Satellites

Predicting Orbital Collisions

There are now well over 20,000 objects larger than a softball in orbit around the Earth. We are poised to get even more objects, as several companies are planning on launching mega-constellations of satellites (750-4,000) to provide internet across the world from low-Earth orbiting platforms. While space appears to be quite empty, there are a lot of satellites (and random stuff) out there. The problem is that there is no one “driving” the majority of these objects, and collisions between objects that are moving at 17,000 MPH typically produce even more pieces of debris that could potentially collide with other objects. And the satellites that are being “driven” have only a certain amount of fuel – once that runs out, the satellite is essentially dead. Therefore, satellite operators are not really wanting to move their satellites all of the time.

The United States Air Force (USAF) is in charge of keeping track of the locations and tracks of all objects in orbit around the Earth. There are several websites and apps where you can see which satellites are going to pass over your location at any time. These are pretty cool, because you can often very clearly see orbiting satellites. If you go outside on a clear day just before sunrise or after sunset, while the sun is still shining in the upper atmosphere, but is not shining on the ground (say 10-45 minutes after sunset), and you look up for a long time, you can see little dots of light moving across the sky that are not airplanes. Those are satellites. An app will tell you exactly which satellites those are and how they are moving. You can even see the shape of the International Space Station (ISS) with good binoculars!

These websites and apps get this orbit information from USAF, which tracks them using radars. There are many tracking stations on the ground that are similar to radar guns that police officers use. Simplistically, the radar sends out a pulse that bounces off the satellite and returns back to the radar. The radar can record the time that it took to go from the radar, to the satellite, and back to the radar, which specifies the distance to the satellite. The radar can also record the Doppler shift of the signal, which indicates the speed of the satellite. The strength of the return signal indicates the size of the satellite. This is all a simplification, but in general it is the case. The radar takes a few of these measurements, and then moves on to another object. With a bunch of measurements, the orbital characteristics of the satellite can be determined. The Air Force does this for the more than 20,000 objects on a daily basis. They make the (rough) orbital characteristics available to the public.

(As an aside, you could actually do the same thing with a telescope at home, just by tracking how the objects that you see move across the sky.  It wouldn’t be as accurate as a radar, but just knowing how the object moves across the sky gives you a pretty good estimate of the orbit. Kepler did this with the moons of Jupiter, and you could do it with the ISS!)

The USAF uses an orbit propagator that takes these radar measurements and calculates the position and velocity of the 20,000 objects every second for the next five days (roughly). They then look for any objects that come within a certain distance from each other. Those satellites are flagged to be examined more carefully. There are multiple ways to do this, but I will discuss the easiest to understand here.

First, let’s back up a bit. Let’s say that you want to predict whether two cars that are moving towards the same intersection will collide. One is traveling east and one is traveling south. Consider some scenarios:

  1. The cars are one block apart and both traveling 30 MPH (44 feet/sec) and are exactly 0.1 miles (528 feet) apart. They will meet in the middle of the intersection. A crash is definite, and will take place in about 12 seconds!
  2. Imagine the same scenario (one), but one car is 0.11 miles away. The cars will miss each other by 0.01 miles (roughly 53 feet).
  3. Imagine scenario one, but one car is moving at a slightly slower speeds (say 29 MPH, or 42.5 feet/sec, instead of 30 MPH). In 12 seconds, the car will fall behind its expected distance traveled by about 18 feet, which is about the length of a car, and therefore will end up missing the other car.
  4. Imagine scenario one again, and both cars start off on a perfect collision course, but one of the cars starts to decelerate due to there being a hill in the way. If that deceleration is enough, the speed that the car is traveling decreases enough, and the collision will be avoided.

In predicting whether the cars will crash, there are many things that cause uncertainty: estimating the initial position and velocity as well as figuring out all of the forces on the cars that can cause accelerations. While there are several differences between collisions between cars and satellites, the concepts are the same. Satellites are moving much faster than cars, and there are significantly fewer objects in orbit than cars in, say, downtown New York City. In addition, satellite operators need a couple of days to figure out whether there will be a collision, so they would like to know well ahead of time if they need to move the satellite out of the way.

It seems like it would be relatively straightforward to predict a collision, since the Air Force has the position and velocity of each of the objects, and it uses a really highly accurate orbit propagator to determine the future positions of the satellites. Except, as illustrated above, if there are uncertainties in the initial position or the initial velocity, then it is not clear if there will be a collision or not. In addition, if there are uncertainties in the accelerations that the objects undergo, it will cause uncertainties in whether there will be a collision.

So, what are the sources of those uncertainties? Well, there are a lot of them. Let’s go through them all one by one:

