The Frame Game

A familiar footstep outside my office, “C’mon in, Vinnie, the door’s open.”

“Hi, Sy, how ya doin’?”

“Can’t complain. Yourself?”

“Fine, fine. Hey, I been thinking about something you said while Al and us were talking about rockets and orbits and such. You remember that?”

“We’ve done that in quantity. What statement in particular?”

“It was about when you’re in the ISS, you still see like 88% of Earth’s gravity. But I seen video of those astronauts just floating around in the station. Seems to me those two don’t add up.”

“Hah! We’re talking physics of motion here. What’s the magic word?”

“You’re saying it’s frames? I thought black holes did that.”

“Black holes are an extreme example, but frame‑thinking is an essential tool in analyzing any kind of relative motion. Einstein’s famous ‘happy thought‘ about a man in a free‑falling elevator—”

“Whoa, why is that a happy thought? I been nervous about elevators ever since that time we got stuck in one.”

“At least it wasn’t falling, right? Point is, the elevator and whoever’s in it agree that Newton’s First Law of Motion is valid for everything they see in there.”

“Wait, which Law is that?”

“‘Things either don’t move or else they move at a steady pace along a straight line.’ Suppose you’re that guy—”

“I’d rather not.”

“… and the elevator is in a zero‑gravity field. You take something out of your pocket, put it the air in front of you and it stays there. You give it a tap and it floats away in a straight line. Any different behavior means that your entire frame — you, the elevator and anything else in there — is being accelerated by some force. Let’s take two possibilities. Case one, you and the elevator are resting on terra firma, tightly held by the force of gravity.”

“I like that one.”

“Case two, you and the elevator are way out in space, zero‑gravity again, but you’re in a rocket under 1-g acceleration. Einstein got happy because he realized that you’d feel the same either way. You’d have no mechanical way to distinguish between the two cases.”

“What’s that mean, mechanical?”

“It excludes sneaky ways of outside influence by magnetic fields and such. Anyhow, Einstein’s insight was key to extending Newton’s First Law to figuring acceleration for an entire frame. Like, for instance, an orbiting ISS.”

“Ah, you’re saying that floating astronauts in an 88% Earth-gravity field is fine because the ISS and the guys share the frame feeling that 88% but the guys are floating relative to that frame. But down here if we could look in there we’d see how both kinds of motion literally add up.”

“Exactly. It’s just much easier to think about only one kind at a time.”

“Wait. You said the ISS is being accelerated. I thought it’s going a steady 17500 miles an hour which it’s got to do to stay 250 miles up.”

“Is it going in a straight line?”

“Well, no, it’s going in a circle, mostly, except when it has to dodge some space junk.”

“So the First Law doesn’t apply. Acceleration is change in momentum, and the ISS momentum is constantly changing.”

“But it’s moving steady.”

“But not in a straight line. Momentum is a vector that points in a specific direction. Change the direction, you change the momentum. Newton’s Second Law links momentum change with force and acceleration. Any orbiting object undergoes angular acceleration.”

“Angular acceleration, that’s a new one. It’s degrees per second per second?”

“Yup, or radians. There’s two kinds, though — orbiting and spinning. The ISS doesn’t spin because it has to keep its solar panels facing the Sun.”

“But I’ve seen sci-fi movies set in something that spins to create artificial gravity. Like that 2001 Space Odyssey where the guy does his running exercise inside the ship.”

“Sure, and people have designed space stations that spin for the same reason. You’d have a cascade of frames — the station orbiting some planet, the station spinning, maybe even a ballerina inside doing pirouettes.”

“How do you calculate all that?”

“You don’t. You work with whichever frame is useful for what you’re trying to accomplish.”

“Makes my head spin.”

~~ Rich Olcott

Climbing Out Of A Well

Al can’t contain himself. “Wait, it’s gravity!”

Vinnie and I are puzzled. “Come again?”

“Sy, you were going on about how much speed a rocket has to shed on the way to some special orbit around Mars, like that’s a big challenge. But it’s not. The rocket’s fighting the Sun’s gravity all the way. That’s where the speed goes. The Earth’s gravity, too, a little bit early on, but mostly the Sun’s, right?”

