Lagrange points

On 25 December 2021, the James Webb Space Telescope (JWST) was successfully launched. It has now reached its destination at the L2 Lagrange point of the Sun-Earth system. For many years I have considered writing a blog about the five L:agrange points, but I was not sure if I could do that in a relatively simple way.

I am still not sure, but in this blog I will give it a try.

Here is a diagram of the Sun-Earth system (not to scale). The five Lagrange points are marked.

Earth and all other planets orbit the Sun because of the gravitational attraction between a planet and the Sun. Earth orbits the Sun in ~365 days at a distance of 150 million km. The other planets do the same, but at different distances and with different periods. Here is the solar system (not to scale).

It was Kepler who studied the planetary motion. He found a relation between the period and the distance, which is now called Kepler’s Third Law: The square of a planetary period is proportional to the third power of its distance. Let’s take Mars as an example. The distance to the Sun is ~228 million km and a Mars year is ~687 days. The distance is a factor 228/150 = 1.52 larger. The third power of 1.52 is 1.52×1.52×1.52 = 3.512. Kepler’s 3rd law predicts that a Mars year will be √ 3,512 = 1.88 times longer than an Earth year. = 1,88 x 365 = 686 days.

Now let us consider a spacecraft in the Sun-Earth system. It’s mass is so small compared to the mass of Sun and Earth, that it will not influence their motion. But it will feel the gravitational attraction from the Sun and also from the Earth. Is it possible that the combined attraction of Sun and Earth will result in a period of 1 year?

The answer is yes, there are exactly 5 points where this is the case, the 5 Lagrange points!

Here is the explanation for L2. This point lies farther away from the Sun than Earth, so the attraction from the Sun is weaker and would result in a longer period. But Earth also attracts the spacecraft in the same direction as the Sun and in L2 they give together enough attraction to let this point orbit in 1 year. Calculation gives that L2 is located 1.5 million km from Earth, 151.5 million km from the Sun

For L1 the explanation is similar. Here the attraction of the Sun is stronger resulting in a shorter period. But now Earth “pulls back” and together they give the right amount of attraction. The location of L1 is also 1.5 million km from Earth, 148.5 million km from the Sun (the figure above is not to scale).

L3 lies at the opposite side of the Sun, Here the attraction from Earth is minimal, it contributes only little to the attraction of the Sun, so L3 lies only slightly further away than 150 million km from the Sun.

Before we describe the points L4 and L5, we will first look in a bit more detail at the solar system. When we say that the planets orbit the Sun, it suggests that the Sun doesn’t move itself, while the planets orbit around it. And that is not true. The Sun and a planet both orbit around their common center of mass, often called their barycenter. In this image the barycenter is shown for the Sun and Jupiter. Because the Sun is much more massive than Jupiter, their barycenter lies close to the Sun.

Here are a few animations for different situations, where the barycenter is marked with a red cross. The animations are not to scale. The first image shows the situation of for example two stars of equal mass. The next one shows minor planet Pluto and its large moon Charon. The last image shows Earth and Sun. The mass of Earth is so small that the barycenter lies within the Sun.

Of course the resulting force in the Lagrange points has to be directed to the barycenter and for L1, L2 and L3 this is automatically the case, because these points lie all three on the line connecting Sun and Earth. These points were already found by the famous mathematician Euler in 1720.

In 1772 Lagrange discovered two more “stable” points, where the attraction of Sun and Earth are not in the same direction, but together point to the barycenter. of the Sun-Earth system. .

The mathematics is complicated, I will use some hand waving to make the existence of L4 (and L5) plausible. In the diagram below, the masses of Sun and Earth are S and E , the barycenter is indicated as b, it lies within the Sun because the Sun is much more massive than the Earth. The location of b depends on the ratio of the two masses S and E;.

L4 is the top of a triangle with all sides equal to the distance between Earth and Sun. Because L4 has an equal distance to Earth and Sun, the gravitational forces on L4 are in the same ratio of S and E. Therefore the resulting force is directed to b ! Note that L4 lies outside Earth’s orbit. Similar to L1, the two combined forces give L4 a period of 1 year, same as Earth.

Actually the barycenter of the Sun-Earth system lies extremely close to the Sun’s center of mass, The radius of the Sun is 670.000km and b lies about 450 km from its center! In this diagram this distance has been strongly exaggerated to show the process. In the usual diagrams of the Lagrange points, L4 and L5 are located so close to the Earth orbit, that it is not possible to see their separation.