  1. The position could be off due to issues with the radar signals. These signals travel up into the atmosphere, hit the orbiting object, then come back through the atmosphere. As the signal goes through the atmosphere, the speed of the signal actually changes, since the propagation speed is dependent on the medium that it travels through. In fact, the path through the atmosphere is dependent on these atmospheric conditions also. So, moisture in the atmosphere and the ionosphere can both change the path of travel, the propagation speed and therefore the delay. If the atmosphere is not modeled correctly, then this will cause the exact position of the orbiting object to have some uncertainty. And remember that the uncertainty only needs to be about as big as the satellite (a few feet) to cause issues in determining whether there will be a collision.
  2. The velocity of the orbiting object needs to be known really, really, really well. As an example, let’s say that there may be a collision 24 hours into the future, and the object that is moving is 10 feet across. That means that the speed needs to be known to within 0.000116 feet/sec, or 0.000079MPH. That is crazy small. Considering that the objects are orbiting at about 17,000 MPH, this is an impossible task. This is a source of very large uncertainty!
  3. The forces that act on the orbiting objects are quite complex and challenging to model. For example, some of the forces include:
    1. Gravity: While most people think of gravity as 9.8 m/s2, it is more complicated than this. The Earth is to first order a flattened sphere, so that the satellites feel more gravity near the equator than near the poles. To higher orders (literally), the orbit propagators need to take into account many of the features of the Earth, such as mountain ranges and oceans. The equation that is used to (accurately) describe the gravity has several hundred terms in it. If you discount these terms, the positions of the objects can be systematically miscalculated, which is bad. The European Space Agency has launched several satellite missions to accurately model the gravity of the Earth, such as CHAMP, GRACE, and GOCE.  Neglecting higher order gravity terms can cause the satellite orbits to be off by several hundred meters after a day.
    2. Sun and moon Gravity: The sun and the moon both exert forces on objects in orbit around the Earth, so they need to be taken into account. Luckily, the orbits of the Earth around the sun and the moon around Earth are pretty well known, so these are easy forces to account for.
    3. Sunlight: Sunlight bouncing off the orbiting object actually imparts momentum to it. While this force is quite small, it actually is very important to model correctly. It is dependent on the materials that the orbiting object is made out of (if the material absorbs sunlight it feels a different force than if it reflects the sunlight), and the orientation of the object (if it has a large area pointed at the sun, then there is a lot of force, but if there is a small area, the force is smaller).  Neglecting this force can cause an error of 10s of meters after a day.
    4. Earth shine: Sunlight reflecting off of the Earth adds pressure to the orbiting object, similar to sunlight described above. This only happens on the dayside, and is pretty weak compared to the direct sunlight, but it still needs to be accounted for. Further, the Earth radiates infrared energy (i.e., the Earth glows). Satellites feel this glow, just like they feel the sunlight, but instead of being directed from the sun, it is directed from the Earth. It is complicated to accurately take into account the Earth shine, but luckily it is a pretty weak force, so for collision avoidance, it can be approximated and not modeled exactly.  Neglecting this results in errors of a couple of meters at most after a day.
    5. Drag: Just like a biker with a headwind feels wind resistance, a satellite in orbit feels a tiny bit of atmospheric drag force that causes it to lose energy all of the time. (See a post on drag here!) The drag force is directly dependent on density of the atmosphere and is dependent on the difference between the velocity of the object and the winds in the atmosphere squared. There is significant uncertainty in both the winds and the density of the atmosphere. As described below, it can be one of the main sources of error in the probability of collision, and can cause the positions to be uncertain to hundreds of meters after a day.

It is impossible to definitively say “there will be a collision between these two objects in 24 hours from now”. What is done instead is determine the probability of collision. This information is then passed on to satellite operators, so they can choose whether they want to move their satellite or not. If the decide to move their satellite, the operators typically speed it up or slow it down to definitively avoid the collision.

A simple method of calculating the probability of collision is to do what is called a “Monte Carlo” simulation of the interaction. By this, the modelers create about a few million versions of object 1 and a few million versions of object 2. They give these objects slightly different initial positions, velocities, and satellite characteristic (using random numbers to perturb these quantities) and see how many of them collide. This number, divided by the few million scenarios, gives the probability of collision.

The satellite characteristics that are perturbed have to do with the drag. Satellites have different shapes and sizes and masses. While the satellites’ shapes are typically well described, the orbital debris is not well described at all. For example, several years ago, the Chinese blew up one of their own weather satellites, resulting in several thousand pieces of debris of unknown size and mass. There are estimates of the size and mass of each of these objects, but there is significant uncertainty. Therefore, the few million objects in the Monte Carlo simulation are given slightly different sizes and masses (technically ballistic coefficients – the ballistic coefficient also depends on the shape of the object and what the object is made out of, but that is a detail.  Well, this whole post is a detail, so it is a detail on a detail!)

The USAF propagates these millions of Monte Carlo satellites through a single atmosphere. While the majority of the time this is fine (since the upper atmosphere is calm most of the time), at times this can give huge systematic errors. For example, when the northern (and southern) lights become active, they add a bunch of energy to the atmosphere, causing it to heat up and expand. This expansion causes the density to increase, which drives a stronger drag force. If this isn’t accounted for, then the probability of collision will be incorrect.

Recently, a paper was published that showed that the behavior of the atmosphere has a large effect on the probability of collision. An event was explored, where the probability of collision was determined to be above the threshold where something should be done. The million objects were then simulated over and over and over again, propagating them through different atmosphere, depending on what was predicted. It was shown that if the sun and the aurora was a tiny bit more active, the probability of collision would be increased, while if the aurora and sun were either a lot more active or any less active, the probability of collision would decrease. It was suggested that this uncertainty in the sun’s brightness and the auroral activity should be taken into account when calculating the probability of collision. (This paper was published by Charles Bussy-Virat, and there is a youtube video of him explaining all of this in a seminar here.)

Finally, the probability of collision that is typically taken as a threshold to do something with the satellites is typically 0.0001%. This is incredibly low! But, considering that a satellite may cost several hundred million dollars and many tens of millions of dollars to launch into space, the operators want to be as cautious as possible.

In summary, calculating the probability of collision between objects in orbit is really hard. When the atmosphere is really calm, the hardest part is figuring out the velocity of the objects – a tiny error in this can cause a large error in the position of the objects at the time of closest approach. When the atmosphere goes a bit crazy, due to the aurora or the sun having more activity than expected, the satellite’s drag force can change pretty dramatically, acting to change the acceleration, velocity, and ultimately the position of the objects at the time of closest approach. While the distribution of the velocity errors is really well understood, so the Air Force can very accurately account for this in determining the probability of collision, the uncertainty in how the atmosphere is behaving is very hard to account for. This lack of knowledge in how to treat the future state of the atmosphere is one of the largest challenges in accurately determining the probability of collision of objects in orbit around the Earth.

