by William Stride – independent author
The beauty and functionality of a solar system is a wonderful thing to behold, with myriad planets, asteroids, and comets in graceful orbits around their home star. The formation of a solar system begins with a nebula collapsing under its own weight and contracting into its center where a star forms. Rotating around the central luminary is a disk of dust and gas called the protoplanetary disk. The material in the protoplanetary disk begins as gas and dust and then coalesces into grains, small pebbles, planetesimals, protoplanets and finally planets.
The physics of planet formation borrows from Classical Mechanics, Gravity, Thermodynamics, Chemistry, Geology and other aspects of the physical sciences all working together to shape and form this cloud of gas and dust into a complete solar system.
We will address some key points of solar system formation and the use of computers to simulate the growth of planets.
Nebular Collapse, Defining a Planet, and Solar System Geometry:
We begin our simulation with a nebula which is a cloud of dust and gas about four light years in diameter. All parts of the nebula are attracted to its center of mass, and it begins to collapse in on itself. As gravitational potential energy is converted into kinetic energy, the nebula’s temperature increases. And as the nebula collapses in on itself, it begins to spin faster and faster and flattens into a disk with a star forming at the center. We now have a protostar with a protoplanetary disk around it.(see Fig. 1).
Figure 1a. Schematic cartoon of a spinning cloud that collapses into a protostar surrounded by a protoplanetary disk. (Credit: W. H. Waller in An Earthling’s Guide to the Solar System [2024], Cambridge Scholars Publishing)
Figure 1b: The HL Tauri protoplanetary system as imaged by the Atacama Large Millimeter/submillimeter Array (ALMA). The system contains the central protostar, HL_Tau, and a surrounding protoplanetary disk of glowing dust and gas. The annular gaps in the disk are thought to trace the paths of protoplanets that have cleared out their orbits. (Credit: ALMA [ESO/NAOJ/NRAO])
The question arises what constitutes a planet? A planet is defined as a self-gravitating body orbiting a star. The planet can contain solids, liquids, gases, or mixes of these three phases. It must have enough surface gravity to assume a spherical shape except for a slight equatorial bulge due to spin. Also, the planet’s “feeding zone” must be cleared of all gas, dust and other objects. As the “feeding” subsides, the change in the planet’s mass over time becomes zero. In this static state we have a finished planet!
If the Sun were scaled down to the size of a basketball, the Earth would be the size of a peppercorn 89 feet away. When one envisions this arrangement, they can immediately see that there is a great deal of “empty space” in our Solar System.
To carry this line of thought further, if the Sun were scaled down to the size of a softball, North America would contain 3-4 such suns, or stars, on its surface (same in the vertical direction). The stars are even more widely spaced relative to their sizes than the planets are, so there is a great deal of empty space in our Galaxy!
Simulating Planet Formation:
To form a solar system, the literature says that in a simulation like this we can ignore the gravity of the Sun and just concentrate on the interaction of the objects in the protoplanetary disk.The gas, dust and other objects rotate around the forming Sun in Keplerian orbital motion. Therefore, a portion of the protoplanetary disk at greater radius will travel at a lower tangential velocity than its inner counterpart. Because of this shearing motion, the gas, dust and other objects in the disk are constantly approaching and receding from each other and sometimes colliding with each other and thus “growing” into larger and larger objects from these collisions. This coalescing process is sped up by the objects’ slightly eccentric orbits as they do in actuality.
The gravitational field of a planetesimal will scoop up the surrounding dust and larger bodies and so become more massive over time. Eventually, some of these planetesimals will reach the status of a protoplanet, if there is enough raw material in the feeding zone to support that.
Planetesimals in the protoplanetary disk have cross sections of about 1 km. Protoplanets have much larger cross sections of 1,000 to 10,000 km. Since the protoplanets are much larger and more massive, they undergo what is called runaway accretion and gain mass very rapidly –therefore clearing out their feeding zone.
Because there is no gas or dust between the planets in our Solar System today, these remnants must have been swept away by the solar wind during the Sun’s T-Tauri phase. When a star is in the T-Tauri phase of its early evolution, it is much more luminous for a short time than its steady state where it spends most of its lifetime. The star in its T-Tauri phase is about 20 to 30 times more luminous and excessively windy.
Objects in the protoplanetary disk eventually undergo collisions with each other. Sometimes they bounce off each other and don’t merge and sometimes they “stick” together. The criterion for collision is when the separation of two planetesimals is less than the sum of their radii. When this is true, the two objects have had some kind of physical contact with each other.
Comment on Algorithms:
Any computer simulation will have to calculate the separation for each pair of objects and check for a collision before incrementing the positions of all objects in the simulation. CPU time for this doubly nested loop goes like N2 where N is the number of objects in the simulation. Luckily modern computers are fast and have lots of memory for this task.
We begin this process of creating a computer simulation by defining an array of data structures that hold information about each object in the simulation. Mass, position, velocity, and acceleration are key to tracking the progress of each object. Along with that, there is radial velocity and angular velocity to consider.
The initial conditions of the simulation can be conveniently derived through the use of random numbers. By defining random positions that fall into a narrow ring (feeding zone) and setting the velocities by the application of Kepler’s third law, we are good to go. The Python computer language has very powerful random number functions that get the job done in just a few lines of code.
Then there is the case of close approach. When the two objects have a great enough mutual gravitational attraction, they are to some degree linked. Their motions relative to each other can be calculated through numerical integration. The numerical integration may or may not result in collision of the two objects but their relative positions and velocities will be altered and carried on through the simulation. This process of numerical integration is CPU intensive and will slow down the overall speed of the simulation depending on how many pairs are found to be in close approach and thus invoking the numerical integration routine.
Further Comments on Solar Systems in Formation:
Our Solar System can be divided into two realms: the inner Solar System where the rocky planets form and the outer Solar System where the giant planets form as gas/ice giants. The so called “ice line” between Mars and Jupiter divides these two parts of the Solar System. The ice line is a function of the intensity of the warming sunlight impinging on the matter found there (see Fig. 2). As the intensity of sunlight diminishes in the outer Solar System, free-floating ice can exist and it can provide raw material for the formation of planets like Jupiter, Saturn, Uranus and Neptune. The giant planets accrete more gas due to their greater mass and corresponding surface gravity.
Figure 2: Schematic cross-sections of an evolving protoplanetary system similar to that of the primitive Solar System. The snow lines refer to radii where various gases freeze out, beginning with water vapor. (Credits Subaru Telescope / NAOJ)
So we have outlined some things concerning Keplerian orbits, capture and close approach in the protoplanetary disk. We have examined some things about computer codes that reflect these processes. The interested reader could begin their own simulation if they wish to do so. The most technical portion of the software is the numerical integration routine used in the case of close approach between two objects in the protoplanetary disk.
It is important to think about the process of planet formation in an intuitive way. Intuitive thinking, when done carefully, can bridge gaps in analytical thinking. All bodies in the protoplanetary disk are attracted to each other and through these gravitational interactions the planets form. The formation of a solar system is truly order and form coming out of chaos.
Bill Stride – January, 2025