Deep Impact Launch Movies

Class 2- The Baratropic Law

Reading: Chapters 1 of both The New Solar Sytem and of Atmospheres (see syllabus for these required textbooks).

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The Baratropic Law or Hydrostatic Equilibrium

Downward gravitational force of a mass is F = m g = density x depth x area x g

Pressure = Force / area = density x depth x g

If a fluid is not compressible - e.g. water (to first approximation) - then pressure (force per unit area due to gravity) just increases linearly with depth:

this is Fig 1-1 of Goody & Walker.

What happens if the fluid is compressible? How do we predict the change in density with pressure of a gas?

This was done in the 19th century by chaps called Boyle and Charles - with Laws named after them.

Boyle's Law: Density is inversely proportional to Temperature (at constant Pressure)

Charles' Law: Density is proportional to Pressure (at constant Temperature)

Together these form the

Ideal Gas Law: P = r/m kT... where r = density and m = molecular mass of the gas (k=Boltzmann constant)

......... more about this in Homework 1. Getting back to atmospheres and pressure.....

The gravitational force due to the mass of the air column is balanced by the vertical pressure gradient.

When there are not vertical motions the vertical forces on a parcel of air are balanced. The small arrows on the right side of the shaded block indicate the pressure excerted on the air in the block by the air above and below it in the column. The heavy arrows represent the vertical forces: the downward weight of the air and the upward force due to the vertical pressure gradient. Note that the incremental pressure change dP is a negative quanitity, since the pressure decreases with height.

Downward force on air (per unit area) F_down= mass x acceleration = rdz x g

Upward force on air (per unit area) F_up= -dP

so dP = - rgdz = -(Pm/kT)g dz ..... here we used the Ideal Gas Law P = r/m kT rearranged as r = Pm/kT

Or dP/P = - (mg/kT) dz.

Integrating the hydrostatic equation yields (assuming that the atmosphere is isothermal, that the molecular weight of the atmosphere doesn’t change with altitude and that the dependence of gravity on altitude can be ignored - T, m and g are all constant with z) from the surface (z=0, P=Po) to height z with pressure P:

Where H = kT/mg is called the atmospheric scale height. Pressure declines exponentially with altitude.

The scale height of the Earth's atmosphere is about 8 km - the height of Everest - so that the pressure at the top of Everest is about 1/e=1/2.7~35% of the pressure at sea level.

With these same assumptions of T, m, g all constant, the air density will also vary with altitude with the same scale height as the pressure.

G&W Fig 1-2

In reality the temperature of the atmosphere is not constant. Nor is the molecular mass - or even g, strictly speaking.

G&W Fig 1-3

We can partly understand why planetary atmospheres are so varied - the scale height shows how atmospheres of different planets have different vertical extents. Compare Earth and Titan:

Earth: P_E =1bar which is not so different from Titan: P_T =1.5 bars

H = kT/mg so H is proportional to T/g. Does this make sense? Increase gravity and the atmosphere is more tightly confined to the planet (H is smaller). Increase T and the atmosphere spreads out - H is bigger.

H_T/H_E=(T_T/T_E)(g_E/g_T) ~(125 Kelvin / 250 Kelvin)(980 cm s^-2 / 135 cm s^-2 )~3.5

This means that the weak gravity of Titan results in a highly-extended atmosphere (compensating strongly for the lower temperatures at Titan than Earth).

90% of the action happens in the lowest scaleheight of the atmosphere.

Homework 1 involves exercises that deal with gravity (g and escape speed, see Class 1) as well as pressure.

In preparation for next week, remind yourself about the Cosmic Abundance of Elements:

Or... putting it another way - the Astronomer's Periodic Table...