A Side Note:

There are a lot of issues in determining the density of the atmosphere. The most accurate models of the upper atmosphere, at this time, are empirical models, meaning that the models were created fitting a ton of data. These models get the mean state of the atmosphere (i.e., the climate) (mostly) correct, but have a hard time with the “weather”. Indeed, the Air Force uses a very old model of the upper atmosphere that is no longer the state of the art, but they have a way of compensating for this. There are over 50 perfect spheres orbiting the Earth at different altitudes. The Air Force can get very good orbital characteristics from these spheres. In addition, they know the shape, mass and what the sphere is made out of, so they can use these to figure out what the drag force actually is, given the change of orbit from one time to another. They can compare the “actual” drag force and the drag force predicted by the model and adjust the model until it matches. Then they use that corrected model to predict the density into the future.

The science community is attempting to improve models of the upper atmosphere all of time.  These are like weather models for the troposphere in that they use fluid dynamic equations to simulate the atmosphere.  Ultimately, the USAF will have to use these types of models if they really want to improve the forecasts when there are large storms in space that can dramatically alter the trajectory of the satellites.  They will make predictions like hurricane forecasters make predictions of landfall – using many different models to better understand the uncertainty.

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Rockets

Escape Velocity

Getting to another planet, or getting to the sun involves getting out of Earth’s gravity well. Essentially, Earth’s gravity extends outwards forever. If Earth were all alone in the universe, its gravity would be felt everywhere. If anything were sitting out in space and not moving at all with respect to Earth, it would feel Earth’s gravity and begin to fall towards the Earth.

escapevelocity_gravitywell
Earth’s gravity well. Everything falls back towards the Earth!

Earth is not alone in the solar system, though.  The Sun is much larger than Earth, and it’s gravity well is much, much bigger than Earth’s.  So, if you get out of Earth’s gravity well, you are still inside of the Sun’s gravity well.  Between the Sun and the Earth, there is a gravitational null, where the gravity wells add together to make a little hill.  If you are on one side of the hill (the left side), you fall towards the Sun, and if you are on the other side of the hill (the right side), you fall towards the Earth.

escapevelocity_sungravitywell
The Earth’s gravity well is inside of the Sun’s gravity well.  In between the Sun and the Earth, the wells add together and sort of cancel each other out, making a little hill.  At the top of the hill is a little gravitational null.  If you are on one side of the hill, you fall towards the Sun.  Of you are on the other, you fall towards the Earth.

If you want to get out of Earth’s gravity well, how do you do it?

Imagine if you were really in a valley and the walls of the valley were not grass or something, but were very, very smooth – like ice.  If you tried to climb up the sides, you would constantly just fall back down into the valley.  Let’s say that you have a ball. If you roll the ball up the side of the valley, it would probably roll up for a ways, then it would slow down, stop eventually, and then come right back down to the bottom of the valley. The ball would be stuck in the gravity well.

But, I am sure that you can imagine, if you were to roll the ball fast enough, it would go up the side of the valley, but still have enough speed at the top of the valley to keep going. It would go over the top of the hill, and escape the valley.  If the sides of the valley were very tall, then you would have to roll the ball really, really fast.  If the sides were not very tall at all, you wouldn’t have to roll the ball very fast at all.

The same thing is true of a planet or the sun or a moon – really any body that has mass.  Let’s consider Earth specifically.  A super simple example is if you throw a ball up in the air: it goes up, slowing down all of the way.  It eventually stops completely, reverses course and falls back to Earth.  Just like the ball in the valley example.  You roll the ball too slow, and it will climb up the side of the valley, and then turn around and come back down. Interestingly, if you ignore air resistance, the ball would hit the ground with the same speed that it was thrown up in the air.

If you switch from a ball to a gun, and fire a gun straight up in the air, the bullet would go up and up, slowing down the whole time.  Eventually, it would stop completely and fall back to Earth.  Luckily, it is incredibly hard to shoot straight up in the air, since the bullet falls back to the ground with the same speed as it leaves the gun.  So, shooting a gun in the air is not too smart, since bullets don’t have enough speed to leave the Earth!  The bullets come down… somewhere. In fact, if you got the biggest gun ever made, it still wouldn’t be able to shoot bullets fast enough to escape into space.  The bullets would travel about 100 miles into space, then stop and fall back to Earth.  This gun, and the guy who made it, are described here. (It is a cool story!)

On the other hand, a rocket that could go a bit over 25,000 miles per hour straight up would never stop moving away from Earth.  The rocket would slow down and slow down and slow down, but would never stop.  Eventually, the rocket would be outside of Earth’s gravity well, with the Sun’s gravity influencing the rocket more than Earth’s gravity.

So, in summary, if you aimed a rocket straight up in the air and got it to go a bit over 25,000 MPH, the rocket would escape Earth’s gravity.  If you want to get to Mars or any other planet, or even the Sun, you have to have a rocket that can go at least this fast, or produce this much “delta-V”.

The term delta-V is used to describe a change in velocity made by a rocket.  So, in this case, the rocket is initially sitting on the ground not moving.  It needs to change velocities from sitting on the ground not moving at all to moving straight up with a speed of over 25,000 MPH.  Delta-V. It would then escape the Earth’s gravity well.

Interestingly, escape velocity works backwards too.  For example, let’s consider trying to land on the moon. (I consider the moon because it has no atmosphere, which complicates things). Let’s say that we have a ship that is on its way from the Earth to the moon,  and that it has just barely slipped from being in Earth’s gravity well to being in the moon’s gravity well, but is moving incredibly slow at that point. If absolutely nothing is done, the ship will fall towards the moon, gaining speed all of the time.  If nothing continues to be done, the ship will splat into the surface of the moon with a speed of about 5,300 MPH. Not coincidently, this is the escape speed of the moon.

