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(#137) The Brutal Math of Mars Trajectories · Mechanical Engineering Made Simple

0:00
Welcome back design junkies from Mechanical Engineering Made
0:02
Simple. If the podcast brings you value,
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help support the show with a like a follow or leave a comment
0:07
below. So today we are taking on a
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massive deep dive. I mean we are literally going to
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space. Oh yeah, we're tackling the
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dynamics of rockets and space vehicles.
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Yeah, and it's not going to be a gentle ride.
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No, definitely not. Imagine you're standing your
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launchpad right. You're looking up at millions of
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pounds of highly explosive fuel and it's all contained in this
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incredibly thin metal cylinder. Right, some parts are literally
0:34
no thicker than a dime. Exactly, and the countdown hit
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0, the clamps release, and for a split second the only thing
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keeping that whole skyscraper of fire from tipping over and, you
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know, vaporizing the pad is this invisible mathematical point.
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Center of thrust. Right.
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Yeah. So today's class, so to speak,
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is going to walk through the absolute brutal mechanics of
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spaceflight. We had a syllabus for this deep
0:56
dive. A syllabus.
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I love it. We're doing this
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chronologically. First, single stage rocket
1:01
dynamics and why early Launchpad saw so many failures.
1:05
Second, the math behind multi stage staging. 3rd Orbital
1:11
mechanics and central force fields.
1:12
Basically, how not to fall out of the sky.
1:14
Exactly. 4th we're looking at electrical propulsion, so like
1:18
ion drives and the small perturbation method.
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And finally #5 designing an actual Earth to Mars trajectory.
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It really is the ultimate engineering Crucible.
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I mean, if you design A car or a train, if something breaks, you
1:31
have a safety net. Right, you can just pull over.
1:33
Exactly. You have friction from the road,
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you have the ground. Even in an airplane you have
1:38
thick air for aerodynamic break, but in aerospace mechanics?
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No safety nuts at all. None.
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The moment you clear that tower, the margin for error
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mathematically approaches 0. So we're going to break down the
1:49
biggest pain points engineers face.
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Yeah, the real world issues, like why rockets spun out of
1:54
control or the crazy misconceptions about gravity.
1:56
And why engineers will aggressively argue over whether
1:59
to use a computer or a pencil for deep space math.
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Oh, I can't wait for that part, but let's start at the very
2:05
beginning. Lesson 1 Beating gravity.
2:08
We are on the launchpad and we have to accept that a rocket is,
2:13
well, it's completely different from a jet engine.
2:15
Completely different. A jet engine is basically a
2:17
parasite on the atmosphere, right?
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It sucks in ambient oxygen to ignite its fuel, but a rocket is
2:24
heading for a vacuum so it has to bring its own environment.
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It's a completely self-contained thermodynamic system.
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It has to carry its fuel like liquid hydrogen and its oxidizer
2:34
like liquid oxygen. And because you're carrying all
2:36
that mass and burning thousands of pounds of it every second,
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the forces are in total chaos. They're constantly changing.
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Which brings us to the three main forces, thrust, gravity and
2:47
air drag. Let's look at the thrust
2:49
equation, because this is where engineers really start to sweat
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right the. Math here is intense.
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The textbook lays it out. T = M dot times VE plus AE
2:58
multiplied by the difference between PE and PA.
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OK, let's unpack that alphabet soup.
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Yeah, please, because reading that out loud strips away how
3:04
violent the physics actually are.
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Well, thrust comes from 2 distinct mechanics.
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The first is pure brute force momentum.
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That's the m.timesvepart. M.beingthemassflowrate, right?
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How many kilograms of fuel you are violently burning every
3:19
second multiplied by the exit velocity VE of that gas?
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It's just a Newton's third law. Throw a ton of mass down really
3:26
fast, the rocket goes up. But then there's the second part
3:29
of the equation, the pressure thrust.
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This is the AE multiplied by PE minus PA, yes?
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AE is the X area of the nozzle, PE is the exhaust pressure, and
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PA is the ambient atmospheric pressure outside.
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So if the pressure inside the exhaust plume is higher than the
3:45
outside air pushing back, you actually get bonus stressed.
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Exactly. The gas expanding against the
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void generates a little extra push.
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Which leads to this wild operational reality.
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As you fly higher, the ambient pressure PA drops.