“Good point, Al. Sun gravity’s what bends the rocket onto a curve instead of a straight line. Okay, Sy, you got a magic equation that accounts for the shed speed? Something’s gotta, ’cause we got satellites going around Mars.”

“Good point, Vinnie, and you’re right, there is an equation. It’s not magic, you’ve already seen it and it ties kinetic energy to gravitational potential energy.”

“Wait, if I remember right, kinetic energy goes like mass times velocity squared. How can you calculate that without knowing how big the rocket is?”

“Good question. We get around that by thinking things through for a unit mass, one kilogram in SI units. We can multiply by the rocket’s mass when we’re done, if we need to. The kinetic energy per unit mass, we call that specific kinetic energy, is just ½v². Look familiar?”

“That’s one side of your v²=2GM/R equation except you’ve got the 2 on the other side.”

“Good eye, Al. The right-hand side, except for the 2, is specific gravitational potential energy, again for unit mass. But we can’t use the equation unless we know the kinetic energy and gravitational potential are indeed equal. That’s true if you’re in orbit but we’re talking about traveling between orbits where you’re trading kinetic for potential or vice versa. One gains what the other loses so Al’s right on the money. Traveling out of a gravity well is all about losing speed.”

Al’s catching up. “So how fast you’re going determines how high you are, and how high you are says how fast you have to be going.”

Vinnie frowns a little. “I’m thinking back to in‑flight refueling ops where I’m coming up to the tanker from below and behind while the boom operator directs me in. That doesn’t sound like it’d work for joining up to a satellite.”

“Absolutely. If you’re above and behind you could speed up to meet the beast falling, or from below and ahead you could slow down to rise. Away from that diagonal you’d be out of luck. Weird, huh?”

“Yeah. Which reminds me, now we’re talking about this ‘deeper means faster‘ stuff. How does the deep‑dive maneuver work? You know, where they dive a spacecraft close to a planet or something and it shoots off with more speed than it started with. Seems to me whatever speed it gains it oughta give up on the way out of the well.”

“It’s a surprise play, alright, but it’s actually two different tricks. The slingshot trick is to dive close enough to capture a bit of the planet’s orbital momentum before you fly back out of the well. If you’re going in the planet’s direction you come out going faster than you went in.”

“Or you could dive in the other direction to slow yourself down, right?”

“Of course, Al. NASA used both options for the Voyager and Messenger missions. Vinnie, I know what you’re thinking and yes, theoretically stealing a planet’s orbital momentum could affect its motion but really, planets are huge and spacecraft are teeny. DART hit the Dimorphos moonlet head-on and slowed it down by 5%, but you’d need 66 trillion copies of Dimorphos to equal the mass of dinky little Mercury.”

“What’s the other trick?”

“Dive in like with the slingshot, but fire your rocket engine when you’re going fastest, just as the craft approaches its closest point to the planet. Another German rocketeer, Hermann Oberth, was the first to apply serious math to space navigation. This trick’s sometimes called the Oberth effect, though he didn’t call it that. He showed that rocket exhaust gets more effective the faster you’re going. The planet’s gravity helps you along on that, for free.”

“Free help is good.”

~~ Rich Olcott

Not Too Fast, Not Too Slow

“Vinnie, those nifty-looking transfer orbits that Hohmann invented but didn’t get to patent — you left something out.”

“What’s that, Sy?”

“The geometry looks lovely — a rocket takes off tangent to its orbit around one planet or something and inserts along a tangent to an orbit around something else. Very smooth and I can see how that routing avoids having to spend fuel to turn corners. But that ignores speeds.”

“What difference does that make?”

“It makes a difference whether or not you can get into the orbit you’re aiming for. Any orbit is a trade-off between gravity’s pull and the orbiter’s kinetic energy. Assuming you’re going for a circular orbit, there’s a strict relationship between your final height and your approach speed when you’re finally flying on the horizontal. You don’t want to come in too fast or too slow.”

“First thing I learned in pilot school. But that relationship’s an equation, ain’t it?”

“A couple, actually, but they’re simple. Let’s back into the problem. Say your mission is to put a communications relay satellite into lunastationary orbit around the Moon—”

“Lunastationary?”

