Wednesday, September 28, 2011

UARS Reentry Location

The NASA UARS assessment says that the reentry location is 14.1°S, 170.2°W.

UARS Reentry reports that UARS reentered the earth's atmosphere at 10.65°S, 168.0°W.

Both locations are near the American Samoa.

The red line marks NASA's limits for the location of the debris field.

Tuesday, September 27, 2011

Lift and Drag for Extremely High Altitudes

Using a simplified model we can derive the formulas for lift and drag for a plate at high altitudes (above that of the mesosphere). The initial assumptions are that the particles present act independently, that in the reference frame of the plate the angle of incidence equals the angle of reflection for a particle bouncing off the plate and that its speed before and after are the same in this frame. The angle of the plate to the direction of motion is α. The direction of the particle after the collision in the frame of the plate is "e".

The initial and final conditions for the collision are as follows from which we can find the initial and final total momentum and kinetic energies.

Equating the momentum and kinetic energy before and after the collision we can determine the work done as the plate moves and the force acting on it.

One can see here that the values are reduced for particles deflected sideways from the those for a plate perpendicular to the direction of its motion. Completing the derivation we find the formulas for the lift and drag. CL and CD are the lift and drag coefficients.

The values for the coefficients are larger than normally encountered. At extremely high altitude the gases present have large separations between them so they have no viscosity and no cohesion. Intermolecular forces do not play a role.

For a more accurate calculation one might try scattering theory. For orbital speeds we can't assume that all collisions are perfectly elastic.

Edit (27 Sept): Changed the subscript of the resultant coefficient from N to R. The resultant is normal to the plate in its frame of reference. I was influenced by the some history in choosing the symbol for it (see Anderson's A History of Aerodynamics pages 104 & 169).

Thursday, September 22, 2011

Recommended Readings

Space Boundary
Kármán line
Stone skipping
THAAD (YouTube video)

Drag Force For Non-interacting Particles

For an extremely rarefied atmosphere we can assume that its particles do not interact with each other and that the collisions with the moving spacecraft are independent. To simplify the problem we represent the spacecraft by a flat plate of mass M perpendicular to the direction of motion and moving with a speed V. Let an individual particle of mass m be initially at rest. From the point of view of the plate the particle initially moves toward it with speed V and after the collision it will move away from it with speed V if no energy is lost in the collision and the mass of the particle is much less than the mass of the plate. To get the view from the perspective of the surrounding atmosphere we have to add the speed of the plate to these values for the speeds of the particle and we get v = 0 before the collision and v = 2V after. If n is the number density of the atmosphere the work, ΔW, done by the plate on N particles as it moves through a distance Δx and the drag force, D, are determined as follows.

The pressure, D/A, is known as ram pressure. If the particles are sent off sideways by the collision the drag force and the pressure are reduced. The corrective factor depends on the geometry an object. If a spacecraft with a flat forward surface is not perpendicular to the direction of motion but at an angle the resulting lateral force can produce lift and cause the spacecraft to skip off the atmosphere.

Wednesday, September 21, 2011

Atmospheric Reentry And Its Effects

At what altitude does the density of the atmosphere become significant and contribute to reentry? To answer this question we can look at atmospheric models such as the US Standard Atmosphere or the International Standard Atmosphere. NRLMSISE-00 models the upper atmosphere and indicates that its density at 100 km about 10-10 g/cm3 decreasing rapidly with increasing altitude.

The ideal gas law tells us that pressure, temperature and density are related by the formula P=ρRT/M. The number density, number of molecules per unit volume, is n=ρ/m where m is the mass of the molecular component. Since the mass of a molecule is quite small the number of particles can still be quite large. This accounts for the atmospheric drag on a satellite. There is still some drag at higher altitudes.

When a meteor or satellite enters the atmosphere at high speeds a shock wave forms in front of it compressing the atmosphere. As the atmosphere is compressed it heats up and becomes luminous. Energy is released in the form of light by the Stefan-Boltzmann law. The pressure from the shock wave acting on the satellite slows it down and drops it out of orbit. Given enough time the satellite will reach a terminal velocity in the atmosphere which is much lower than its orbital velocity.

Meteors and spacecraft can skip off the atmosphere. During meteor showers the bolides pass through the upper atmosphere and can skip off it.

Tuesday, September 20, 2011

UARS Is Definitely Descending From Orbit

UARS is predicted to make its reentry sometime this weekend possibly late Friday evening or early Saturday morning depending on one's time zone. I have been studying the changes in the satellite's orbital elements and did a polynomial fit to the UARS's perigee as a function of the UTC day of the year. The plot below is the "altitude" as defined by the difference between UARS's perigee and the Earth's equatorial radius. The eccentricity is also decreasing which means that the orbit is becoming more circular.

It is predicted that UARS will reenter off the east coast of Africa but the exact location and time are somewhat uncertain. UARS probably has its greatest interaction with the Earth's upper atmosphere near the Equator because of the equatorial bulge and greater height of the atmosphere there.

If you have ever seen a meteoric bolide they are truly impressive. UARS should be heated to luminescence on reentry. As it passes overhead at night UARS is not alway visible since it is usually in the Earth's shadow but one may be able to capture an IR image if it is heated enough by atmospheric friction.