Planetary Processes

ORBITS

Kepler’s First Law

The orbit of planets are ellipses with the sun at one of the foci.

Kepler’s Second Law

As a planet orbits the Sun, it sweeps out equal areas in equal times.
Conservation of angular momentum

Kepler’s Third Law

Distant planets orbit slower
P in yrs, a in AU, M in solar masses

\( P^2 = \frac{a^3}{M} \)

In reality, both the planet and the star orbit their common centre of mass:

\( P^2 = \frac{(a_1 + a_2)^3}{M_1 + M_2} \)

Orbital Elements

Semi-major axis
Eccentricity
Orbital Period
Epoch of periastron passage
Inclination angle
Position angle of nodes
Argument of periastron

Vis-Viva Equation

Velocity at any point in an elliptical orbit:

\( v = \sqrt{GM_{\odot} (2/r - 1/a)} \)

Escape Velocity

Set the gravitational PE equal to KE and solve for v

Hohmann Transfer Orbit

Most time-efficient way to transfer orbits

ANOMALIES AND TIDAL FORCE

Anomalies

Mean anomaly: fraction of a period since the last periastron passage
Eccentric anomaly: angle between periastron, centre of ellipse and planet (sort of)
True anomaly: angle between periastron, star and planet.

Tidal Force

Derivation of the tidal force

\( F = \frac{2GMmr}{d^3} \)

Earth – moon: Effects of tidal forces

Friction with the rotating Earth pulls the tides slightly ahead of the Earth-Moon line.
Moon’s gravity tries to pull the tides back, slowing Earth’s rotation.
The gravity of the bulges pulls the moon ahead, increasing orbital distance.

Hill Sphere

Gravitational sphere of influence on a secondary body:

\( R_H = \frac{M_2}{3(M_1 + M_2)^{1/3}} a(1-e) \)

ORBITAL RESONANCE, HILL SPHERE, ROCHE LIMIT

Orbital Resonance

Energetically preferred for tightly packed moons around a planet.
Tidal forces cause the moons to move outward
Io : Europa : Ganymede = 1 : 2 : 4
TRAPPIST-1 System

Lagrange Points

Points where the centripetal and gravitational forces are in equilibrium where small masses feel no forces
Stable: L4 and L5
Unstable: L1, L2, L3
Hill sphere extends between L1 and L2

Roche Limit

Minimum orbital distance of a moon (or other orbiting body) so it doesn't get torn apart
Roche Limit derivation:
\( d = \frac{2Mr}{m}^{1/3} \)
In terms of densities
\( d = 1.26 (\frac{\rho_M}{\rho_m})^{1/3} \)

DISSIPATIVE PROCESSES

Radiation Process

Felt by micron sized particles
Derivation of radiation process
\( F_{rad} = (\frac{L_{\odot} R^2 Q_{PR}}{4cd^2}) \)

\( \frac{F_{rad}}{F_{grav}} = \beta = (\frac{3 L_{\odot} Q_{PR}}{16 \pi GM_{\odot}cR \rho}) \)
β depends only on the particle’s size, density, and reflectivity.

Poynting Robertson Drag

Valid for cm sized particles.
Particles experience a headwind due to photons – since the speed of light is finite.
Radiation pressure vs PR Drag

Zodiacal Light

Dust clouds in the orbital plane of the solar system scatters sunlight
Dust in debris disks possibly come from collisions between planetesimals.
PR drag and radiation pressure should be removing this dust.

Yarkovsky Effect

Valid for larger bodies (m to km)
Surface temperature difference causes the evening side to emit more than the morning side, increasing or decreasing orbital radius.
Asteroids with low obliquity have diurnal Yarkovsky Effect, and asteroids with high obliquity have a seasonal Yarkovsky effect.

YORP

Second order effect from uneven heating that changes the rotational rate of an asteroid.

Corpuscular Drag

Interaction of the solar wind with small particles orbiting the Sun.
Sub-micron sized particles.
The momentum of the solar win particles is smaller than that of a photon but the collisions are more “head on” as a result of the their lower velocity.

Gas Drag

Not so important for objects orbiting the sun. More so for ring particles or spacecraft orbiting close to a planet with an atmosphere.

\( F_D = -0.5 C_D A \rho_g v^2 \)

CRATERS

Simple Craters

Circular raised rim
No central peak
Breccia: Sedimentary rock created in the event
Maximum boundary diameter depends on g

Complex Craters

Central peak composed of uplifted material from the centre
Depth to diameter ratio decreases
Flat bottom
Land slides around the outer rim
More melting of surface material

Transition from simple to complex

Simple craters are in the strength regime. Size of the crater limited by the strength of rock.

Complex craters are in gravity regime.