“Like geostationary, but with the Moon. The satellite’s supposed to hover permanently above one spot on the Moon’s equator, so its orbital period has to equal the Moon’s ‘day,’ <pulling out Old Reliable, tapping> which is 27.322 days. Your satellite must loop around the Moon in exactly that much time. Either it’s scooting at low altitude or it’s ambling along further up. If we knew the speed we could find the radius, and if we knew the radius we could find the speed. We need some math.”

“I knew it. You’re gonna throw calculus at me.”

“Relax, Vinnie, it’s only algebra and we’re only going to combine two formulas and you already know one of them. The one you don’t know connects the speed, which I’m calling v, with the radius, R. They’re tied together by the Moon’s mass, M and Newton’s gravitational constant G. The formula is v2=2G×M/R. You can handle that, right?”

“Lessee … that says if I either double the mass or cut the distance by two, the speed has to be four times larger. Makes sense ’cause that’s about being in a deeper gravity well or getting closer in. Am I on track?”

“Absolutely. Next formula is the one you know, the circumference of a circle or in this case, the distance around that orbit.”

“That’s easy, 2πR.”

“And that’s also speed times the time, T so I’ll set those equal. <tapping on Old Reliable> Okay, the first formula says v2 so I square the circumference equation and solve that for v2 . You still with me?”

“You’re gonna set those two v-squareds equal, I suppose.”

“You’re still on track. Yup and then I gather the Rs on one side and everything else on the other. That gives me something in R3 but that’s okay. Plug in all the numbers, take the cube root and we get that you need to position that satellite 111 megameters out from the Moon’s center, flying at 296 meters per second. Think you can manage that?”

“Given the right equipment, sure. Seventy thousand miles out from the Moon … pretty far.”

“It’s about ¾ of the way to the Moon from Earth.”

“Cool. Does that R3 formula work for the planets?”

“Sure. Works for the Sun, too, but that’s so massive and spins so fast the sol‑stationary orbit’s half way to Mercury. An orbiter would have to fly 205 000 miles an hour to keep up with an equatorial sunspot. Flying something‑stationary over other planets offers problems beyond targeting the orbit, though.”

“Besides how long the trip would be?”

“Well, that, yes, but here’s another one. Suppose you’re going to Mars, aiming at an ares‑stationary orbit. It’ll be 20 megameters, 12500 miles from the center. You need to make your tangential injection at a Mars‑relative speed of 1439 meters per second. Problem is, you left Earth from a geostationary orbit at 3075 m/s relative to Earth. At the classic Hohmann positions, Earth’s going 5710 m/s relative to Mars, Somehow you’re going to have to shed 7346 m/s per second of excess speed.”

~~ Rich Olcott

Carefully Considered Indirection

“C’mon, Vinnie, you’re definitely doing Sy stuff. I ask you a question about how come rockets can get to the Moon easier than partway and you go round the barn with ballistics and cruisers. Stop dodging.”

“Now, Al, Vinnie’s just giving you background, right. Vinnie?”

“Right, Sy, though I gotta admit a lot of our talks have gone that way. So what’s your answer?”

“Nice try, Vinnie. You’re doing fine, so keep at it.”

“Okay. <deep breath> It has to do with vectors, Al, combination of amount and direction, like if you’re going 3 miles north that’s a vector. You good with that?”

“If you say so.”

“I do. Then you can combine vectors, like if you’re going 3 miles north and at the same time 4 miles east you’ve gone 5 miles northeast.”

“That’s a 3-4-5 right triangle, even I know that one. But that 5 miles northeast is a vector, too, right?”

“You got the idea. Now think about fueling a rocket going up to meet the ISS.”

“Sy said it’s 250 miles up, so we need enough fuel to punch that far against Earth’s gravity.”

“Not even close. If the rocket just went straight up, it’d come straight down again. You need some sideways momentum, enough so when you fall you miss the Earth.”

“Miss the Earth? Get outta here!”

“No, really. Hey, Sy, you tell him.”

“Vinnie’s right, Al. That insight goes back to Newton. He proposed a thought experiment about building a powerful cannon to fire horizontally from a very tall mountain. <sketching on paper napkin> A ball shot with a normal load of powder might hit the adjacent valley. Shoot with more and more powder, balls would fly farther and farther before hitting the Earth. Eventually you fire with a charge so powerful the ball flies far enough that its fall continues all around the planet. Unless the cannon blows up or the ball shatters.”

