Frontiers Equations

Question 1

Flux of a point source

Answer 1

F = L/4πr²

L is luminosity
r is radial distance from source

Question 2

Arcminute and Arcsecond

Answer 2

1’ = (1/60)ᵒ
1’’ = (1/3600)ᵒ

Question 3

Kepler’s Third Law

Answer 3

T² ∝ a³

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

Question 4

Virial Theorem

Answer 4

2KE + PE = 0

True for bound orbits (systems in equilibrium)

Question 5

Distance to a nearby star using parallax

Answer 5

θ = 1AU / D

D is distance to nearby star
θ is the parallax angle

Question 6

Apparent magnitude of star

Answer 6

m = constant - 2.5log₁₀F

F is the flux

Question 7

Absolute magnitude of star

Answer 7

M = m - 5log₁₀(D/10pc)

D is the distance to the star

Question 8

Wein’s Displacement law

Answer 8

λₘₐₓ = 2.9X10⁻³ / T(K)

T(K) is temperature in kelvin

Question 9

Stefan Boltzmann Law (The luminosity of a spherical blackbody)

Answer 9

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

Question 10

class of elliptical galaxies

Answer 10

= 10(1- b/a)

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

Question 11

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

Answer 11

= Φ(L)dL

Question 12

Number of Galaxies per unit volume

Answer 12

N = ∫ Φ(L)dL

Question 13

Total Galactic Luminosity

Answer 13

Lₜₒₜ = ∫ L Φ(L)dL

Question 14

Schechter function

Answer 14

Φ(L) = KL⁻¹ e⁻ᴸ/ᴸ*

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

Question 15

Galactic Velocity and M(r)

Answer 15

M(r) = rv²/G

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

Question 16

Vₗₒₛ and doppler

Answer 16

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

Question 17

Vₗₒₛ and inclination angle

Answer 17

Vₗₒₛ = Vₜᵣᵤₑ sin(i)

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

Question 18

Doppler broadening

Answer 18

Δ λ / λ = σ / c

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

Question 19

Approximate energy of a particle at the black hole event horizon

Answer 19

Δ E ≈ 1/2 Δmc²

Question 20

Luminosity of a particle at the event horizon

Answer 20

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

Question 21

Hubble Equation

Answer 21

V = H₀ D

Question 22

Hubble age of universe

Answer 22

t₀ = 1 / H₀

Question 23

Force and Potential Energy

Answer 23

F(x) = - dU/dx

Question 24

Potential energy for simple harmonic oscillator

Answer 24

U(x) = 1/2 kx²

Question 25

Scaling for area and volume

Answer 25

A ≈ L² V ≈ L³

Question 26

Reynold's Number

Answer 26

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

Question 27

Eₙ for particle in a box

Answer 27

Eₙ = n²h² / 8mL²

Question 28

Heisenberg uncertainty principle

Answer 28

Δx Δp ≥ ℏ

Question 29

Tunnelling current

Answer 29

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

Question 30

resonant frequency of cantilever in dynamic AFM

Answer 30

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

Question 31

Kₑբբ (effective spring constant) in dynamic AFM

Answer 31

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

Question 32

wavenumber k

Answer 32

k = 2π/λ

Question 33

Acoustic impedance

Answer 33

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

Question 34

Intensity of a sound wave

Answer 34

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

Question 35

Speed of sound in fluid

Answer 35

√B / √ρ B is the bulk modulus ρ is the density

Question 36

Bulk modulus B

Answer 36

ΔP = - B (ΔV/V) B links pressure change to relative change in volume

Question 37

Speed of sound in gases

Answer 37

v = √ (γRT/m) γ is the adiabatic constant; ratio of heat capacity at constant pressure to heat capacity at constant volume

Question 38

adiabatic constant

Answer 38

γ = B/P

Question 39

Ratio of intensities of transmitted waves and incident waves T

Answer 39

T = (4 Z₁Z₂) / (Z₁+Z₂)² significant transmission only occurs when Z₁ ≈ Z₂

Question 40

Ratio of intensities of reflected waves and incident waves R

Answer 40

R = (Z₂ - Z₁)² / (Z₁+Z₂)²

Question 41

Intensity level in Decibels

Answer 41

dB = 10log₁₀(I / I₀)

Question 42

Maximum Pulse repetition frequency PRF

Answer 42

PRF < v / 2L

Question 43

Intensity and attenuation

Answer 43

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

Question 44

Relative frequency change of ultrasound reflected by a moving source

Answer 44

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

Question 45

Radioactive decay

Answer 45

dN/dt = -rN N = N₀ e⁻ʳᵗ r is the decay rate N is the number of radioactive atoms

Question 46

Activity Equation

Answer 46

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

Question 47

Dose equivalent in sieverts

Answer 47

(Sv) = absrobed dose in Gy x RBE RBE is relative biological effectiveness

Question 48

Absorbed radiation dose (Gy)

Answer 48

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

Question 49

magnetic moment of the nucleus

Answer 49

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

Question 50

NMR signal

Answer 50

S = S₀ e ⁻ᵀᴱ/ᵀ₂ S is the signal S₀ is the original signal TE is the measurement time T₂ is the relaxation time

Question 51

Larmor frequency

Answer 51

ω = γ B B is the magnetic field strength γ is the gyromagnetic ratio

Question 52

Flux across cellular membrane

Answer 52

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

Question 53

Magnetic Field Gradient

Answer 53

B = Gx ω = γ Gx B is the field at x G is the field gradient x is position

Question 54

Energy in an applied magnetic field

Answer 54

Energy = - μ . B ΔE = γ ℏ B

Question 55

Nernst Equation

Answer 55

ΔV = RT/zF ln[Cₒᵤₜ - Cᵢₙ] F is faraday's constant z is the relative charge of the ion (i.e z(k⁺) = +1)

Question 56

Biot Savart Law

Answer 56

μ₀/4π . I₀ Δs X rhat /r²