Transition depends on material strength, density of material and gravity.

Stages of crater formation

Contact and compression’
Excavation
Modification

Contact and compression

Jetting: Highest speed ejecta thrown out as the impactor edge penetrates the surface. Can eject upto 80% of impactor mass.

Short-lived: ~ τ/2

Shock waves travel through the impactor and target, with speed similar to that of the impactor.

When the shockwave reaches the top of the impactor, pressure is released as a rarefaction wave, vaporising the impactor.

Total duration: ~1.5τ

Excavation

Material is removed upward and outward from the crater

Ejecta speed less than impact speed

End of excavation phase – transient crater

Modification Stage

Shock wave is now weaker and no longer shapes the material it passes through

This phase ends when stuff stops falling.

For simple craters,

Gault Scaling Relation

Weird equation, can’t be bothered typing it.

Planet density and gravity have negative powers

Equilibrium

At equilibrium, no more age information is added since each new crater destroys an old one.

At any given point, craters below a certain diameter exhibit equilibrium.

Geometric Saturation

The theoretical limit where every patch of surface area has a crater on it.

Atmospheric Limit

An atmosphere limits the size of projectiles that can penetrate to the surface.

On Earth, rocky meteors need to be atleast 60 cm in diameter o1-10 m: Break up in the atmosphere, fragments all at terminal velocity. o10-100 m: Continue at higher speed, disrupted by ram pressure o>100m: Reach surface after colliding with total amount of gas that is less than its own mass

SPECTRA

Planetary Spectra

Most planetary spectra have two components: oReflected sunlight oThermal emission in the IR

Electronic energies

Wavelength of emission/absorption

Fine structure: Interaction between electron spin and electron orbital angular momentum

Hyperfine structure: interaction between electron spin and nuclear spin

Rotational Energies

Transitions: ΔJ = 1

Vibrational Energies

Energy levels are evenly spaced
There is a zero-point energy
Vibrational states can jump multiple states at once.

Rotational + Vibrational Transitions

R branch: Molecule moves to a higher energy in both rotation and vibration (or lower)
P branch: Molecule moves to a higher energy in rotation and lower in vibration (or vice versa)

ATMOSPHERE

Energy Transport

Conduction – Dominates in compressed areas
Radiation – Dominates in less dense regions
Convection – Dominates when temperature gradient allows

Planetary Boundary Layer (0 – 0.3 to 3 km)

Lowest layer of the troposphere
Influenced by convection

Troposphere (0 – 10km)

Energy transport by convection and radiation

Stratosphere (10 – 50 km)

Home to the ozone layer
Ozone absorbs solar photons, bonds break, and the excess energy heats the atmosphere. This results in a temperature inversion.

Mesosphere (50 – 85 km)

Increase in cooling due to CO2
At high densities, CO2 absorbs infrared radiation and transfers it to other molecules through collisions transferring energy.
At low densities, there aren’t enough molecules for collisions. The energy is radiated to space, resulting in cooling.

Thermosphere (85 – 800 km)

Heating due to O2 photolysis and absorption oMost XUV and X rays are absorbed in the thermosphere.

Satellites orbit in the thermosphere

Exosphere (800 – 3000 km)

Beginning of space

Some satellites orbit here

Pressure and Scale Height

Derivation

\( P(r) = P(r_0) e^{-(r-r_0)/H} \)

LAPSE RATES AND STABILITY

Convection

Transports heat from lower to upper layers.

Dry Adiabatic Lapse Rate

Assumptions

The air parcel is assumed to: oBe thermally insulated from its surroundings. Temperature changes are adiabatic. oRemain at the same pressure as the surrounding air pressure. oMove slowly enough that the KE of a particle in the parcel is a negligible fraction of the total energy.

Derivation

\( \frac{dT}{dz} = -\frac{g}{C_P} = \frac{\gamma - 1}{\gamma} g \frac{\mu}{k_B} \)

Typical values of γ:

Monoatomic – 5/3
Diatomic – 5/7
Polyatomic – 4/3

Wet adiabatic Lapse Rate

The latent heat released by the water vapour in the atmosphere offsets some of the dry adiabatic cooling.

We need to consider the extra vaporisation term: L – Latent heat of vaporisation ws – mixing ratio – mass of water vapour per unit mass of air CP is for the atmosphere as a whole, not just the condensing gas Potential Temperature

The temperature a parcel of gas would have if expanded (or compressed) adiabatically from its existing temperature and pressure to a pressure of 1 bar.

The potential temperature is a conserved quantity as an air parcel moves through the atmosphere adiabatically.