Until now we have described the 5 Lagrange points as points that orbit the Sun in one year, same as the Earth. Another description is often used, a rotating coordinate system. In such a coordinate system, centered in the barycenter and rotating once a year, Sun, Earth and the 5 Lagrange points are stationary. But it comes at a cost. Because such a coordinate system is not an inertial system, fictitious forces have to be introduced, for example the centrifugal force,

In the diagram below the Lagrange points are indicated, in such a rotating frame. The contour lines give the gravitational field energy. Compare it with the contour lines on a topo map. The blue and red arrows indicate the direction of the force (the direction of the slope in a topo map). In topo map terminology L4 and L5 are located on the top a hill, while the other three are located in so-called saddle points. On first sight it would seem that all Lagrange points are unstable, For the L1-L3 points a small displacement in the x-direction, and for L4 and L5 a small displacement in any direction would be enough to disturb the balance (like a pencil on its tip).

Careful and complicated mathematical analysis (see for example here) leads to a surprising result: the regions around L4 and L5 are actually stable, objects in a large region around these Lagrange points will move in orbits and stay in that region. The regions around the other three Lagrange points are unstable, objects can orbit for a while, but will eventually escape. That is illustrated in the two diagrams below. The left diagram. shows the Sun-Earth system in an inertial frame, the right one in a rotating frame The 5 Lagrange points are marked in red.

Notice the moving tiny points, They are test masses. released near the various Lagrange punts. Look carefully and you will see that the test masses released near L1 and L2 quickly move away. For L3 it takes a bit longer. All these three Lagrange points are unstable. But around L4 and L5 the test masses do not “escape”, these points are stable.

In the introduction of this blog I wrote that the JWST had reached its destination at the L2 Lagrange point of the Sun-Earth system. Actually the space telescope is not positioned in :the Lagrange point itself but orbiting L2. And what an orbit it is! Elliptical, the distance to L2 varies between 250.000 km and 832.000 km. One period takes about 6 months. The orbit is not stable, about every 21 days the thrusters of the JWST must perform minor course corrections.

A more detailed explanation of the WEBB launch and orbit can be found in this brilliant YouTube video: How James Webb Orbits “Nothing”

There also satellites orbiting L1. At the moment for example the SOHO satellite to study the Sun and the DSOVR to study the Earth. Here are two pictures taken by these two spacecraft.

In 1978 the International Sun-Earth Explorer-3 (ISEE-3). was the first spacecraft that went into an orbit around a Lagrange point. It studied the Sun and Earth for 4 years and also here the unstable orbit had to be corrected regularly. Here is a diagram of the launch process.

After its mission was completed, the spacecraft got a new target, to study comets! It was renamed International Cometary Explorer, left its orbit and via amazingly complicated manoeuvres went on its way to a comet., Click on the screenshot to see an animation of the mission. Very informative and fascinating..

What about L3? This Lagrange point is permanently behind the Sun, as seen from the Earth. No scientific use, but it has played a role in science fiction. . Here is an example, a science fiction movie Journey to the Far Side of the Sun, released in 1969 (the same year that humans landed on the Moon). Click on the screenshot to watch the movie.

Synopsis of the movie: In 2069 a planet is discovered in L3 and the director of Eurosec (named Jason Webb !) organises a mission to what turns out to be a mirror-earth. Very interesting to watch.

We now know that L3 is unstable, with a “decay time” of about 150 year. It would be a suitable location for alien enemies to hide, while preparing for an attack 😉

L4 and L5 are stable (under certain conditions) but have no use for science. Possibly in the far future, these regions could be used to build human colonies.

Until here we have concentrated on the Lagrange points of the Sun-Earth system, but the Earth-Moon system has also its Lagrange points and so do for example the Sun and Jupiter.

Jupiter has collected thousands of asteroids around its L4 and L5 points. They are called trojans because they are named after heros of the Trojan war. Here is an animation. The asteroids in front of Jupiter are called the Greeks and the ones trailing Jupiter are called the Trojans.

The name trojan is now generally used for objects in the L4 and L5 points of other planets. In the L4 and L5 points of the Earth until now “only” two Earth trojans. have been observed.