What can be done to save the poor astronauts who are on board the possibly doomed rocket ship? Well, the ship can fall faster and faster and faster towards the surface of the moon, gaining almost all of the 5,300 MPH, and at the last possible second, the rocket can fire, slowing the rocket down to almost zero, for a nice touch down.  This implies that the rocket needs to be capable of producing a delta-V of 5,300 MPH (slowing from 5,300 MPH to zero).  There would be an incredibly strong acceleration at the end, though. This might flatten down the poor astronauts! A much better alternative, especially of there are astronauts on board, is that the rocket can continuously fire the rockets, keeping the speed to some manageable level, so that it doesn’t have to produce so much acceleration at the very end. Interestingly, from the rocket’s point of view, these scenarios are almost identical to each other – each needs the rocket to produce a total of 5,300 MPH of delta-V. In one case, it does it all at once, and in the other case, it does it slowly, letting the rocket ship speed up a bit (due to gravity), then slowing it down using the rocket, then repeating until the rocket is safely on the ground.

So, landing on a planet/moon with no atmosphere requires a delta-V of the escape velocity, just like taking off from the planet.  I find that really cool!

Two asides here:

The first is that the moon landings were not like this at all.  The Apollo capsules went to the moon and got into orbit around the moon.  Then the landing module (which was much smaller than the Apollo capsule), took two of the astronauts down to the surface.  This small module had to burn away the orbital velocity of the Apollo capsule (plus a bit extra).  Because the landing module was so much smaller than the Apollo capsule, it saved a huge amount of fuel.  If the astronauts had landed the Apollo capsule, the Saturn-V rocket would have had to have been much bigger.  That was not possible at the time.

The second is that when we land on planets such as the Earth, Venus, and Mars, we don’t need to have the delta-V to slow all the way down, since each has an atmosphere.  We use the atmosphere to slow us way, way, way down, which saves a huge amount of fuel.  When ships return from the International Space Station, they do a very small burn to put them on a trajectory that takes them into the atmosphere a tiny bit.  This slows them down to the point where they get dragged into the atmosphere and de-orbit because of this.  If the Earth had no atmosphere, it would take about 17,000 MPH of delta-V to get from the ISS to the ground.  Since we have a robust atmosphere, it only takes a few hundred MPH of delta-V.  Mars, on the other hand, has a really weak atmosphere, and when we try to land rovers and such on Mars, the atmosphere doesn’t stop them very well (i.e., terminal velocity on Mars is too fast, and stuff will break if it hit the ground traveling at terminal velocity).  So, there needs to be some way to reduce the speed of the object a bit.  NASA comes up with crazy ways to do this type of stuff, and you should read about the Mars rovers to see these crazy ideas!

Finally, every object has an escape velocity.  For the Earth, this is about 25,000 MPH.  For the moon, it is about 5,300 MPH.  For Jupiter, it is 135,500 MPH (wow!).  Jupiter has a HUGE gravity well! But, it is nothing compared to the Sun’s gravity well. If you escaped from the Earth, then wanted to escape the Sun, you would have to go about 95,000 MPH away from the Sun – even being 93 million miles away from the Sun!  If you were near Mercury, and wanted to escape the Sun (not Mercury), you would have to go 152,000 MPH!  Solar Probe Plus, a mission to the Sun, is going to 10 solar radii.  If it wanted to fully escape the Sun at that point, it would have to be going 440,000 MPH!

Just FYI, my escape velocity is 0.1 mm/s.  So, anything that is moving away from me with a speed of about 0.1 mm/s will escape. That is pretty small!  But, I am happy that my escape velocity is so small, since otherwise I would accumulate a lot of stuff!

Anyways, escape velocity is an important concept to understand when one is trying to get off of a planet and get to some other body.  I hope this helps explain it a bit!

Uncategorized

How to Get to Mars

This is going to be the first in a series on issues surrounding colonizing Mars.  I will talk about why it is so incredibly difficult to actually get there and get back as well as some ideas on how we should realistically be looking at minimizing the costs to do this.

Ever since the 1960s, we have been trying to get to Mars and take pictures and explore.  Mars has always captured our imagination, since it seems to tantalizing that it could contain life. It is right there within our grasp.  But, still, it is so far away.

Almost 2/3 of all missions that have been slated to go to Mars have failed.  Some of these include rockets that have blown up.  Others include a Russian lander that returned about half of one image before it stopped working.  Viking was the first lander that actually took pictures and really worked.

We can go to Mars about every other year.  This is because a Mars year is pretty close to two Earth years, so Mars and Earth have the correct positions once every Mars year.

In this post, I will walk through how we actually get to Mars right now, independent of cost or any real consideration – just the basic facts.  In the next post, I will walk through some of the costs in terms of rocket fuel needed to actually do this. Here we go!

The first thing that the spacecraft has to do is to get off the Earth.  The easiest way to do this is to pick a good direction and just accelerate up to just over the escape velocity of the Earth.  This is about 11.2 km/s (about 25,000 MPH) on the surface of the Earth.  That means if you launch something with >11.2 km/s, it will escape Earth’s gravitational field and won’t return. (Hmmm, I need to write a post on escaping Earth and other gravity wells.)

Slide1
To escape the Earth gravitational well, pick a direction and go at a speed larger than 11.2 km/s. You will get away with it!

Now that the spacecraft has escaped Earth, let’s switch frames of reference.  It may seem like 11.2 km/s (25,000 MPH!) is super fast and our spacecraft is definitely on its way to Mars.  Nope.  It has escaped Earth, but that just means that it is going around the sun with the Earth.  Take a look at the illustration below.  This shows that if a spacecraft ONLY escapes Earth, it will just orbit the sun with the Earth, staying in roughly the same position with respect to the Earth. Interestingly, the Earth is moving at 29,800 m/s, which is about 66,650 MPH.  So, our spacecraft went from moving 0 MPH with respect to the Earth to moving 66,650 MPH with respect to the sun.

Slide2
If the spacecraft escapes the Earth, but doesn’t do anything else, it will just orbit the sun next to the Earth.  Something more is needed!

