Δt ≈ 2L/c²

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21
Q

Hubble Equation

A

V = H₀ D

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22
Q

Hubble age of universe

A

t₀ = 1 / H₀

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23
Q

Force and Potential Energy

A

F(x) = - dU/dx

24
Q

Potential energy for simple harmonic oscillator

A

U(x) = 1/2 kx²

25
Q

Scaling for area and volume

A

A ≈ L²
V ≈ L³

26
Q

Reynold’s Number

A

Re = pvL / η

p is fluid density
v is velocity
L is linear dimension of object
η is the fluid viscosity

Reynolds number is the ratio of inertial forces to viscous forces for an object travelling through a fluid

27
Q

Eₙ for particle in a box

A

Eₙ = n²h² / 8mL²

28
Q

Heisenberg uncertainty principle

A

Δx Δp ≥ ℏ

29
Q

Tunnelling current

A

I = I₀ e⁻²ᵏᵈ
K is 1/η , η is the penetration distance
d is the width of the barrier

30
Q

resonant frequency of cantilever in dynamic AFM

A

ω = √kₑբբ / m
where k=kₑբբ is the effective spring constant
m is the cantilever mass

31
Q

Kₑբբ (effective spring constant) in dynamic AFM

A

Kₑբբ = k - d/dx Fₛᵤᵣբ
where k is the natural resonant frequency
Fₛᵤᵣբ is the force on the cantilever due to pauli exclusion forces.

32
Q

wavenumber k

A

k = 2π/λ

33
Q

Acoustic impedance

A

Z = p / u = ρ v

p is pressure
u is the medium velocity
ρ is the density
v is wave speed

34
Q

Intensity of a sound wave

A

I = p₀² / 2Z
p₀ is the pressure amplitude
Z is acoustic impedance

35
Q

Speed of sound in fluid

A

√B / √ρ

B is the bulk modulus
ρ is the density

36
Q

Bulk modulus B

A

ΔP = - B (ΔV/V)

B links pressure change to relative change in volume

37
Q

Speed of sound in gases

A

v = √ (γRT/m)

γ is the adiabatic constant; ratio of heat capacity at constant pressure to heat capacity at constant volume

38
Q

adiabatic constant

A

γ = B/P

39
Q

Ratio of intensities of transmitted waves and incident waves T

A

T = (4 Z₁Z₂) / (Z₁+Z₂)²

significant transmission only occurs when Z₁ ≈ Z₂

40
Q

Ratio of intensities of reflected waves and incident waves R

A

R = (Z₂ - Z₁)² / (Z₁+Z₂)²

41
Q

Intensity level in Decibels

A

dB = 10log₁₀(I / I₀)

42
Q

Maximum Pulse repetition frequency PRF

A

PRF < v / 2L

43
Q

Intensity and attenuation

A

I = I₀ e ⁻²ᵃˣ
a is the attenuation coefficient
x is the penetration distance

44
Q

Relative frequency change of ultrasound reflected by a moving source

A

f / fᵢ = 2 X (reflector speed) / (v+ reflector speed)

45
Q

Radioactive decay

A

dN/dt = -rN
N = N₀ e⁻ʳᵗ

r is the decay rate
N is the number of radioactive atoms

46
Q

Activity Equation

A

A = RN
R is the rate
N is the number of undecayed nucleons

47
Q

Dose equivalent in sieverts

A

(Sv) = absrobed dose in Gy x RBE

RBE is relative biological effectiveness

48
Q

Absorbed radiation dose (Gy)

A

E / m

E is ionising radiation absorbed (J)
m is the mass (kg)

49
Q

magnetic moment of the nucleus

A

μ = γ J

The gyromagnetic ratio γ (gamma) is the ratio of the magnetic moment to angular momentum J.

50
Q

NMR signal

A

S = S₀ e ⁻ᵀᴱ/ᵀ₂

S is the signal
S₀ is the original signal
TE is the measurement time
T₂ is the relaxation time

51
Q

Larmor frequency

A

ω = γ B

B is the magnetic field strength
γ is the gyromagnetic ratio

52
Q

Flux across cellular membrane

A

J = -D dC/dx

J is the number of ions crossing unit area in unit time
D is the diffusion coefficient
C is the # of ions per unit volume

53
Q

Magnetic Field Gradient

A

B = Gx
ω = γ Gx

B is the field at x
G is the field gradient
x is position

54
Q

Energy in an applied magnetic field

A

Energy = - μ . B
ΔE = γ ℏ B

55
Q

Nernst Equation

A

ΔV = RT/zF ln[Cₒᵤₜ - Cᵢₙ]

F is faraday’s constant
z is the relative charge of the ion (i.e z(k⁺) = +1)

56
Q

Biot Savart Law

A

μ₀/4π . I₀ Δs X rhat /r²

Year 1 (6 decks)

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