But there may have been one in the early history of Earth!. I will end this blog with a fascinating theory about the origin of the Moon! The theory is called the Giant-Impact Hypothesis. When the Sun and the planets were born, about 4.5 billion year ago, Earth was not alone. It had a Mars-sized sister planet in the L4 (or L5) Lagrange point. About 20-30 million year later, this hypothetical planet, named Theia, possibly disturbed by the other planets, left the L4 region and collided with Earth. It must have been a cataclysmic event From this collision the Moon was born.. Here is the scenario.

And a visualisation

Here is the Wikipedia List of Objects at Lagrange Points

All the images are taken from the Internet, many from Wikipedia.

The DART mission

Two years ago I published a detailed blog post: Will an asteroid hit Earth? In that post I discussed the scenario that an asteroid had been discovered on a collision course with Earth and what could be done to avoid such a possibly catastrophic collision. One option is to send a spacecraft to the asteroid and let it crash with it. The impact should change the course of the asteroid, so it would no longer hit Earth.. The DART mission will test the feasibility of this “kinetic impactor” technique. DART will be launched on 24 November, so it is time for an update.

The acronym DART stands for Double Asteroid Redirection Test. Target for DART is the minor asteroid Didymos, discovered in 1996. It has a diameter of 780 meter and orbits the sun in 2.11 year. In 2003 it was discovered that Didymos has a small moon with a diameter of 160 meter. This moon has been named Dimorphos , it orbits Didymos in about 12 hour at a distance of 1.2 km. DART will crash into this moon at a speed of 6.6 km/s. and change its orbit slightly. In the infographic this change is hugely exaggerated. It is estimated that the crash will change the speed of Dimorphos only about 0,4 mm/s and its orbital period about 10 minutes

Originally DART was part of the much more ambitious AIDA mission. The crash will take place at about 11 million km from Earth. How to observe the effects of the crash? The solution was to launch another spacecraft earlier than DART, which would reach Didymus and go into orbit around the asteroid. This AIM spacecraft , to be developed by the European Space Agency (ESA), would observe the crash and send data back to Earth. It would even deploy a small lander, MASCOT2 to study the properties of Dimorphos.

But in December 2016, AIM was cancelled by ESA, after Germany withdrew the 60 million Euro funding for the project. I commented in the above mentioned blog:

As an European I feel rather ashamed that Europe has acted this way.

NASA decided to continue with DART., which will be launched by a SpaceX Falcon 9 rocket. A fascinating feature from the Falcon 9 is that part of it (the first stage) will return to Earth, land vertically (!) and can be used again for other missions. It will land on a so-called drone ship, an unmanned platform in the ocean. There are three of these drone ships active at the moment, all with poetic names. The Falcon 9 will land on “Of Course I Still Love You” Here is the ship.

And here is a video of the take off and landing. You must see it to believe it ;-).

DART will arrive at the asteroid end of September 2022. The spacecraft will use autonomous navigation to point itself to the moon. It has a camera on board, the DRACO that takes high-resolution photos. On-board software will analyse these photos, be able to distinguish between Dimorphos and Didymos and point Dart to Dimorphos.

About 10 days before reaching its destination, DART will deploy a tiny spacecraft, a so-called CubeSat . This LICIACube has been developed by ASI, the Italian Space Agency and will take pictures of the crash. So at least images of the collision will be sent to Earth.

Here is a short YouTube video of the DART mission. I will point out a few details.

  • 0:07 The nose cone of the Falcon 9 opens to deploy DART
  • 0:15 the solar arrays are unrolled, a new technique. Each one is 8,5 m ;long
  • 0:22 The lens cover of DRACO opens
  • 0:26 Didymos in the center, Dimorphos to the right
  • 0:32 The orbits of Earth and Didymos. They comes close, but are still 11 million km away from each other when DART crashes.
  • 0:37 The Xenon thruster will steer the spacecraft
  • 0:41 The LICIACube is deployed
  • 0:54 DRACO will find the target
  • 0:58 Found the target
  • 1:02 On collision course
  • 1:04 The end of DART

My next update about DART will probably be in October next year.