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It approaches 0 in the vacuum of space, so that pressure
4:02
difference gets larger, meaning the exact same rocket engine
4:05
actually gets more efficient and produces more thrust the higher
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it goes. Because the atmosphere isn't
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pushing back against the exhaust.
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That is so cool. But Speaking of the atmosphere
4:15
pushing back, we need to talk about a huge misconception.
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Oh, the drag misconception. Yes, you see this on engineering
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forms all the time. People watch a rocket launch,
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they see those massive vapor cones as it breaks the sound
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barrier, and they think, WOW, air drag must be this
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insurmountable force dominating the fuel budget.
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It's totally intuitive to think that.
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I mean, stick your hand out of car condo at 70 miles an hour.
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It hurts. Right.
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And a rocket is going thousands of miles an hour.
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But mathematically, yeah, for a heavy launch, vehicle drag is
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almost a complete distraction in early mission planning.
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Wait, really? A distraction.
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Because our atmosphere is actually incredibly shallow, the
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density drops off exponentially by the time you hit an altitude
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of just 100,000 feet. Which takes what, a couple of
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minutes? Exactly at that height, the air
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density is only 1% of its sea level value.
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So 99% of the aerodynamic resistance is just gone in 2
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minutes. The total cumulative drag over
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the whole flight is just dwarfed by the sheer magnitude of the
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engines thrust and the weight of the fuel.
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So when engineers calculate burnout velocity, they just
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ignore drag. In first order calculations,
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yeah, the drag term is frequently entirely omitted.
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It just doesn't change the final number enough to justify doing
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the incredibly complex differential math.
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That blows my mind. The raw power of the engine just
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makes the air irrelevant. But OK, let's go back down to
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the launchpad ignition, because early on engineers couldn't even
5:44
get past the 1st 10 seconds. Oh, the Vanguard program early V
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twos, yeah. You watch archival footage and
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the rocket lifts off a few feet, tilts over and just detonates.
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Why did that keep happening? It comes down to a really
5:56
unforgiving rule of static stability.
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When a rocket first lifts off, it's moving incredibly slowly,
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maybe 10 miles an hour. So it has 0 aerodynamic
6:06
stability. Fins don't work if there's no
6:08
air rushing past them. Right.
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You can put giant fins on it, wouldn't matter.
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Yeah. So you are basically trying to
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balance a skyscraper on a single column of fire.
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Like balancing a heavy broom on the palm of your hand.
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That's the perfect analogy. If you balance a broom from the
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bottom, your hand is the thrust. The heavy bristles at the top
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are the center of mass. And if you waver even a
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millimeter, gravity pulls the heavy top down.
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It flips. Exactly.
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But if you pinch the top of the broom handle and let it hang,
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it's totally stable your hand the thrust is above the mass.
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So the fundamental rule is the center of thrust must be located
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above the center of mass. Must be.
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Finding the center of mass is easy.
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Just weigh the fuel, the metal, the payload.
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But finding the exact center of thrust?
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That is a massive pain point. And here's another misconception
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we need to call out for the students.
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People look at that beautifully machined bell nozzle at the
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bottom of the rocket and they assume the shame of that metal
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perfectly dictates the thrust direction.
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Oh, I wish it were that simple. Right, like I pointed the nozzle
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straight down in CAD so my thrust is perfectly straight up.
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Yeah, fluid dynamics rarely cooperates with simple geometry
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because as the supersonic exhaust gas hits the lip of that
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bill nozzle, it meets the outside atmosphere.
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Not contained by the metal anymore.
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Right. And it experiences what we call
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expansion waves. Expansion waves.
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What does that actually look like physically?
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Well, the flow changes direction at the very exit corner.
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It fans out radially. The gas doesn't shoot straight
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down in a neat little cylinder. Oh, so some of the momentum
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vectors, like the force of the gas molecules, are actually
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pointing outward away from the center line?
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Precisely. The expansion waves actively
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change the direction and the magnitude of the velocity O.
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You can't just draw a line U from the middle of the engine.
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You have to calculate the vector sum of all the leaving particle
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momentums. Yes, and if you fail to account
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for those expansion waves, your calculated center of thrust
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might be inches off from your software prediction.
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And inches off means a rotating moment 3 seconds into flight,
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the rocket veers and the automated safety system has to
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blow it up. Exactly.