In order to go towards Mars, you have to do what is called a Hohmann transfer.  This is where you go from having a roughly circular orbit to having an elliptical orbit, with one side of the ellipse being at Earth’s orbit, and the other side being at Mar’s orbit. In order to do this, you have to go to a higher orbit, which requires the spacecraft to accelerate and increase its velocity.  To get to Mars, the spacecraft has to speed up to 32,700 m/s, or 72,150 MPH.  This is a difference of 2,900 m/s or 6,500 MPH, so the spacecraft has to speed up by this amount in order to change trajectories towards Mars.

Slide3
In order to get on an Earth-Mars transfer orbit, the spacecraft has to increase its velocity once it escapes the Earth.

Time passes.  The worlds and our spacecraft move.  We don’t need to use any fuel at all, since the spacecraft is just coasting towards where Mars is supposed to be in a few months.Slide4

After about 8 months, the spacecraft arrives at Mars!  Yeah!

Slide5
The spacecraft arrives!

Now, if we do nothing, the spacecraft will continue to be on an elliptical orbit, and will fall back towards the Earth.  This is very bad, so we have to do something about it!

Slide6

Interestingly, in order to stay in Mars’s orbit, the spacecraft has to accelerate again.  This is because Mars’s orbit is above the elliptical orbit, so it demands an increase in velocity. Mars is moving at about 24,100 m/s or about 54,000 MPH. Our spacecraft, when it arrives at Mars, is moving at about 21,500 m/s or 48,100 MPH.  This means that the spacecraft has to speed up by about 2,600 m/s or 5,900 MPH.

Slide7

In order to not fall back toward Earth when the spacecraft arrives at Mars, it needs to speed up again!

Now, the spacecraft is in a very similar circumstance as when it was near the Earth.  If we do nothing, it will just orbit the Sun next to Mars.  We want to either have the spacecraft orbit Mars, or we want to have it land on Mars.  In order to do that, the spacecraft has to slow way down. If it orbits Mars, it has to slow down a fair bit, but less than if it were going to land.

The change in velocity needed to get into orbit is a bit complicated to figure out.  First we have to figure out the escape velocity of Mars.  The reason for this is that if the spacecraft is way far away from Mars and doesn’t do anything at all, it will smash into the surface at the escape velocity, which is 5,000 m/s or about 11,200 MPH.  To stop this from happening, the rocket has to slow down by this much (assuming that there is no atmosphere!)

Orbital velocity at 400 km above Mars’s surface is about 3,360 m/s or 7,500 MPH. From a long way away from Mars, the spacecraft can fall toward Mars, and slow down to about 3,360 m/s from the hypothetical escape velocity of 5,000 m/s.  That means that the spacecraft has to slow down by about 1,640 m/s or about 3,700 MPH. The spacecraft will then be in orbit around Mars.

To get to the surface (assuming that there is no atmosphere), the spacecraft has to slow down another 3,360 m/s or 7,500 MPH.

One of the cool things about spacecraft around Mars is that they often use aerobraking to change their orbit or slow down enough to land.  Aerobraking is where the spacecraft enters the upper atmosphere a tiny bit to experience drag and will slow down.  It is a time consuming, but very inexpensive way to slow down enough to get into Low Mars Orbit, or to land.  The spacecraft could save about 5,000 m/s by using aerobraking.  But it is complicated to do that.

Slide8
Now, when it is close to Mars, it has to slow way down in order to not be slammed into the planet.

In summary, the total delta-V that is needed to get to Mars, including escaping from Earth, getting into an elliptical orbit towards Mars, then getting out of the elliptical orbit near Mars, and landing on the surface (not using aerobraking!), is close to 22,000 m/s or about 49,000 MPH. This is a huge amount of delta-V, and almost all spacecraft end up using aerobreaking to save around 5,000 m/s.

In the following post, I will talk about how much fuel is needed in each of these steps and we can figure out how large of a spacecraft we can land on Mars.

Rockets

Max-Q and Why Does it Matter?

In a rocket launch, they almost always talk about Max-Q, which is the point in the launch sequence where the rocket experiences the maximum dynamic pressure. First, let’s talk about what that actually means.

The rocket, as it is going up into space is experiencing three forces: gravity (down), thrust (up), and atmospheric drag (down).  Gravity is probably obvious, since you are probably sitting in a chair or on the couch and are feeling its effects (well, you are probably feeling the normal force from the chair that is pushing up on your backside, but let’s just say that you are feeling gravity). Thrust is what the rocket does as it expels fuel to make it go up. I have a post on that.

I also have a post on drag and terminal velocity if you would like a refresher on what all of that means. But, as a tiny backstory, atmospheric drag is like friction that an object feels as it moves through a fluid. So, when you are riding a bike, you feel drag, which makes it so you have to pedal harder into a headwind than a tailwind. This force is proportional to the density of the medium (it is MUCH harder to ride a bike underwater than it is in the atmosphere!), and the velocity squared (when you go twice as fast, you have to pedal four times as hard).

Returning to the subject at hand:

When the countdown ends and the rocket starts to fire, but before it starts to move, there are two forces acting on the rocket: gravity and thrust.  Thrust has to be a bit bigger than gravity for the rocket to start moving, otherwise the rocket will just sit there.  When the rocket starts moving upwards, it gain speed, and starts to experience a drag force.

Now, if the rocket flew horizontally, or if the atmosphere extended upwards forever, the drag force would increase and increase and increase as the rocket got faster and faster and faster.  As it happens in reality, the rocket launches mostly in a vertical state, and the atmospheric density decreases quite rapidly with altitude.  By about 23 km altitude, the atmosphere has decreased down to about 10% of its density.  For reference, airplanes fly at about 1/2 of this altitude, so the density is about 33% of its sea-level value.

The situation with the rocket, then, is that it starts to accelerate and the drag force starts to grow dramatically, since the force is related to velocity squared:  whenever the speed doubles, the force quadruples.  But, at the same time, the density is decreasing, so the force is weakening because of this.  At some altitude, the decreasing density wins out over the increasing velocity, and the drag force starts to decrease.