“That’s my point, Al. See, Newton’s cannon balls started out going flat, not up. To get up and into orbit you need up and sideways velocity, like on the diagonal. You gotta calculate fuel to do both at the same time.”

“So what’s that got to do with easier to get to the Moon than into orbit?”

“‘S got everything to do with that. Not easier, though, just if you aim right the vectors make it simpler and cheaper to carry cargo to the Moon than into Earth orbit.”

“So you just head straight for the Moon without going into orbit!”

“Not quite that simple, but you got the general idea. Remember when I brought that kid’s top in here and me and Sy talked about centrifugal force?”

“Do I? And I made you clean up all those spit-wads.”

“Yeah, well, suppose that cannon’s at the Equator <adding dotted lines to Sy’s diagram> and aimed with the Earth’s spin and suppose we load in enough powder for the ball to go straight horizontal, which is what it’d do with just centrifugal force.”

“If I’m standing by the cannon it’d look like the ball’s going sideways.”

“Yup. Basically, you get the up‑ness for free. We’re not talking about escape velocity here, that’s different. We’re talking about the start of a Hohmann orbit.”

“Who’s Hohmann?”

“German engineer. When he was a kid he read sci‑fi like the rest of us and that got him into the amateur rocketry scene. Got to be a leader in the German amateur rocket club, published a couple of leading‑edge rocket science books in the 1920s but dropped out of the field when the Nazis started rolling and he figured they’d build rocket weapons. Anyhow, he invented this orbit that starts off tangent to a circle around one planet or something, follows an ellipse to end tangent to a circle around something else. Smooth transitions at both ends, cheapest way you can get from here to there. Kinks in the routing cost you fuel and cargo capacity to turn. Guy shoulda patented it.”

“Wait, an orbit’s a mathematical abstraction, not a thing.”

“Patent Office says it’s a business method, Sy. Check out PAS-22, for example.”

“Incredible.”

~ Rich Olcott

  • Thanks to Ken, who asked another question.

The Cold Equation

Afternoon break time. I’m enjoying one of Al’s strawberry scones when he plops one of his astronomy magazines on Vinnie’s table. “Vinnie, you bein’ a pilot and all, could you ‘splain some numbers which I don’t understand? It’s this statistics table for super‑heavy lifter rockets. I think it says that some of them can carry more cargo to the Moon than if they only go partway there. That’s nuts, right?”

Vehicle Payload to LEO GSO Payload TLI Payload
Energia 100 20 32
Falcon Heavy 64 27 28
NASA’s SLS 1b 105 42
SpaceX Starship 100
Yenisei 103 26 28
Yenisei Don 140 30 33
LEO=Low Earth Orbit, GSO = Geosynchronous Orbit, TLI=Trans Lunar Injection
Payloads in metric tons (megagrams)

“Lemme think … LEO is anywhere up to about 2000 kilometers. GSO is about 36000 kilometers out, so it makes sense that with the same amount of fuel and stuff you can’t lift as much out there. TLI … that’s not to the Moon, that’s to a point where you can switch from orbiting the Earth to orbiting the Moon so, yeah, that’s gonna be way farther out, like a couple hundred thousand kilometers or more depending.”

“Depending on what?”

“Oh, lots of things — fuel, orbit, design philosophy—”

“Now wait, you been taking Sy lessons. Philosophy?”

“No, really. There’s two basic ways to do space travel, either you’re ballistic or you’re cruising. All the spacecraft blast‑offs you’ve seen are ballistic. Use up most of your fuel to get a good running start and then basically coast the rest of the way to your target. Ballistic means you gotta aim careful from the get‑go. That’s the difference between ballistic and cruise missiles. Cruisers keep burning fuel and accelerating. That lets ’em change directions whenever.”

“Cruisers are better, right, so you can point at different asteroids? I read about that weird orbit they had to send the Lucy mission on.”

“Actually, Lucy used the ballistic‑and‑coast model. NASA spent a bucketful of computer time calculating exactly where to point and when to lift off so Lucy could visit all those asteroids.”

“Why not just use a cruise strategy and skip around?”