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OK, so let's assume the engineering team nailed the
8:20
math. They tracked the expansion
8:22
waves, used engine gambling to keep the thrust perfectly above
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the mass, and we clear the tower.
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We survive the launchpad. But now we hit the next massive
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hurdle. We have to reach orbital
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velocity. Which brings us to the syllabus
8:36
item to the staging dilemma. Yeah, the fundamental limitation
8:40
of a single stage rocket. Because a single stage vehicle
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carries all its dead weight all the way to orbit.
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Or at least it tries to. Let's talk about the final
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velocity equation for a single stage flying vertically.
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The math here is brutal. It really is.
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The equation dictates that V equals VE times the natural log
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of the ratio of initial mass to burnout mass.
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And then you subtract the gravity penalty which is G
9:07
naughty times DB or burnout time.
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It's that natural log function that is the absolute bane of
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aerospace engineers. Oh completely.
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Let's breakdown why Initial mass is everything on the pad.
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Fuel tanks, engines, payload, burnout.
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Mass is what's left when the fuel is gone.
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Which is the payload, but also all the empty metal tanks, the
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turbo pumps, the wiring, pure dead weight.
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Right. And because the velocity is tied
9:30
to a natural logarithm, you run into terrifying diminishing
9:33
returns. If you want to go twice as fast,
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you can't just double the fuel. You have to exponentially
9:38
increase the fuel. Yes, and more fuel means bigger
9:41
tanks. Bigger tanks mean heavier empty
9:44
metal, which means you need bigger engines to lift the
9:47
tanks, which need more fuel. It's a vicious cycle.
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You end up dragging 10s of thousands of pounds of empty
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metal tubing at supersonic speeds.
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Wasting precious thrust accelerating a hollow tin can.
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Which is why single staged orbit is essentially an engineering
10:00
dead end for heavy payloads. The solution is staging multi
10:05
stage rockets. Exactly.
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You ditch the dead weight as you go.
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When stage 1 is empty, explosive bolts fire, you drop the massive
10:12
empty tanks into the ocean and stage 2 ignites.
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And crucially, stage 2 doesn't have to push stage 1's dead
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weight, so it's initial mass is way lower.
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That natural log ratio resets, it becomes highly favorable, and
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your terminal velocity dramatically increases.
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But wait, how do engineers mathematically optimize this?
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You can't just guess how big each stage should.
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Be no. If stage 1 is too big, you
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suffer the dead weight problem early.
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If it's too small, stage 2 becomes too heavy.
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So they use some heavy mathematical artillery, the
10:39
LaGrange multiplier. Yes, often denoted by the letter
10:43
alpha. This is a classic optimization
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problem. You want to maximize your final
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velocity, which is VN divided by VE while constrained by fixed
10:51
payloads and initial mass limits.
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I'm. Going to play the skeptical
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student for a second because staging is incredibly dangerous.
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You're detonating ordinance mid flight.
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If the second stage doesn't ignite in the vacuum, the
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mission is over. Does staging really make that
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much of A mathematical difference to justify the risk?
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It's the critical question. Complexity breeds risk, but the
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math in the textbook's Examle 51 proves we literally have no
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choice. OK, let's look at the numbers
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for the 500 kilogram payload example.
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Right. So we have an initial mass of 75
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oh oh kilograms, pillow to 500 kilograms, a structure mass of
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1000 kilograms, and an exhaust velocity of 3000 meters per
11:30
second. OK, so if we stubbornly build
11:32
that as a single stage rocket, what's the terminal velocity?
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At burnout, the math calculation AT400827 meters per second.
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Four hundred 827 meters per second, which is fast, but low
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Earth orbit requires roughly 7800 meters per second, so you
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fall way short. You basically just made a very
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expensive suborbital firework. Exactly.
11:52
It crashes into the ocean, but let's redesign that exact same
11:55
vehicle. We don't have a single drop of
11:57
fuel, same payload, same engines, but we use the LaGrange
12:01
multiplier method to perfectly optimize it as a three stage
12:04
rocket. The textbook says the optimal
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alpha value for this is 0.68203. Right.
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So using that exact mass distribution, we stage it twice,
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we drop the dead weight, the final velocity of that third
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stage skyrockets to 16411 meters per second.
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Wow, six 511 meters per second. You gain almost 1600 meters per
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second of speed, absolutely for free.
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Just by mathematical optimization, you didn't add
12:27
energy, you just stopped wasting it on empty metal.