It is this point – where the increasing velocity is balanced by the decreasing density of the atmosphere and the drag force starts to decrease that is called Max-Q.

Screen Shot 2018-06-24 at 4.27.21 PM
A plot of the drag on the rocket as a function of altitude (in meters). The drag force is negative at this stage, since it is towards the ground.  Max-Q happens around 11 km in altitude in this simulation.

Now the question is – why do people care?

Well, rockets are really unstable. The thrust is coming out the back end and they are extremely long and narrow.  The thrust vector has to be right through the center of mass of the rocket in order for it to not rotate.  If the thrust vector is off, it could easily start to roll over.  If thrust and gravity were the only forces, it would be a bit complicated to control, but when you add in aerodynamic forces, it becomes even more complicated.  Older rockets (and model rockets!) used to have fins in order to help with this.  The fins made it so that if the rocket tipped at all, the aerodynamic forces on the fins would help correct for the tipping and apply a restorative force back into a non-tipping orientation. See the picture below.  There are aerodynamic forces all along the rocket, but the fins have the largest forces, so those forces win, and the rocket will rotate around the center of gravity and restore back to vertical.  With no fins, this restorative force doesn’t exist, since the forces along the rocket are all roughly equal, which makes it so that the rocket may or may not rotate one way or the other around the center of gravity.  Because of this, the rocket motors have to solve the issue all on their own.

Slide1

So, the drag force makes is a bit harder to control the rocket. Most rockets use computers to figure out how the rocket is tilting and adjust the thrust vector of the engine to rotate the rocket back to vertical.

If the computer stopped working and the rocket couldn’t adjust back to vertical, what would the problem be? The problem is that the rockets are not exactly super rigid and made out of super-strong materials, since they are designed to be as lightweight as possible.  It is designed to fly like an upright pencil through the atmosphere.  If you were to turn the pencil over and try to shove the pencil in a sideways configuration through the atmosphere, the rocket would probably shred to bits.  This is very bad.  So, the rocket needs to be aimed upwards as much as possible.  Any deviation from this upright position, and the aerodynamic forces could rip the rocket apart.

Slide2
If the rocket tilts too far, the forces on both sides of the center of gravity with cause the rocket to break apart. That is not good.

Max-Q is the time during the rocket’s flight in which the aerodynamic (drag) forces are the strongest.  So, you really don’t want the rocket to tip or do anything wonky during this time.  After Max-Q, the force decreases quickly, and the engineers can relax a bit.

It is around Max-Q, where the rocket starts to tilt a bit and rotate towards the horizontal direction.  This is because rockets only go upwards to get to their correct altitude.  They really need to go horizontally at about 7.6 km/s (that is fast).  If the rocket were to go straight up to like 500 km, then tilt over, it wouldn’t work very well, since it takes a long time to accelerate up to 7.6 km/s. Turning down low allows the rocket to take a while to get up to 500 km and take a while to get up to 7.6 km/s speed.  The rocket times it just right so that both are met at about the same time.  The location of Max-Q is where this tilting starts.

 

 

 

 

Satellites, Space Stations

It Fell From the Sky

Recently, a news article talked about China’s “Failed” Space Station (Tiangong-1) re-entering Earth’s atmosphere.  There are several aspects of this that are interesting:

  1. China launched a Space Station!  That is sort of cool and crazy.  The article calls it a “failed” space station, but really, it was just about as successful as any country’s first space station. Really, only three countries have launched space stations, besides the International Space Station: Russia, USA, and now China.  All of them had problems with their first stations. But, this post isn’t about space stations.  I will write about those some other time!
  2. The space station is going to re-enter Earth’s atmosphere, as most objects in low Earth orbit do.
  3. Researchers don’t know exactly when it will re-enter the atmosphere.  They gave an estimate of around April 2nd, with a two-week window size.  It seems like this is a really large window size.
  4. The article discusses that the space station could land in the United States! Should you run for the hills and hide for the entire two-week period that it could re-enter the atmosphere?
Tiangong-1
Tiangong-1 Space Station

First, let’s talk about how many objects there are in low Earth orbit.  There are a LOT – about 22,000 that are larger than a softball.  There are even more that are smaller than this size, but we don’t really have the ability to track those objects.  Very few of these objects are operational satellites, like less than 1,000.  You can find the most objects at an altitude around 700 km.  We put a lot of satellites in this range for pretty much the same reason that you find a lot of stuff there – the atmosphere above about 700 km is super weak, so that the drag is extremely low.  Objects at 700 km altitude will take well over 50 years to be pulled back down into the atmosphere.  This means that anything put up there will stay for a really long time.

 

Objects below about 500 km altitude will re-enter the atmosphere within about 10 years.  Anything put there or lower is just sort of swept into the atmosphere relatively quickly. This is why there is not much stuff at these altitudes – it all re-enters the atmosphere.

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Altitude distribution of objects in low Earth orbit around the Earth. NASA Orbital Debris Program Office (ODPO).

About one object re-enters the atmosphere every day.  Most of these objects are really small and burn up completely.  Others are very big and make it through the re-entry process and land on Earth.  Things like first and second stages of rockets are examples of relatively large objects that don’t always burn up in the atmosphere.

The Department of Defense (DoD) tracks all of these objects, and attempts to predict exactly when and where they will enter the atmosphere.  This is really difficult to do.  First, the orientation of the objects are not really well known, and they could be tumbling. Therefore, it is really hard to predict the area that they are presenting to the incoming air, so the drag is difficult to calculate.  Also, the objects may have lots of protrusions, like antennas and solar panels.  If the satellite is tumbling, the area could change dramatically.  Or, if there is enough force to rip the panels off, then the area could change quite suddenly and stay at a lower value.  Think about driving a minivan down the road with a mattress strapped onto the top.  The minivan feels a lot of drag while the mattress is attached, but suddenly feels quite unburdened when the mattress flies off onto the cars behind it.  The same happens when satellites or space stations are unburdened of their solar panels as they enter the atmosphere.  Just like a minivan with a mattress strapped onto the top, it is difficult to predict if and when this unburdening event may happen, so determining the exact drag for the last few weeks of the object’s life is quite difficult.