“Cruisers are just fine once you’re up between planets. NASA’s Dawn mission to the Vesta and Ceres asteroids used a cruise drive — but only after the craft rode a boostered Delta‑II ballistic up to low Earth orbit. Nine boosters worth of ballistic. The problem is you’re caught in a double bind. You need to burn fuel to get the payload off the planet, but you need to burn fuel to get the fuel off, too. ‘S called diminishing returns. Hey, Sy, what’s that guy’s name?”

“Which guy?”

“The rocket equation guy, the Russian.”

“Ah. Tsiolkovsky. Lived in a log cabin but wrote a lot about space travel. Everything from rocket theory to airlocks and space stations. What about him?”

“I’m tellin’ Al about rockets. Tsiol… That guy’s equation says if you know how much you need to change velocity and you know your payload mass, you can figure how much fuel you need to burn to do that.”

“With some conditions, Vinnie. There’s a multiplier in there you have to calibrate for fuel, engine design. even whether you’re traveling through water or vacuum or different atmospheres. Then, the equation doesn’t figure in gravity. Oh, and it only works with straight‑line velocity change. If you want to change direction you need to use calculus to figure the—”

“Hey, I just realized why they use boosters!”

“Why’s that, Al?”

“The gravity thing. Gravity’s strongest near the Earth, right? Once the beast gets high enough, you’re not fighting as much gravity. You don’t need the extra power.”

“True, but that’s not the whole picture. The ISS orbit’s about 250 miles up, which puts it about 4250 miles from the planet’s center. Newton’s Law of Gravity says the field all the way up there is still about 88% of what’s at the surface. The real reason is that a booster’s basically a fuel tank. Once you’ve burned the fuel you don’t need the tank and that’s a lot of weight to carry for nothing.”

“Right, tank and engine don’t count as payload so dump ’em.”

“Seems cold‑hearted, though.”

~~ Rich Olcott

DARTing to A Conclusion

The park’s trees are in brilliant Fall colors, the geese in the lake dabble about as I walk past but then, “Hey Moire, I gotta question!”

“Good morning, Mr Feder. What can I do for you?”

“NASA’s DART mission to crash into Diddy’s mos’ asteroid—”

“The asteroid’s name is Didymos, Mr Feder, and DART was programmed to crash into its moon Dimorphos, not into the asteroid itself.”

“Whatever. How’d they know it was gonna hit the sunny side so we could see it? If it hits in the dark, nobody knows what happened. They sent that rocket up nearly a year ago, right? How’d they time that launch just right? Besides, I thought we had Newton’s Laws of Motion and Gravity to figure orbits and forces. Why this big‑dollar experiment to see if a rocket shot would move the thing? Will it hit us?”

“You’re in good form today, Mr Feder.” <unholstering Old Reliable> “Let me pull some facts for you. Ah, Didymos’ distance from the Sun ranges between 1.01 and 2.27 astronomical units. Earth’s at 1.00 AU or 93 million miles, which means that the asteroid’s orbit is 930 000 miles farther out than ours, four times our distance to the Moon. That’s just orbits; Earth is practically always somewhere else than directly under Didymos’ point of closest approach. Mm… also, DART flew outward from Earth’s orbit so if the impact has any effect on the Didymos‑Dimorphos system it’ll be to push things even farther away from the Sun and us. No, I’m not scared, are you?”

“Who me? I’m from Jersey; scare is normal so we just shrug it off. So why the experiment? Newton’s not good enough?”

“Newton’s just fine, but collisions are more complicated than people think. Well, people who’ve never played pool.”

“That’s our national sport in Jersey.”

“Oh, right, so you already know about one variable we can’t be sure of. When the incoming vector doesn’t go through the target’s center of mass it exerts torque on the target.”

“We call that ‘puttin’ English on it.'”

“Same thing. If the collision is off‑center some of the incoming projectile’s linear momentum becomes angular momentum in the target object. On a pool table a simple Newtonian model can’t account for frictional torque between spinning balls and the table. The balls don’t go where the model predicts. There’s negligible friction in space, you know, but spin from an off‑center impact would still waste linear momentum and reduce the effect of DART’s impact. But there’s another, bigger variable that we didn’t think much about before we actually touched down on a couple of asteroids.”