Vertical Stability

Dotted line is the adiabatic lapse rate and the solid line is the environmental lapse rate.
At A, the parcel is hotter than the surrounding air and rises (unstable atmosphere).
Till B, the environmental lapse rate is more negative than the adiabatic lapse rate, which implies the air parcel is hotter and continues to rise.
At B, the atmosphere is stable and the air parcel is returns to with small vertical displacements.

SCATTERING

Radiative Transfer

Specific Intensity – Amount of energy per unit area, per unit solid angle, per second, per frequency of light.

The specific intensity changes along a path ds by dIν given by:

If the scattering into the beam and the emission is negligible:

The solution to this is an exponential: This is called Lambert’s exponential Law. No sources, just sinks.

Scattering Phase Function

Scattering phase function is given by p(cosθ) where θ is the scattering angle
Molecular scattering is mostly isotropic.
Molecules with size comparable to the wavelength of light mainly forward scatter.
The largest objects back-scatter.

Rayleigh Scattering

Scattering by molecules much smaller than the wavelength of light.
Mostly by gasses.
The strong wavelength dependence enhances Rayleigh scattering for the short wavelengths, giving us a blue sky.
Scattering at right angles is half the forward intensity.

Mie Scattering

Scattering by particles the same size particles as the wavelength of light.
Mostly forward scattering.

Scattering Parameter

Rayleigh Scattering: x << 1
Mie Scattering: x ~ 1
Geometric Optics: x>>1

Scattering Albedo

Fraction of light that undergoes scattering

Brightness Temperature

The temperature a blackbody would have in order to produce radiation of the same frequency.

Effective Temperature

The temperature a blackbody would have to produce the integrated flux from a source over all frequencies.
For an object that is mostly a blackbody, the effective temperature mostly matches the physical temperature of the emission layer.

Equilibrium Temperature

The temperature of a body derived by balancing the incoming flux to the outgoing flux.

EQUILIBRIUM TEMP

Albedo

Fraction of incoming radiation that is reflected.
Bond Albedo: Monochromatic albedo integrated over all the frequencies.
Geometric Albedo: Ratio of actual brightness at zero phase angle to that of a Lambertian surface

Scattering Angle

Emissivity

Ratio of energy radiated by a surface to the energy radiated by a blackbody at the same temperature.

Equilibrium Temperature

Derivation

Greenhouse Effect

To calculate the equilibrium temperature, assume radiative equilibrium in each layer of the atmosphere.

Result: – optical depth

Clouds

Venus
Sulfur compounds released from the surface interact with photo-dissociated oxygen and water in the atmosphere to form sulfuric acid.

Mars

3 main cycles – dust, CO2 and water.
Injection of airborne dust serves as a source of opacity for solar radiation.
Dust also serves as nuclei for cloud formation.

Titan

Methane subject to a cycle of sublimation and condensation.
Possible outgassing of methane through cryovolcanism.
Hazes in the upper atmosphere are a result of photochemical reactions.

ATMOSPHERIC MOTIONS

Clouds

Saturation vapour pressure: Pressure at a given temperature where a given condensable condenses. Dashed line represents the saturation vapour pressure of water

Clouds form where the saturation vapour pressure curve is at a higher temperature than the environmental lapse rate. Atmospheric Motions

Motions in planetary atmospheres are described by: oNavier-Stokes equation oLaws of thermodynamics oMass continuity equation oIdeal gas law (equation of state)

Atmospheric motions can be described in two views: oEulerian: Flow is studied at a fixed point oLagrangian: Motions follow atmospheric flow oThey can be related as: Navier-Stokes Equation

Derivation For a non-viscous incompressible fluid: Geostrophic Balance

Derivation Where

Wind speed is proportional to the pressure gradient. Rossby Number

The Rossby number is the ratio of the characteristic horizontal wind speed and the Coriolis force.

\( R = \frac{U}{fL} \)

In the geostrophic balance, R<<1.

Geostrophic Wind

Balance between the Coriolis force and the pressure gradient.

An air parcel moves from the high pressure region at the equator to the poles but is deflected to the right (in the Northern Hemisphere) by the Coriolis force. When the Coriolis force is equal to the pressure gradient, wind flows parallel to the isobars. Cyclostrophic balance For slowly rotating bodies, the Coriolis force is negligible. Wind is a result of the balance between the pressure gradient and the centrifugal force.