Another thing that adds to the difficulty of predicting the drag is that there are a lot of aspects of the atmosphere between about 100 and 150 km that we don’t really understand that well.  For example, in this region, the temperature goes from being the coldest part of the atmosphere (about -75°C at 100 km) to the hottest part of the atmosphere (about +700°C at about 200 km).  That exact transition is not well understood.  Part of the reason is that it is impossible for a satellite to survive there and take measurements.  It is also impossible for an airplane to fly there, or a balloon to ascend to there.  Really, the only way to take measurements is either with rockets (so about 10-20 measurements a year at most), or through remote sensing, which has a lot of assumptions.

One of the great ironies in NASA is that when the Upper Atmosphere Research Satellite (UARS) was going to re-enter the atmosphere, NASA researchers also had a window of a couple of weeks.  One would think that given the name of the satellite, we would be able to specify the atmosphere well enough to predict when it was going to de-orbit!  But, not so much.  It is a hard problem!  Another interesting thing about UARS is that it actually re-entered the atmosphere over the United States! UARS was about the size of a school bus, so a lot of pieces may not have burned up in the atmosphere and may have made it all the way to the ground. No one was hurt.

NASA has come computers codes that you can run that will predict what will make it to the ground.  You actually have to runs these codes before you can get permission to launch a satellite. Most stuff like aluminum and plastics burn up,  but things like tungsten and other really dense metals may not.  You can then predict how fast the objects will hit the ground by estimating the surface area and mass, and predicting the terminal velocity.

Given how much stuff has re-entered the atmosphere, why has no one ever died due to getting hit my something? Well, the surface of the Earth is really, really large.  One website estimated that the percentage of area that we humans have covered with artificial surfaces is about 0.6%.  That is not much.  If the objects were falling randomly over the Earth, then one might expect about 2 objects a year to re-enter the atmosphere over a populated area.  Since the vast majority of objects burn up in the atmosphere, there is not too much to worry about.

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A rocket stage that made it back to Earth. And not the way Space-X get’s their stages back.

There have been objects like meteors that have recently entered the atmosphere over populated areas.  For example, on January 16, 2018, a meteor landed just north of the University of Michigan.  This meteor was large enough to cause a huge boom and could be seen by thousands of people.  Still, no one was hurt. Even in a relatively dense population center, there is a lot of empty land.

 

So, there is not very much chance that when the Chinese Space Station re-enters the atmosphere and some of the bits make it back to Earth, they will land on anyone or cause much damage.  Most likely the pieces will land in the ocean or over land where there are not too many humans.  But, I guess there is always a first time.

Engines, Rockets

Why the Falcon Heavy Makes Good Business Sense

This week, Space-X had a test launch of the Falcon Heavy, the largest rocket ever launched besides the Saturn-5. It was a great success, putting a Tesla Roadster into trans-Martian orbit. Space-X has taken a fantastic first step. If you have not seen the launch, you should definitely watch it – it is fantastic.

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Falcon Heavy at liftoff (Elon Musk via SpaceNews).

While the Falcon Heavy is very cool and a great achievement, it is also a good business investment.  There are two reasons for this.

The first reason is that the Falcon Heavy is a good business decision is that the Falcon Heavy is simply three Falcon-9s strapped together. Other rocket companies have different lines for different weight classes.  For example, Orbital/ATK has the Pegasus rocket, which last launched the eight CYGNSS satellites.  The next mission that will use the Pegasus is ICON, which will launch later this year.  That is about 1.5 years between launches.  The Pegasus launches off of an airplane, which is super cool.  The next size up from the Pegasus that Orbital makes is the Minotaur-C, which is capable of of carrying about four times as much mass to orbit as the Pegasus.  The Minotaur-C uses some of the same components as the Pegasus, but they are pretty different.  This means that Orbital needs to keep up two lines of production, which is quite difficult and costly.

The Falcon-9 uses 9 Merlin 1D engines in the first stage and 1 Merlin 1D (Vacuum) engine in the second stage. So, it uses essentially 10 of the same engines.  The Falcon Heavy uses 3 Falcon-9 first stages for a total of 27 Merlin engines in the first stage, with a second stage that is identical to the Falcon-9 second stage, for a total of 28 Merlin engines.  The engines and first stages can be used as Falcon-9s or Falcon Heavies.  This makes the production costs much lower, since they don’t have to maintain different manufacturing lines for different rockets.

What would be super awesome is if they could have a rocket that used one Merlin engine (a Falcon-1 rocket, which used to exist) that could compete with the super small rockets, like the Pegasus or the Electron (as discussed in a previous post).  But, Space-X made a choice that their smallest rocket would be the Falcon-9, which was a good decision, since the Falcon-1 could not really be expanded, like the Falcon-9 to the Falcon Heavy.

(Side Note: The Falcon Heavy is a relatively small rocket compared to the BFR that Space-X is planning for going to Mars.  There are plans for more, extremely large, rockets. The BFR is going to be a fundamentally different design than the Falcon Heavy, which is somewhat sad, since I just wrote a bunch of words above about how awesome it is that Space-X are combining the same rockets to get bigger rockets. I have no real idea how much development has gone into the BFR yet.  I will find out and get back to you!)