Perseverance

Are you staying up tonight?, a friend asked me a few weeks ago. No, why? , I replied. He knows about my interest in space travel and expected that I was aware of the landing of a spacecraft on Mars that night. But I was not 😉

I checked the timing, the Perseverance would land at 4:55 am in the morning of 19 February (Malaysian time). In this blog I will explain why I decided to enjoy my sleep and check the next morning if the landing had been successful 😉

In 2018 I wrote a blog Landing on Mars, in which I described the various Mars missions, concentrating on the Curiosity Mission of 2012. The procedure to land the Curiosity was new, using a so-called sky-crane for the last phase.

Here is a diagram of what is called the Entry, Descent and Landing (EDL) process. The spacecraft enters the (thin) Martian atmosphere with a velocity of ~ 20.000 km/h. About 7 minutes later it must land on the surface with a velocity of less than 1 m/s. As signals between Earth and Mars take about 11 minutes, EDL can not be controlled from Earth, the whole process must have been programmed in the computers on board. Mission Control can only wait and see. That’s why these 7 minutes have been called the seven minutes of terror.

In my 2018 blog I describe the three phases in more detail, here is an very informative animation.

In 2012 everything went well, the Curiosity is actually still operational at the moment, much longer than originally planned.

The Perseverance that landed last week, has followed the same EDL procedure. Of course it must have been a relief for Mission Control that it was again a smooth process, but to keep calling it seven minutes of terror is exaggerated. That’s why I decided to enjoy a good night’s sleep. Here is the EDL process for the Perseverance. As you see it is basically the same as for Curiosity.

The two rovers also look the same. To the left Curiosity, the official name of the mission is Mars Science Laboratory (MSL). The Perseverance, to the right, is part of the Mission 2020 project.

Of course there are differences. The wheels have been redesigned, the robotic arm is heavier and the rover carries more cameras, 23 in total. Notice the “hazcams” at the front and the back of the rover, to avoid obstacles. Sherloc, Watson and Pixl are science cameras, I will tell a bit more about them later.

Some of the cameras have not a real science function, but have been added mainly to please the general public 🙂 . The back shell has a camera looking up to see how the parachute deploys. The camera of the sky-crane is looking down and can follow how the rover is being lowered to the ground. And the rover has a camera looking upwards to see the sky-crane. And a camera looking downward to the ground. That one is important, the spacecraft has a digital map of the surface and uses the camera images with a lot of AI to steer to the right location.

Keep in mind that all these images can only be transmitted back to Earth, after the spacecraft has landed. During the EDL, Mission Control only receives telemetry signals (altitude, speed etc). NASA has published a spectacular video where those messages are combined with the camera images. This is a YouTube video your really should watch (several times!). No wonder that this video has already been viewed more than 14 million times.

This map of Mars gives the location of the NASA missions. Insight and Curiosity are still operational. For a list of all Mars missions, click here.

Perseverance has landed in the Jezero crater. In this picture, taken by the Mars Express orbiter, I have marked the landing location with a cross.

This amazing photo has been taken by the Mars Reconnaissance Orbiter, one day after the landing. During the EDL the heat shield, the parachute and the sky-crane (descent stage) have to be jettisoned away from the rover. When you enlarge the picture above, you can just see the two small craters.

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Perseverance’s mission

Now that Perseverance has landed successfully on the Red Planet, what is it going to do? The missions of Curiosity and Perseverance are basically the same, to determine whether Mars ever was, or is, habitable to microbial life.

When Mars was a young planet, billions of years ago, water was abundant, there were lakes and rivers, similar to young Earth. On Earth life started about 3,5 billion years ago in the form of microbes. Fossil remains of these microbial colonies are called stromatolites. Here is an example, found in Australia, ~ 3.4 billion year old.

Could primitive microbial life have started on Mars in a similar way? Curiosity landed in the Gale crater, created about 3.7 billion year ago by a gigantic meteor impact. The crater became a lake, rivers deposited sediments. Curiosity collected surface material with its robotic arm, pulverised and heated it, before using a variety of analysing tools. Many organic molecules were found, for example thiophenes. which, on Earth at least, are primarily a result of biological processes.

Mars Mission with the Perseverance will continue this research with advanced technology.

Here is an artist impression how the Jezero crater may have looked like, when it was filled with water. Notice the river, top left, flowing into the lake. That river deposited a lot of sediments in the lake and it is near these sediments that Perseverance has landed.