“And that is…?”

“Texture. We’re used to thinking of an asteroid as just a solid lump of rock. It was a surprise when Ryugu and Bennu turned out to be loose collections of rocks, pebbles and dust all held together by stickiness and not much gravity. You hit that and surface things just scatter. There’s little effect on the rest of the mass. Until we do the experiment on a particular object we just don’t know whether we’d be able to steer it away from an Earth‑bound orbit.”

“Okay, but what about the sunny‑side thing?”

“Time for more facts.” <tapping on Old Reliable> “Basically, you’re asking what are the odds the moonlet is in eclipse when DART arrives on the scene. Suppose its orbit is in the plane of the ecliptic. Says here Dimorphos’ orbital radius is 1190 meters, which means its orbit is basically a circle 3740 meters long. The thing is approximately a cylinder 200 meters long and 150 meters in diameter. Say the cylinder is pointed along the direction of travel. It occupies (200m/3740m)=5% of its orbit, so there’s a 5% chance it’s dark, 95% chance it’s sunlit.”

“Not a bad bet.”

“The real odds are even better. The asteroid casts a shadow about 800 meters across. Says here the orbital plane is inclined 169° to the ecliptic so the moonlet cycles up and down. At that tilt and 1190 meters from Didymos, 200‑meter Dimorphos dodges the shadow almost completely. No eclipses. DART’s mission ends in sunlight.”

~~ Rich Olcott

  • Thanks to my brother Ken, who asked the question but more nicely.

Why I Never Know What Time It Is

It’s always fun watching Richard Feder (of Fort Lee, NJ) as he puts two and two together. He gets a gleam in his eye and one corner of his mouth twitches. On a good day with the wind behind him I’ve seen his total get as high as 6½. “I wanna get back to that ‘everybody has their own time‘ monkey‑business where if you’re moving fast your clock slows down. What about the stardates on Star Trek? Those guys go zooming through space at all different angles and speeds. How do they keep their calendars in synch?”

Trekkie and Astronomy fan Al takes the bait. “Artistic license, Mr Feder. The writers can make anything happen, subject to budgets and producer approval. The first Star Trek series, they just used random four‑digit numbers for stardates. That was OK because the network aired the episodes in random order anyway so no‑one cared about story arc continuity. Things were more formal on Captain Picard’s Enterprise, as you’d expect — five‑digit stardates, first digit always ‘4‘ for 24th Century, thousands digit was ‘1‘ for season one, ‘2‘ for season two and so on. Working up the other way, the digit right of the decimal point was tenths of a standard day, the units place counted days within an episode and the tens and hundreds they just picked random numbers.”

“I suppose that’s what they did, but how could they make it work? You guys yammer on about time dilation. Say a ship’s running at Warp Whoop‑de‑doo, relativity should slow its calendar to a crawl. You couldn’t get a whole fleet into battle position when some of the ships had to get started years ahead of time. And that’s just the dilation slow-down, travel time’s on top of that.”

“Travel time measured how, Mr Feder, and from where?”

“Well, there you go, Cathleen, that’s what I’m talking about!”

“You know that Arthur C Clarke quote, ‘Any sufficiently advanced technology is indistinguishable from magic‘? The Enterprise crew’s always communicating with ‘sub‑space radio’, which sure looks like magic to me. They could send sync pulses through there along with chatter. When you drop out of warp space, your clocks catch the pulses and sync up, I suppose.”

“There’s a deeper issue than that, guys.”

“What’s that, Sy?”

“You’re talking like universal time is a thing, which it isn’t. Hasn’t been since Einstein’s Special Relativity used Minkowski’s math to stir space and time together. General Relativity scrambles things even worse, especially close to a strong gravity center. You remember about gravity forcing spacetime to curve, right? The curvature inside a black hole’s event horizon gets so tight that time rotates toward the geometric center. No, I can’t imagine what that looks like, either. The net of it, though, is that a black hole is a funnel into its personal future. Nothing that happens inside one horizon can affect anything inside another one so different holes could even have different time rates. We’ve got something like 25000 or more stellar black holes scattered through the Milky Way, plus that big one in the center, and that’s just one galaxy out of billions. Lots of independent futures out there.”

