Arden stood up, tossed Channing's napkin at him, and started toward the door. Channing watched her go, his hand making motions on the tablecloth. His eyes fell to the table and he took Franks' pencil and drew a long curve from a spot of gravy on one side of the table to a touch of coffee stain on the other. The curve went through a bit of grape jelly near the first stain.
"Here goes the tablecloth strategist," said Franks. "What now, little man?"
"That spot of gravy," explained Don, "is Mars. The jelly is the Empress of Kolain. Coffee stain is Venus, and up here by this cigarette burn is Venus Equilateral. Get me?"
"Yop, that's clear enough."
"Now it would be the job for seventeen astronomers for nine weeks to predict the movement of the jelly spot with respect to the usual astral standards. But, fellows, we know the acceleration of the Empress of Kolain, and we know her position with respect to the orbit of Mars at the instant of take-off. We can correct for Mars' advance along her—or his—orbit. We can figure the position of the Empress of Kolain from her angular distance from Mars! That's the only thing we need know. We don't give a ten-dollar damn about her true position."
Channing began to write equations on the tablecloth. "You see, they aren't moving so fast with respect to us. The course is foreshortened as they are coming almost in line with Venus Equilateral, curving outward and away from the Sun. Her course, as we see it from the Station here, will be a long radius upward curve, slightly on the parabolic side. Like all long-range cruises, the Empress of Kolain will heist herself slightly above the plane of the ecliptic to avoid the swarm of meteors that follow about the Sun in the same plane as the planets, lifting the highest at the point of greatest velocity."
"I get it," said Franks. "We get the best beam controller we have to keep the planet on the cross hairs. We apply a spiral cam to advance the beam along the orbit. Right?"
"Right." Don sketched a conical section on the tablecloth and added dimensions. He checked his dimensions against the long string of equations, and nodded. "We'll drive this cockeyed-looking cam with an isochronic clock, and then squirt a beam out there. Thank the Lord for the way our beam transmitters work."
"You mean the effect of reflected waves," asked Chuck.
"Sure. They're like light—only they ain't. We're going to use a glorified meteor detector. We'll control the spread and dispersion so that we cover a healthy hundred miles or so, which will give us sufficient power, I believe. If not, we'll have to tighten the beam. At any rate, spreading from a point source to an object of a given dimension, the waves rebound as though the object were a plane mirror. That will give us a dispersion of twice the dimension of the Empress of Kolain's planar projection through this axis. Twelve hundred feet isn't much, but once we get her on the beam and have confirmation, we can forget the rebound. We'll have her pinned."