9m_BFR_vs_12m_ITS_vs_NG_vs_SLS
The future of rockets compared to the Saturn V. The Falcon Heavy is the biggest rocket available today, but it is very small compared to what is needed to go to the moon or Mars with humans. (https://en.wikipedia.org/wiki/BFR_(rocket))

The other thing that Space-X has done is to make the Falcon-9s reusable. The fact that they can fly back down and land makes them very valuable.  While the reusability of the rocket engines, and the number of times that they can be reused, is still questionable, it is quite certain that they will get there and the engines will be able to be used many times.

The other reusable space vehicle has been the Space Shuttle.  The problem with the shuttle was that it was extremely costly to reuse it – about $500,000,000 to launch the shuttle. (I will write a post on this soon!) The Space-X rockets are fundamentally different things.  They do not have a super complicated heat shield, or relatively complicated solid rocket boosters.

Interestingly, the primary reason that the Falcon-9 lands the way that it does, standing up using thrust, is that this is the way that it would have to land on Mars. Because Mars has such a weak atmosphere, it is very hard to land with parachutes or with wings, like the shuttle. Space-X therefore designed the Falcon-9 so that it could land vertically.  Not that there will be any Falcon-9s on Mars, but the vertical landings are great tests of the technology so that when rockets do land on Mars, they will have undergone significant real-world tests. That is in addition to making the rockets reusable, which drives the price down significantly.

In summary, Space-X is making really good business decisions in its Falcon-9 and Falcon Heavy lines: using the same engines and the same structures is really smart, and making the rockets reusable is genius.

Satellites

Two Reasons Why the Humanity Star is Not a Complete Waste of Time

If you have not seen this, you should look at this web page that describes The Humanity Star.  It is basically a nearly spherical object that was launched into space in January of 2018.  It has 65 reflective surfaces that will reflect sunlight while it is in orbit. The general idea is that whenever it is in the sun, it will be so bright in the sky that you can see it.

Normal satellites can be seen in orbit around the Earth from the ground.  What happens is that when it is dark on the ground, but still light at orbital altitudes (around 250 miles high and above), sunlight can reflect off the satellite and it can look like a star in the sky.  This happens just after sunset or just before sunrise.  If you are very patient and look up at the sky during these times (preferably from an outdoor hot tub), you can sometimes see objects that look like stars that are moving from south to north or north to south. To give you an idea, it should take them about 10 minutes to go from horizon to horizon.

The Humanity Star is so bright that it should be be visible during the day.  The web page talks about how this will be a beacon to draw humanity back together and to make them look to the stars.  I personally don’t think that a tiny star-like object in the sky will bring humanity back together unless the star-like object is getting bigger and bigger every day and has the potential to wipe out humanity.  Other people that I have talked to have a similar feeling, and so it seems like The Humanity Star really has no real value. Except it does. There are a few good reasons, some intended, and others maybe not.

Humanity-Star-2
The Humanity Star. It is not in orbit in this picture. It is sitting on the ground. (From the website).

The true purpose of The Humanity Star was really to test the Electron rocket by Rocket Lab. This was the first flight of the Electron.  While Space-X just launched the Falcon Heavy, Rocket Lab launched a small rocket that can take only 150-225 kilograms to orbit for an estimated price of $5M.  This is a huge deal because constellation missions would like to spread out satellites.  It is incredibly difficult to truly distribute a constellation of satellites from a single launch vehicle (rocket).  If you could buy 8 tiny rockets that could take one or two satellites to orbit for the price of one medium sized rocket that could take 8 satellites, it would allow you to distribute the satellites immediately.

When you test a rocket for the first time, the probability of failure is quite high (like, explosively high).  Some companies give a special deal to satellite companies to launch their satellite on a very risky rocket launch. If it blows up, then everyone loses, but they are not out a huge amount of money.  If it doesn’t blow up, everyone wins – the rocket is proven to work, and the satellite gets to orbit for cheap. Other companies just launch dummy payloads in order to prove that the rocket works.  If it works, then they have a proven rocket.  If it doesn’t, no one is harmed.  This path doesn’t make the company any money (if the rocket works), but also doesn’t make people really angry (if it doesn’t work).

The Humanity Star was a dummy payload for the first test launch of the Electron rocket.  This is similar to Space-X launching a Tesla on the Falcon Heavy (another dummy load with an actual dummy in the driver’s seat). Instead of just saying that it was a test load, Rocket Lab made a big deal about The Humanity Star instead of talking about their super cool and super small rocket.

The second interesting thing about The Humanity Star is that it can actually be used to do science, even though it has no power or sensors or anything. The Air Force has many dummy spheres like this in orbit. The reason for this is that all objects in low Earth orbit feel atmospheric drag.  Since the projected area of a nearly spherical object is known exactly and basically never changes (since it looks exactly the same from every angle) the only change in the drag force that the object feels is due to changes in the atmospheric density. Normal satellites are strange shapes and have lots of protrusions, like antennas and such.  If the orientation of the satellite changes, the drag changes. It is often extremely difficult to model this behavior accurately.  So, simple spheres are used and are tracked with radars from the ground.

The Humanity Star will allow us to more accurately track the thermospheric density since it is really big (about 1 meter across) and pretty light (about 8 kg).  Its area to mass ratio means that the drag that it feels will be pretty big, so it will reenter the atmosphere pretty quickly (less than a year). Because it feels such a large drag, the drag force will be easy to determine and any changes will be caused by only by changes in the thermospheric density.  This is the type of research that I do!

Another really minor thing about The Humanity Star is that because it can be visible from just before sunrise to just after sunset, including the whole day, it could be used for educational purposes.  You see, a satellite’s orbit can be determined just by tracking how it moves across the sky.  If you point a telescope at the satellite in the sky and mark down the direction that the telescope is pointed, and do this over and over again as the satellite moves across the sky, the math is relatively easy to do to determine the orbit (well, students do this in Junior-level Aerospace Engineering classes). This is a great real life example that students could use to put their education to use! In the daylight!

Hopefully this has convinced you that The Humanity Star is not a complete waste of time and money!