“What about the past, Sy? I’d think the Big Bang would provide a firm zero for time going forward and it’s been one second per second since then.”

“Nup. Black holes are an extreme case. Any mass slows down time in its vicinity, the closer the slower. That multi‑galaxy gravitational lens that lets us see Earendel? It works because the parts of Earth‑bound light waves closest to the center of mass see more time dilation than the parts farther away and that bends the beam toward our line of sight.”

“Hey, that reminds me of prisms bending light waves.”

“Similar effect, Vinnie, but the geometry’s different. Prisms and conventional lenses change light paths abruptly at their surfaces. Gravitational lenses bend light incrementally along the entire path. Anyhow, time briefly hits light’s brakes wherever it’s near a galaxy cluster, galaxy or anything.”

“So a ship’s clock can fidget depending on what gravity it’s seen recently?”

“Mm-hm. Time does ripples on its ripples. ‘Universal Time‘ is an egregious example of terminology overreach.”

~~ Rich Olcott

The Red Advantage

“OK, Cathleen, I get that JWST and Hubble rate about the same for sorting out things that are close together in the sky, and I get that they look at different kinds of light so it’s hard to compare sensitivity. Let’s get down to brass tacks. Which one can see farther?”

“An excellent question, Mr Feder. I’ve spent an entire class period on different aspects of it.”

“Narrow it down a little, I ain’t got all day.”

“You asked for it — a quick course on cosmological redshift. Fasten your seat belt. You know what redshift is, right?”

“Yeah, Moire yammers on about it a lot. Waves stretch out from something moving away from you.”

I bristle. “It’s important! And some redshifts don’t have anything to do with motion.”

“Right, Sy. Redshift in general has been a crucial tool for studying everything from planetary motion to the large‑scale structure of the Universe. Your no‑motion redshift — you’re thinking of gravitational redshift, right?”

“Mm-hm. From a distance, space appears to be compressed near a massive object, less compressed further away. Suppose we send a robot to take up a position just outside a black hole’s event horizon. The robot uses a green laser to send us its observations. Space dilates along the beam’s path out of the gravity well. The expanding geometry stretches the signal’s wavelength into the red range even though the robot’s distance from us is constant.”

“So, that’s gravitational redshift and there’s the Doppler redshift that Mr Feder referred to—”

“Is that what its name is? With p‘s? I always heard it as ‘doubler’ effect and wondered where that came from.”

“It came from Christian Doppler’s name, Al. Back in the 1840s he was investigating a star. He noticed that its spectrum was the overlap of two spectra slightly shifted with respect to each other. Using wave theory he proposed that the star was a binary and that the shifted spectra arose from one star coming towards us and the other moving away. Later work confirmed his ideas and the rest is history. So it’s Doppler, not doubler, even though the initial observation was of a stellar doublet.”

“So what’s this cosmo thing?”

“Cosmological redshift. It shows up at large distances. On the average, all galaxies are moving away from us, but they’re moving away from each other, too. That was Hubble’s big discovery. Well, one of them..”

“Wait, how can that be? If I move away from Al, here, I’m moving toward Sy or somebody.”

“We call it the expansion of the universe. Have you ever made raisin bread?”

“Nah, I just eat it.”

“Ok, then, just visualize how it’s made. You start with a flat lump of dough, raisins close together, right? The loaf rises as the yeast generates gas inside the lump. The dough expands and the raisins get further apart, all of them. There’s no pushing away from a center, it’s just that there’s an increasing amount of bubbly dough between each pair of neighboring raisins. That’s a pretty good analogy to galactic motion — the space between galaxies is expanding. The general motion is called Hubble flow.”

“So we see their light as redshifted because of their speed away from us.”

“That’s part of it, Al, but there’s also wave‑stretching because space itself is expanding. Suppose some far‑away galaxy, flying away at 30% of lightspeed, sent out a green photon with a 500‑nanometer wavelength. If the Doppler effect were the only one in play, our relative speeds would shift our measurement of that photon out to about 550 nanometers, into the yellow. Space expansion at intermediate stations along its path can cumulatively dilate the wave by further factors out into the infrared or beyond. Comparing two galaxies, photons from the farther one will traverse a longer path through expanding space and therefor experience greater elongation. Hubble spotted one object near its long‑wavelength limit with a recognizable spectrum feature beyond redshift factor 11.”