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That is profound. OK, so we've staged perfectly.
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The final engine shuts down. We are officially in space.
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Which brings us to lesson 3, Orbital Mechanics 101.
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The paradigm totally shifts here.
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We fought gravity on the pad, but now we're in a central force
12:43
field. The Earth's mass is so much
12:45
greater than the satellite that we treat the Earth as a
12:47
stationary point, and the fundamental law here is
12:50
conservation of angular momentum.
12:52
Right, the formula is m * r ^2 times Theta dot equals a
12:57
constant lowercase L. So M is our vehicle mass, R is
13:01
our radius from the planet, and Theta dot is our angular
13:04
velocity. Exactly.
13:05
It's the classic figure skater analogy.
13:07
Oh right. Arms extended, large radius,
13:10
they spin slowly. Pull the arms in the radius
13:13
shrinks, the spin rate dramatically increases.
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That is exactly how an elliptical orbit works.
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The orbit is dictated by an inverse square law for
13:20
gravitational force. The force F of R equals the
13:23
universal gravitational constant G times the planet mass M times
13:28
the vehicle mass M all divided by r ^2.
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So as you fall closer to Earth, R shrinks, gravity pulls harder
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and you whip around the planet super fast.
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As you swing back out, R expands and you slow down.
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And that brings up maybe the most pervasive misconception of
13:42
Popular Science. Escape velocity and 0.
13:45
Gravity. Yes, if you ask a random person,
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they think escape velocity means flying high enough to cross a
13:51
border where gravity magically turns.
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Off it is a complete fallacy. Gravity extends infinitely.
13:56
It gets weaker because of the inverse square law, but it never
13:59
reaches 0. So what is the true mechanical
14:01
definition of escape velocity? We have to look at energy.
14:05
A spacecraft has positive kinetic energy from its speed,
14:09
and it has negative potential energy because it's trapped deep
14:13
inside a gravity wall. So it's like being in a massive
14:15
hole, you would gravity a debt. Exactly.
14:18
In a normal orbit, your total mechanical energy E is a
14:22
negative number. Escape velocity is the exact
14:24
speed required to make your total mechanical energy equal
14:27
perfectly to 0. So you're positive kinetic
14:29
energy perfectly cancels out your negative potential debt.
14:32
Yes, and geometrically the very second E equals 0.
14:36
The orbital eccentricity E changes to exactly 1.
14:40
Which means the orbit transitions from a closed
14:42
ellipse into an open parabolic curve.
14:45
Right, you don't lose gravity, you just have exactly enough
14:47
kinetic energy that gravity can never slow you down enough to
14:51
pull you back. The formula is V escape equals
14:54
the square root of the quantity 2 * g * m / r.
14:58
It's so elegant, but let's look at a real world example from the
15:02
text. Example 5.3 A weather satellite.
15:06
A geosynchronous satellite, specifically.
15:07
Right. Engineers need it to stay over
15:10
one exact spot on the equator, so it has to rotate at the
15:14
Earth's exact angular speed. Which is Omega or 7.2722 * 10 to
15:20
the negative 5th radians per second.
15:21
So how do they calculate the altitude for that?
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It's simple balancing act. You set the circular velocity
15:26
equation, which is sqrt g * m / r equal to R times Omega.
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So you're setting the velocity dictated by gravity equal to the
15:34
velocity dictated by the Earth's rotation.
15:37
Exactly. And when you solve for R, you
15:39
get an orbital radius of 42,238 kilometers from the center of
15:43
the Earth. It is totally unforgiving.
15:45
If your math is slightly off, your satellite drifts away,
15:47
Yeah. You don't get to choose the
15:48
parking spot. Physics dictates the
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coordinates. OK, so circular and elliptical
15:52
orbits assume we are coasting, we burn the engines, and now
15:56
we're just falling. But what if we want to
15:58
constantly apply thrust? That takes us to lesson 4,
16:02
continuous propulsion. Electrical propulsion
16:05
specifically. Like ion drives?
16:07
Yes, they collect electrons emitted from the sun, strip and
16:11
inert gas like xenon, and accelerate the ions out the
16:14
back. It allows for continuous thrust.
16:17
But the physical force is microscopic, right?
16:19
Like the thrust couldn't even push a piece of paper across a
16:21
desk. Right.
