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Lesson 3: Presentation Materials February 16, 2010

Posted by drspaceshow in Uncategorized.
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Space Show Classroom Lesson 3: Orbital and Flight Dynamics. Tuesday, February 16, 2010.  

 I.  Presentation Material From Guest Panelist: Dan Adamo, Retired NASA Mission Control Flight Dynamics Officer.

Here are several precepts applicable to managing flight dynamics near relatively massive objects like the Earth and Moon.  They are collectively referred to as “ARRGH” (or Adamo’s Rules of the Road for Gravitational Harmony) because any student of flight dynamics in space will sooner or later find them frustrating.

Departure from these precepts will likely consume additional propellant.  Any such “gas guzzling” must therefore be justified as contributing to achieved mission objectives, including mission assurance requirements such as crew safety.  To illustrate propulsive consequences associated with ARRGH precepts, each is accompanied by a “LEOrot” value.  A LEOrot is a rule-of-thumb valid in a near-circular low Earth orbit (or LEO) context.  The LEO for which LEOrot values are computed is at approximately 200 km altitude, where geocentric inertial speed is near 7800 m/s.

ARRGH #1)  Do not raise or lower both ends (or apses) of a circular or elliptical orbit if only one end (or apsis) needs to be raised or lowered.  Likewise, do not raise AND lower the same apsis of an orbit.  Applications include orbit insertion following launch, rendezvous, and deorbit.  LEOrot #1: to change an apsis by 1 km requires a change in speed of 0.30 m/s at the other apsis.  This type of maneuver is known as a Hohmann transfer.

ARRGH #2)  To efficiently achieve escape or capture, apply thrust at the lowest possible altitudes where speed is greatest.  Align thrust with velocity as closely as possible during escape or capture.  Applications include interplanetary departure or arrival.  LEOrot #2: to achieve minimal Earth escape requires a speed increase of 3200 m/s.

ARRGH #3)  To efficiently rotate a trajectory plane, do so when speed is minimal.  This may not be possible if one trajectory plane is to be brought into coincidence with another because thrust must be applied at “the line of nodes”, where the two planes intersect.  Applications include launch window definition and rendezvous.  LEOrot #3: to rotate an orbit plane by 1 deg requires a change in velocity of 140 m/s.

II.  Presentation Material from Co-host Dr. John Jurist:

 Effects of Earth’s Spin on Suborbital Point to Point Transportation

The figure below shows the minimum Delta-V needed to go a specified distance around the Earth in a ballistic trajectory.  The horizontal axis shows the range traveled in degrees (out of 360) and the vertical axis shows the Delta-V expressed as a percentage of surface circular velocity (about 7,900 meters per second).  Atmospheric drag is ignored and a spherical Earth and impulsive boost are assumed in order to simplify the analysis.

 

For a fixed Delta-V and range, there are two ballistic trajectories that can attain lesser ranges.  This is easily understood with a simple thought experiment involving a water hose and nozzle. Water comes out of the nozzle at some speed.  If the hose is held to direct the water stream upwards at about 45 degrees, the water hits the ground at a maximum range.  At lesser ranges, there are two angles for the water to hit the desired spot.  One angle is more than 45 degrees and the other is less than 45 degrees.  If the water pressure is reduced to slow its ejection speed from the hose, the lesser range can be attained with the water stream directed upwards at 45 degrees.  The two angles converge to a single angle as water speed is reduced to the minimum to hit that range.  The same holds true for the ranges and Delta-V’s shown in the graph except the optimum angle varies with range because the Earth is spherical.

To get half way around the Earth, the required minimum Delta-V is circular velocity.  To get about ¼ of the way around the Earth, about 90 percent of circular speed is needed.  Therefore, for ranges of more than approximately 6,000 miles, 90 percent of orbital velocity is required.  The Earth’s spin adds some speed to eastward launches.  At 45 degrees latitude (Billings, Montana), that eastward boost is about 330 meters per second.  For westward launches from that latitude, a similar amount needs to be added to the Delta-V shown in the graph.  In order to go half way around the Earth launching to the east, about 7,600 meters per second is required, but to go the same distance to the west requires about 8,200 meters per second.  This difference implies that going more than about 94 degrees to the west in a ballistic trajectory requires as much Delta-V as launching to the east into a circular orbit.  In terms of miles, 94 degrees corresponds to about 6,500 miles.

These simplified approximations ignore our pesky atmosphere and some ogeometric issues but still illustrate two points:

  • Surface suborbital point to point ballistic transportation over ranges that are significant in terms of the Earth’s circumference require Delta-V’s closely approaching orbital speeds.

 

  • If one wants to go west about 6,500 miles or so in a suborbital ballistic trajectory, one may as well go east in a fractional orbital trajectory (not counting de-orbit Delta-V).  If one wants to go further west but not half way around, it is easier to go east.

These two points have profound implications for people hoping to evolve suborbital up and down launches to 100 kilometers or so into point to point suborbital transportation.  The impulsive Delta-V without atmosphere for an up and down trip to 100 kilometers is roughly 1,400 meters per second compared to surface circular orbital velocity of about 7,900 meters per second.  The suborbital point to point transport Delta-V is much closer to orbital Delta-V than it is to the up and down flights – especially to the west as discussed above.

Except for affecting prevailing wind direction, the Earth’s rotation does not have an appreciable role in the difficulty of flying aircraft substantial distances around our planet.

I suspect that commercial suborbital point to point transportation will evolve downward from orbital launch capability rather than evolve up from commercial vertical launches to 100 kilometers or so.  I also suspect that suborbital point to point transport to the west will actually result from eastward launches – at least in the early stages.  If commercial space transport vehicle Delta-V capability is much larger than that required for launch to LEO, then the issues described above become less relevant.  Until then, Los Angeles to Hong Kong suborbitally, anyone?

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