“Hey, that’s the answer to Mr Feder’s question!”

“So what’s the answer, smart guy?”

JWST will be able to see farther, because its infrared sensors can pick up distant light that’s been stretched beyond what Hubble can handle.”

~~ Rich Olcott

Lord Rayleigh Resolves

Mr Feder just doesn’t quit. “But why did they make JWST so big? We’re getting perfectly good pictures from Hubble and it’s what, a third the size?”

Al’s brought over a fresh pot and he’s refilling our coffee mugs. “Chalk it up to good old ‘because we can.’ Rockets are bigger than in Hubble‘s day, robots can do more remote stuff by themselves, it all lets us make a bigger scope.”

Cathleen smiles. “There’s more to it than that, Al. It’s really about catching photons. You’re nearly correct, Mr Feder, the diameter ratio is 2.7. But photons aren’t captured by a line across the primary mirror, they’re captured by the mirror’s entire area. The important JSWT:Hubble ratio is between their areas. JWST beats Hubble there by a factor of 7.3. For a given source and the same time interval, we’d expect JWST to be that much more sensitive than Hubble.”

“Well,” I break in, “except that the two use photon detectors that are sensitive to different energy ranges. The two scopes often won’t even be looking at the same kinds of object. Hubble‘s specialized for visible and UV light. It’s easy to design detectors for that range because electrons in solid‑state devices respond readily to the high‑energy photons. The infrared light photons that JWST‘s designed for don’t have enough energy to kick electrons around the same way. Not really a fair comparison, although everything I’ve read says that JWST‘s sensitivity will be way up there.”

Mr Feder is derisive. “‘Way up there.’ Har, har, de-har. I suppose you’re proud of that.”

“Not really, it just happened. But Cathleen, I’m surprised that you as an astronomer didn’t bring up the other reason the designers went big for JWST.”

“True, but it’s more technical. You’re thinking of resolution and Rayleigh’s diffraction limit, aren’t you?”

“Bingo. Except Rayleigh derived that limit from the Airy disk.”

“Disks in the air? We got UFOs now? What’re you guys talking about?”

Portrait of Sir George Airy
licensed under the Creative Commons
Attribution 4.0 International license.

“No UFOs, Mr Feder, I’ll try to be non‑technical. Except for the big close objects like the Sun and its planets, telescopes show heavenly bodies as circular disks accompanied by faint rings. In the early 1800s an astronomer named George Airy proved that the patterns are an illusion produced by the telescope. His math showed that even the best possible apparatus will force lightwaves from any small distant light source to converge to a ringed circular disk, not a point. The disk’s size depends on the ratio between the light’s wavelength and the diameter of the telescope’s light‑gathering aperture. How am I doing, Al?”

“Fine so far.”

“Good. Rayleigh took that one step further. Suppose you’re looking at two stars that are very close together in the sky. You’d expect to see two Airy patterns. However, if the innermost ring from one star overlaps the other star’s disk, you can’t resolve the two images. That’s the basis for Rayleigh’s resolvability criterion — the angle between the star images, measured in arc‑seconds, has to be at least 252000 times the wavelength divided by the diameter.”

After a diagram by cmglee
licensed under the Creative Commons
Attribution 3.0 International license.

“But blue light’s got a shorter wavelength than red light. Doesn’t that say that my scope can resolve close-together blue stars better than red stars?”

“Sure does, except stars don’t emit just one color. In visible light the disk and rings are all rimmed with reddish and bluish fuzz. The principle works just fine when you’re looking at a single wavelength. That gets me to the answer to Mr Feder’s question. It’s buried in this really elegant diagram I just happen to have on my laptop. Going across we’ve got the theoretical minimum angle for resolving two stars. Going up we’ve got aperture diameters, running from the pupil of your eye up to radio telescope coalitions that span continents. The colored diagonal bands are different parts of the electromagnetic spectrum. The red bars mark each scope’s sensor wavelength range. Turns out JWST‘s size compared to Hubble almost exactly compensates for the longer wavelengths it reports on.”

~~ Rich Olcott