16:22
The thrust is incredibly low compared to the rocket's weight.
16:26
We use a parameter called nu to represent the ratio of
16:30
electrical thrust to gravitational force.
16:32
And nu is on the order of 10 to the power of -3.
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Or even smaller. It's a tiny perturbation.
16:37
Which leads to a huge debate among engineers.
16:40
Because the trajectory is constantly altering, the
16:43
equations of motion become highly complex.
16:45
So why don't we just plug the differential equations into a
16:47
computer and use a numerical solver like the runj CUDA
16:51
method? This is a major engineering pain
16:55
point. Yes.
16:56
Runji CUDA will solve it numerically and give you a graph
17:00
of the trajectory. OK, so what's the problem?
17:02
The problem is that numerical methods hide the relationships
17:05
between the physical parameters. Wait, I need you to break that
17:08
down. How does a computer hide the
17:10
relationships? Imagine the computer graph shows
17:13
your orbit isn't reaching the destination because the computer
17:17
just brute force the numbers you don't inherently know.
17:20
Why should you double the thrust?
17:22
Have the payload? The graph doesn't tell you the
17:24
causal relationship, you're just guessing numbers and rerunning
17:27
simulations. So the computer shows you the
17:30
destination, but it doesn't give you the map of how the steering
17:33
wheel connects to the tires. Precisely.
17:36
That's why the text advocates for an analytical method, the
17:39
small perturbation method. Because the thrust ratio new is
17:42
so microscopically small, we treat it like a tiny crosswind
17:46
on an other otherwise perfect circular orbit.
17:48
Exactly. You slice the problem into a
17:50
simple base layer, the unpowered orbit plus a tiny mathematical
17:54
correction, and applying this analytically yields an equation
17:58
where the parameters are visible.
18:00
The text gives the solution up to the 1st order of nu.
18:03
It's P = 1 plus nu times the quantity 2 Tau -2 sine Tau.
18:08
And the physical result of that equation, The trajectory isn't a
18:11
closed loop anymore, it's a spiral orbit.
18:13
And because it's an analytical equation, engineers can look at
18:17
it and instantly see that the increment of the radius is
18:20
directly proportional to the tangential thrust and the
18:23
initial angular speed. Right.
18:25
If you need to spiral out faster, the equation tells you
18:28
exactly how much to turn up the power.
18:30
No guessing required. That is a brilliant defense of
18:33
doing the math by hand. But spiraling out slowly is only
18:37
good for probes. If we want to send a crude
18:39
vehicle to Mars, we don't have three years to spiral out.
18:42
No, the crew would run out of life support.
18:44
We need massive impulsive thrusts.
18:47
Which brings us to the final phase, Lesson 5.
18:50
Interplanetary trajectories and Mars capture.
18:52
This is where things get truly complicated because Earth is
18:55
moving, Mars is moving, and the Sun is warping space.
18:59
Trying to calculate all three at once is the three body problem.
19:02
Which is mathematically impossible to solve perfectly.
19:05
So how do we do it? We cheat.
19:07
We use a method called patched conics.
19:10
We divide the trip into 3 mathematically isolated 1 body
19:14
segments. OK, walk me through the patches.
19:16
First, near Earth we ignore the Sun and Mars.
19:19
We treat it as A2 body problem spacecraft and Earth we
19:22
calculate the escae velocity right.
19:24
Once we cross a boundary called the sphere of influence, we
19:27
patch that to segment 2, the Sun's field.
19:30
Now we pretend Earth and Mars are gone and calculate the mass
19:34
of transfer orbit. And finally approaching the
19:36
destination segment 3 Mars is field.
19:39
Exactly. Now for that middle segment, the
19:41
transfer orbit, there are two main methods.
19:44
The Homan transfer and the O birth method.
19:46
Let's start with Homan. It's 3 impulses entering from an
19:49
established Earth circular orbit.
19:51
It's very fuel efficient. You burn to leave Earth coast on
19:55
ellipse that kisses Mars orbit and burn to circular eyes at
19:58
arrival. But the textbook table says the
20:00
coasting time. Is 256 days.
20:02
Wait, wait, is that right? Yes, 256 days.
20:05
OK, that's over 8 months in deaf space.
20:07
If we want to go faster, we have to look at the O birth method.