CIRCULATION

Hadley Circulation

For slow/no rotation, there are two large Hadley cells stretching from the equator to the poles.
For faster rotation, the cells are split into three: Hadley cell, Ferrel cell and Polar cell

Acoustic Waves

Derivation

Pressure Amplitude: Where, is the speed of sound

Sound wave amplitude: Where, is the displacement of the air particles due to the perturbation is the angular frequency o the wave

Sound Intensity:

Circulation and Vorticity

Circulation: Macroscopic measure of rotation for a finite area of the fluid.
Vorticity: Microscopic measure of rotation at any point in the fluid.
Anticlockwise rotation – positive vorticity

GRAVITY WAVES

Vorticity

Vorticity can be defined as the curl of the velocity vector

For a purely 2D flow, vorticity only has a z-component

is highly correlated with synoptic scale weather disturbances. A highly positive implies: o – the northward wind increases with x o – the eastward wind decreases with y

Absolute vorticity is the sum of the relative and planetary vorticities:

Potential Vorticity

Absolute vorticity of an air parcel changes with compression/expansion.
Dividing the absolute vorticity by the vertical spacing of levels of constant potential temperature gives potential vorticity – a conserved quantity.

Gravity Waves

Air parcel that is displaced vertically from equilibrium will undergo oscillations.
Derivation When m>0, the wave propagates in the vertical direction
Trapped gravity waves occur when wind speeds rise sharply above the mountain and the stability decreases in the layer above the mountain.
Vertically propagating waves occur when vertical stability increases and wind speed does not vary significantly with height.

JET STREAMS

Jet Streams

Narrow bands of strong wind in the upper levels of the atmosphere.

Rossby Waves

These are disturbances in the jet streams due to temperature difference, leading to pressure differences.

Counterclockwise flow around the low-pressure area brings warm tropical air to the warm air column, resulting in further heating. Similarly, clockwise flow around the high-pressure region brings cooler polar air to the low temperature column, resulting in further cooling.

At (1), But as the air column moves south, decreases which causes the air column to spin faster than the surrounding air which produces cyclonic curvature . Vice versa at (2).

Direct Detection of Exoplanets

Involves detecting the light from exoplanets
First detection of an exoplanet orbiting a brown dwarf: 2003
First detection of an exoplanet orbiting a star: 2008

Indirect Detection

Two types of detection of exoplanets:

From its gravitational influence
From it blocking out the star’s light

Pulsar Planets

Pulsar give off radio signals as they rotate
As an object orbits the Pulsar, the pulsar moves away and toward from the Earth causing a delay in the radio signal.

RADIAL VELOCITY

A star with a planet will orbit a common centre of mass of the star-planet system.
Part of the star’s motion will be along the line of sight to the observer. This radial velocity can be detected through a shift in the stellar absorption lines.
The first exoplanets detected this way were hot jupiters, which produce an RV signal of ~50 m/s.
The velocity semi-amplitude K is:
\( K = (\frac{2 \pi G}{P})^{1/3} \frac{M_2 sin(i)}{(m_1 + m_2)^{2/3} } \frac{1}{\sqrt{1-e^2}} \)
This method can measure:
- Orbital period
- Eccentricity
- Minimum planet mass ()

GIANT PLANETS

Exoplanet Demographics

Planet Fraction: # of stars with planets / total # of stars
Planet Occurrence: total # of planets / total # of stars

Planet Metallicity Correlation

Planets about the mass of Jupiter or larger orbiting FGK stars with periods less than 4 years are more likely to be found around higher metallicity stars.

Stellar Mass Correlation

Jupiter mass stars with periods less than 4 years are also more likely to be found orbiting high mass stars. Formation of giant planets

Core Accretion

Form a ~10 Earth mass core of solids
At 10 Earth masses, the core is strong enough to accrete gas from the protoplanetary disk
But gas in the protoplanetary disk only lasts a few million years.

Gravitational Instability

Gas giants are formed from overdensities in the protoplanetary disk.
Much more efficient than core accretion

Hot Jupiters

Jupiter mass stars orbiting close to the star
Likely migrated inward from the outer stellar system
Most Hot Jupiters are in tidally circularised

TRANSIT METHOD

Can be used to measure:

Planet size
Inclination
Period
Atmospheric spectra (for the best cases)

Impact Parameter

b=0: Centre of the planet passes through the centre of the star, w.r.t. the observer
b=1: Centre of the planet grazes the top/bottom of the star

Planet Density

By measuring the same planet with both the radial velocity and transit methods, we can obtain the density of the planet.
- Radial velocity gives us mass * sin(i)
- Transit method gives the inclination angle and planetary size

Transit Timing Variations

TTVs allow us to measure the masses of planets in multi-planet systems without RVs

TTVs are strongest for planets in resonance.

Habitable Zone

The range of distances from a star such that an Earth-like planet with an Earth-like atmosphere would have a surface temperature such that water can remain in liquid form.

Short Period Exoplanet Demographics

Kepler is more complete to habitable zones around M stars.
90% of stars are M stars and 25% of M stars have a planet in its habitable zone

Fulton Gap

Gap in the occurrence rate of planets at ~1.8 Earth radii.
Likely due to the photoevaporation of the planet’s atmosphere.