Frontiers Equations Flashcards
Flux of a point source
F = L/4πr²
L is luminosity
r is radial distance from source
Arcminute and Arcsecond
1’ = (1/60)ᵒ
1’’ = (1/3600)ᵒ
Kepler’s Third Law
T² ∝ a³
T is the orbital period (years)
a is average distance from sun (AU)
Virial Theorem
2KE + PE = 0
True for bound orbits (systems in equilibrium)
Distance to a nearby star using parallax
θ = 1AU / D
D is distance to nearby star
θ is the parallax angle
Apparent magnitude of star
m = constant - 2.5log₁₀F
F is the flux
Absolute magnitude of star
M = m - 5log₁₀(D/10pc)
D is the distance to the star
Wein’s Displacement law
λₘₐₓ = 2.9X10⁻³ / T(K)
T(K) is temperature in kelvin
Stefan Boltzmann Law (The luminosity of a spherical blackbody)
L = 4π R² T⁴
L is luminosity
R is radius of star
T is temperature
class of elliptical galaxies
= 10(1- b/a)
b/a is the ratio between the semi minor and semi major axis
Number of Galaxies per unit volume in (L, L+dL)
= Φ(L)dL
Number of Galaxies per unit volume
N = ∫ Φ(L)dL
Total Galactic Luminosity
Lₜₒₜ = ∫ L Φ(L)dL
Schechter function
Φ(L) = KL⁻¹ e⁻ᴸ/ᴸ*
K is the normalisation constant
L* is the knee of the function
Galactic Velocity and M(r)
M(r) = rv²/G
M(r) is the tot mass enclosed by the galaxy in radius r
v is tangent rotational velocity
Vₗₒₛ and doppler
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
Vₗₒₛ and inclination angle
Vₗₒₛ = Vₜᵣᵤₑ sin(i)
Vₜᵣᵤₑ is the actual tangent rotational velocity
i is the inclination angle
Doppler broadening
Δ λ / λ = σ / c
σ is the RMS of random stellar velocities
Δ λ is the spectral line width
Approximate energy of a particle at the black hole event horizon
Δ E ≈ 1/2 Δmc²
Luminosity of a particle at the event horizon
L = ΔE / Δt = 1/2 Δmc² / Δt
Δm /Δt ≈ 2L/c²
Hubble Equation
V = H₀ D
Hubble age of universe
t₀ = 1 / H₀
Force and Potential Energy
F(x) = - dU/dx
Potential energy for simple harmonic oscillator
U(x) = 1/2 kx²
Scaling for area and volume
A ≈ L²
V ≈ L³
Reynold’s Number
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
Eₙ for particle in a box
Eₙ = n²h² / 8mL²
Heisenberg uncertainty principle
Δx Δp ≥ ℏ
Tunnelling current
I = I₀ e⁻²ᵏᵈ
K is 1/η , η is the penetration distance
d is the width of the barrier
resonant frequency of cantilever in dynamic AFM
ω = √kₑբբ / m
where k=kₑբբ is the effective spring constant
m is the cantilever mass
Kₑբբ (effective spring constant) in dynamic AFM
Kₑբբ = k - d/dx Fₛᵤᵣբ
where k is the natural resonant frequency
Fₛᵤᵣբ is the force on the cantilever due to pauli exclusion forces.
wavenumber k
k = 2π/λ
Acoustic impedance
Z = p / u = ρ v
p is pressure
u is the medium velocity
ρ is the density
v is wave speed
Intensity of a sound wave
I = p₀² / 2Z
p₀ is the pressure amplitude
Z is acoustic impedance
Speed of sound in fluid
√B / √ρ
B is the bulk modulus
ρ is the density
Bulk modulus B
ΔP = - B (ΔV/V)
B links pressure change to relative change in volume
Speed of sound in gases
v = √ (γRT/m)
γ is the adiabatic constant; ratio of heat capacity at constant pressure to heat capacity at constant volume
adiabatic constant
γ = B/P
Ratio of intensities of transmitted waves and incident waves T
T = (4 Z₁Z₂) / (Z₁+Z₂)²
significant transmission only occurs when Z₁ ≈ Z₂
Ratio of intensities of reflected waves and incident waves R
R = (Z₂ - Z₁)² / (Z₁+Z₂)²
Intensity level in Decibels
dB = 10log₁₀(I / I₀)
Maximum Pulse repetition frequency PRF
PRF < v / 2L
Intensity and attenuation
I = I₀ e ⁻²ᵃˣ
a is the attenuation coefficient
x is the penetration distance
Relative frequency change of ultrasound reflected by a moving source
f / fᵢ = 2 X (reflector speed) / (v+ reflector speed)
Radioactive decay
dN/dt = -rN
N = N₀ e⁻ʳᵗ
r is the decay rate
N is the number of radioactive atoms
Activity Equation
A = RN
R is the rate
N is the number of undecayed nucleons
Dose equivalent in sieverts
(Sv) = absrobed dose in Gy x RBE
RBE is relative biological effectiveness
Absorbed radiation dose (Gy)
E / m
E is ionising radiation absorbed (J)
m is the mass (kg)
magnetic moment of the nucleus
μ = γ J
The gyromagnetic ratio γ (gamma) is the ratio of the magnetic moment to angular momentum J.
NMR signal
S = S₀ e ⁻ᵀᴱ/ᵀ₂
S is the signal
S₀ is the original signal
TE is the measurement time
T₂ is the relaxation time
Larmor frequency
ω = γ B
B is the magnetic field strength
γ is the gyromagnetic ratio
Flux across cellular membrane
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
Magnetic Field Gradient
B = Gx
ω = γ Gx
B is the field at x
G is the field gradient
x is position
Energy in an applied magnetic field
Energy = - μ . B
ΔE = γ ℏ B
Nernst Equation
ΔV = RT/zF ln[Cₒᵤₜ - Cᵢₙ]
F is faraday’s constant
z is the relative charge of the ion (i.e z(k⁺) = +1)
Biot Savart Law
μ₀/4π . I₀ Δs X rhat /r²