A detailed map of the landing region, with the various geological structures in different colors. The “valley” of the former river and the delta are clearly visible The location of the rover again marked with a cross. The scientists have already made a proposal how the rover will explore the region (yellow line). The mission will take at least one Mars year (687 Earth days). If you want more information why the Jezero crater was chosen, click here.

The robotic arm has three scientific instruments, the PIXL, SHERLOC and WATSON. PIXL stands for Planetary Instrument for X-ray Lithochemistry . SHERLOC is an acronym for Scanning Habitable Environments with Raman & Luminescence for Organics & Chemicals and WATSON represents a Wide Angle Topographic Sensor for Operations and eNgineering . Engineering sense of humor.

PIXL is the main instrument. It points a very narrow X-ray beam at a piece of rock and detects the reflected light (fluorescence ), which is characteristic for the chemical elements in the rock. By analysing this reflected light, PIXL hopes to find biosignatures. Here is an artist impression of PIXL in action.

SHERLOC searches for organics and minerals that have been altered by watery environments and may be signs of past microbial life . Its helper Watson will take close-up images of rock grains and surface textures.

Suppose that Perseverance finds promising locations during its traveling. Wouldn’t it be wonderful if scientists on Earth could study the material at these locations in greater detail?

Well, that is exactly the most ambitious part of the Mars 2020 project, to bring back rock and regolith back to Earth. When you are a follower of my blog, you may remember that Hayabusa2 has brought back material from the asteroid Ryugu, and of course moon rocks have been brought back. But never yet material from a planet.

The robotic arm of Perseverance contains a drill, which can collect core samples. Here it is ready to start drilling.

The core sample (comparable in size with a piece of chalk) is put in a sample tube and taken over to the body of the rover where a few measurements are made. Then it is hermetically sealed to avoid any contamination, and temporally stored in a cache container. The container has space for 43 tubes. Here is an example of a sample tube.

Watch the video to follow the complicated process. Three robotics arms are used!.

How to get these sealed tubes back to Earth? NASA and ESA (the European equivalent will work together in what at first sight looks almost like science fiction. Actually it is still partly fiction at the moment! Here is the plan.

In July 2026 (!) a spacecraft will be launched, consisting of a lander and a rover. In August 2028 it will land near the Perseverance.

Here the spacecraft has landed on the surface of Mars, the rover still has to be deployed.

The only function of the rover is to fetch the sample tubes and bring them back to the lander. In this artist impression it is suggested that the sample tubes are scattered around, but that doesn’t make sense to me. Probably Perseverance will have created a few depots, or even kept all tubes in its own storage. The various descriptions I have found on the Internet, are not clear about this. The whole Return Mission is very much work in progress.

Here the tubes are handed over by the “fetch rover” to the lander, where they are put in the Sample Return Container.

The Sample Return Container might look like this. It will be designed so that the temperature of the samples will be less than 30 degrees Celsius.

The container will be loaded in a rocket, the Mars Ascent Vehicle, which will be launched in spring 2029.

The rocket will bring the container in a low Mars orbit and release it there..

In the meantime In October 2026 the Earth Return Orbiter has been launched, it will arrive at Mars in 2027 and lower its orbit gradually to reach the desired altitude in July 2028. There it will wait to pick up the container.

After the Earth Return Orbiter has caught the container, it will “pack” it in the Sample Return Capsule (SRC) and then go back to Earth, where it will arrive in 2031, ten years from now. It is this SRC that will will be released and finally land on Earth.

Here is a simulation of the procedure.

The primary mission of Mars2020 is to determine if Mars was habitable in the past. But there are also secondary missions. On board of the Perseverance there is one experiment, called MOXIE, that will produce oxygen from the carbon dioxide in the Martian atmosphere. Just a proof of concept experiment, important for future human missions to Mars.

Quite spectacular is that Perseverance is bringing a small helicopter, the Ingenuity. The Mars atmosphere is thin, but the helicopter should be able to fly. A bit similar to a drone, flying a few meter high, and maximum 50 m away. At the moment it is still hanging under Perseverance, planning is to test it after a few months. Here is an animation

At the moment Perseverance is testing al its components. It has made its first test drive, only a few meters. Here is a picture, you can clearly see the tyre tracks.

If there is more news about the Mars2020 mission, I will update this blog or write a new one.

Let me end this blog with an animation created in 1988 (!) , describing a Sample Return Mission to Mars. Fascinating to watch.