GRADES 9–12
NGSS: Motion and Stability:
Analyze data to support the claim that Newton’s second law of motion describes the mathematical relationship among the net force on a macroscopic object, its mass, and its acceleration.
Nowadays the positions of the planets are known very precisely. At the beginning of the Space Age, though, this was not the case. When the Mariner mission planners first approached astronomers asking for the positions of Venus and Mars, the answer
came back as the angles required to point a telescope at the planets given to an accuracy of about a hundredth of an arcsecond and distances in terms of the Astronomical Unit, which is the average distance from the Earth to the Sun. Since an arcsecond is one sixtieth of an
arcminute, which is one sixtieth of a degree, and since an object’s distance is irrelevant to looking at it through a telescope, this was plenty of accuracy for any astronomical need. After flying one hundred million miles through space, though, an error of one hundredth of
an arcsecond would put a spacecraft off course by almost five miles.
As bad as a fivemile miss is, that is not the biggest problem. The size of an Astronomical Unit, which formed the basis of all distance measurements in the Solar System, was not known to better than one part in ten thousand. Astronomers with their
telescopes could measure angles very precisely, but measuring the distance to any astronomical body beyond the Moon is very difficult. While an error of one part in ten thousand is quite small, at a distance of one hundred million miles the resulting error is
about ten thousand miles, which is more than the diameter of any of the inner planets. This simply was not good enough for the space mission planners.
Early in the Space Age, in 1964, scientists bounced radar waves off the planet Venus and timed how long it took for them to travel there and back again. This improved the accuracy of the measurement of the size of the Solar System by an order of magnitude, to about one part in
150,000. Measuring the trajectories of interplanetary spacecraft, knowing their speeds, and measuring the time required for radio transmissions to travel between them and the Earth has increased the accuracy further so that now we know the size of the Solar System
(and the distances to the planets) to about one part in fifty billion. For something one hundred million miles away, that is an uncertainty of about ten feet.
With modern highspeed computers, planners of space missions can solve the equations of motion that describe a spacecraft’s path through the Solar System directly. These equations account for the gravitational attraction on the spacecraft from the Earth, the Moon, the Sun, and any
other relevant astronomical bodies. Early in the Space Age, though, before powerful computers were developed, mission planners had to use approximations to allow them to solve the equations of motion.
The primary approximation that mission planners used is called the
“patched conic” approximation. To use the patched conic approximation, one assumes that when the spacecraft is influenced primarily by one astronomical body the effects of all other astronomical bodies can be neglected. This simplifies the problem to a “twobody problem” which has an elliptical orbit as its
solution. The ellipse is the “conic” of the “patched conic” approximation.
When the spacecraft passes from the influence of one astronomical body to another, its position and velocity at the point it makes the transition are used as the initial conditions for an elliptical orbit around the second astronomical body. This is the “patch” part of the “patched conic” approximation. This is not exact, obviously, but the
gravitational attraction of the second body as the spacecraft is in orbit around the first body pulls in one direction, the gravitational attraction of the first body as the spacecraft is in orbit around the second body pulls in the opposite direction, and the two effects counteract each other. The errors introduced by the approximation
are corrected by a midcourse correction (see above) after the spacecraft has moved beyond the transition point.
Sixty Years Ago in the Space Race:
February 21: The Soviet Union launched a singlestage sounding rocket carrying 3,340 pounds of scientific instruments to an altitude of 294 miles.
February 28: The American Department of Defense put the Air Force in charge of developing the nation's intercontinental ballistic missiles (ICBMs) and intermediaterange ballistic missiles (IRBMs).