20:10
But I have to say, the O birth effect sounds like absolute
20:13
voodoo to me. It really is deeply
20:15
counterintuitive. O Birth optimizes fuel by
20:18
realizing that a rocket engine is drastically more efficient at
20:22
adding energy if it burns deep inside a gravity well when the
20:27
spacecraft is moving at its maximum velocity.
20:29
And this is where my brain stalls.
20:32
A rocket engine outputs a fixed amount of energy based on the
20:35
chemical reaction. How can burning the exact same
20:38
amount of fuel give you more kinetic energy just because you
20:40
were already moving fast? It's all in the mathematical
20:43
definition of kinetic energy. The total energy per unit mass
20:48
equates to 1/2 times the square of delta Vt.
20:51
It's the squared term that changes everything.
20:53
The velocity squared OK if. You're going 10 meters per
20:55
second, and your engine gives you a 10m per second boost.
20:58
Your speed goes to 20. Yeah.
20:59
Your kinetic energy, which is velocity squared, goes from 100
21:02
to 400. You gain 300 units.
21:04
Right, 400 -, 100. But if you are already screaming
21:08
through a gravity well at 1000 meters per second and do the
21:11
exact same 10m per second burn. Your speed goes from 1000 to
21:15
1010, and 1000 ^2 is 1,000,000. A 1010 ^2 is a little over
21:20
1,020,000. Precisely, you gained over
21:22
20,000 units of kinetic energy from the exact same tank of
21:26
fuel. That is literal orbital jiu
21:28
jitsu. You are stealing energy from the
21:30
gravity well. Exactly so, Ober says.
21:33
Launch directly from Earth's surface, dive deep to maximize
21:36
speed, and fire the main engines right at the fastest point.
21:39
Amazing. OK, so we've crossed the void
21:42
and we are approaching the climax.
21:44
Example 5.5 The Mars Capture. The ultimate final exam We're
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approaching Mars on a an elliptical transfer orbit with a
21:51
major axis of 4.0 * 10 to the 11th meters.
21:55
And if the engine doesn't fire, we fly right past it into deck
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space. Right, so we have to calculate
21:59
the exact intersection point. The math says it's at a radius
22:02
of 2.278 * 10 to the 11th. Meter and what's our speed
22:06
relative to the transfer orbit? The probe is moving at 400 and
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767 meters per second, tangentially and -2218 radially.
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But Mars is moving too. So when we calculate our
22:16
relative velocity from the perspective of Mars, we enter
22:19
with a hyperbolic orbit velocity.
22:21
Yes, our kinetic energy is too high.
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Mars can't capture us so. We have to spin the ship around,
22:26
point the engines forward and brake.
22:28
We have to force the probe into a long term spiral observation
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orbit. And we need a specific orbit, an
22:34
altitude of 3500 kilometers. The tangential velocity needed
22:38
to balance centrifugal force For a circular orbit there is 3489
22:43
meters per second. So the engineers look at our
22:46
screaming hyperbolic approach, they look at the target of three
22:49
hundred 489 meters per second, and they calculate the breaking
22:52
impulse required. The exact delta V breaking
22:55
impulse needed is 25164 meters per second two. 1564 meters per
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second and Mission Control can't joystick this.
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The communications delay is too long.
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They upload the commands and pray the math is flawless.
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It is entirely autonomous. The valves have to open, the
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propellants must ignite and bleed off exactly 25164 meters
23:14
per second of speed. It's just breathtaking.
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Real world missions like the Mars Reconnaissance Orbiter
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execute their orbital insertions exactly like this.
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Precise velocity reductions at the exact intersection angle.
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It's the pinnacle of predictive mechanics, truly.
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Well, before we wrap up this deep dive, I want to leave you
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all with a final provocative thought.
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A little brain teaser based on the texts.
23:35
Problem set. Oh, I like where this is going.
23:37
We know a planet's radius vector sweeps out an equal area at a
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constant rate in an elliptical orbit.
23:44
But think about this. What would mathematically happen
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to the motion of that planet if the minor axis of the ellipse
23:51
approach 0 under the exact same inverse square force law?
23:56
That is a dark thought experiment.
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Right. If the Oval flattens out
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entirely, what physical reality does that describe?
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Think about what happens when it stops being in orbit entirely.
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And just turns into a very fast, very fatal plunge straight down.
24:09
Exactly. Chew on that math until next
24:11
time. If you got something useful out
24:13
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24:16
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