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In our solar system, MOIDs (minimum orbital intersection distance) of different planets reach a minimum of ~30 million miles (Mercury and Venus). However, other star systems have more compact planets. So I am wondering, what is the absolute minimum for planets of different sizes (list below)

- 1 Earth mass
- 5 Earth masses
- 10 Earth masses (Neptune-sized)
- 50 Earth masses (Super-Neptune)
- 100 Earth masses or 0.3 Jupiter mass (Sub-Jupiter)
- 1 Jupiter mass
- 5 Jupiter masses

P.S. Sorry if the list is too big, some can be omitted for sake of writing all of that.

I don't know how to calculate the minimum possible forbidden region for a planet of a specific mass orbiting a star of a specific mass at a specific semi-major axis. So I can't say what the absolute theoretical minimum separation between the orbits of two planets in the same planetary system can be.

A *reductio ad absurdum* calculation suggests that the minimum possible separation between planetary orbits could be about 2,500 to 5,000 kilometers. Probably astrophysicists should be able to calculate a minimum possible separation of orbits which would be many, many times that.

There are also some theoretical configurations of planets which would result in two or more planets sharing the same orbit, in which case one could be facetious and claim there is a zero separation between their orbits.

More Detailed Answer:

Part One of Six: Some "Dole ful" Calculations.

I do know that there have been formulas to calculate the minimum possible spacing between planets in a star system for decades.

*Habitable Planets for Man*, 1964. 2007, by Stephen H. Dole, is a scientific discussion of the parameters of planets which are habitable for humans.

https://www.rand.org/content/dam/rand/pubs/commercial_books/2007/RAND_CB179-1.pdf

Chapter Three: introduction to General Planetology, has a section "Spacing of Planets in the Solar System", on pages 49 to 52, discusses the spacing of planets in our solar system.

Dole bases his discussion of forbidden regions on his own paper:

"Limits for stable Near-circular Planetary or Satellite Orbits in the Restricted Three-Body Problem". ARS J, 31, no. 2 (February, 1961), pp. 214-219.

Dole calculated the limits of the "forbidden" regions of all the planets in the solar system, based on the masses of each planet and the Sun, the eccentricity of each planet's orbit, and the semi-major axis of each planet's orbit around the Sun.

The "forbidden" region is a ring around the Sun that a planet orbits within and which no other planet should be able to have a stable orbit within.

According to Dole's calculations, our solar system out to Neptune or Pluto is approximately half full of forbidden regions around planetary orbits:

This pattern of regularity should also be found in other planetary systems. Forbidden regions take up about 50 per cent of our solar system, and if this is true of other planetary systems (or multiple star systems); then it would be a simple matter to design any number of stable planetary systems by random mechanical processes.

So, according to Dole's calculations, a star like the Sun might have the same number and masses of planets as the Sun does, if its planetary system was about 50 per cent filled by planets and their forbidden regions. If planets could be spaced close enough together that the limits of their forbidden regions touched, then our solar system could have perhaps twice as many planets with the same masses, about 16 to 18.

More importantly to me, if there are one, two, or three planets within the circumstellar habitable zone of the Sun, in the planetary system of a star like the Sun with the forbidden zones of the planets just touching, there could be only two to six planetary orbits within the circumstellar habitable zone of that star. As a kid who liked science fiction stories where many planets in a single solar system were habitable, I found that idea rather "Dole ful".

Part Two: Some Early Ideas to Increase the Number of Habitable Planets in a system.

A number of science fiction stories have habitable planets in Trojan orbits, in which a planet orbits around a star in the L4 or L5 Lagrange point of another planet or another star, 60 degrees ahead of or behind the other astronomical object.

Such Trojan orbits are known to work when the primary, secondary, and tertiary objects differ in mass by hundreds or thousands of times. In our solar system, the Sun is thousands of times as massive as planets which are thousands of times as massive as the asteroids in their Trojan positions. And Saturn is thousands of times as massive as its moons Tethys and Dione, which in turn are thousands of times as massive as their Trojan moons.

But could two planets in the size range to be habitable share the same orbit, one planet being in the L4 point of the second one, which would be in the L5 point of the first one? I don't know.

But if that was possible, and if there can be two to six planetary orbits within the circumstellar habitable zone of a star like the Sun, then if there are two Trojan planets in each orbit that solar system could have four to twelve potentially habitable planets in its circumstellar habitable zone.

Another way to increase the number of habitable planets within the circumstellar habitable zone would be to replace each Earth mass habitable planet with a double planet having the same total mass as the Earth. According to Dole, the minimum mass for a habitable planet would be about 0.4 Earth mass, so a double planet with twin planets of about 0.42 Earth mass would have a total mass of 0.84 Earth, and thus a slightly smaller forbidden region than Earth.

So theoretically about four to twelve habitable planets could orbit in the circumstellar habitable zone of a star identical to the Sun, if arranged as two to six sets of twin planets.

Part Three: Co-Orbital Planets?

In astronomy, a co-orbital configuration is a configuration of two or more astronomical objects (such as asteroids, moons, or planets) orbiting at the same, or very similar, distance from their primary, i.e. they are in a 1:1 mean-motion resonance. (or 1:−1 if orbiting in opposite directions).1

There are several classes of co-orbital objects, depending on their point of libration. The most common and best-known class is the trojan, which librates around one of the two stable Lagrangian points (Trojan points), L4 and L5, 60° ahead of and behind the larger body respectively. Another class is the horseshoe orbit, in which objects librate around 180° from the larger body. Objects librating around 0° are called quasi-satellites.

An exchange orbit occurs when two co-orbital objects are of similar masses and thus exert a non-negligible influence on each other. The objects can exchange semi-major axes or eccentricities when they approach each other.

https://en.wikipedia.org/wiki/Co-orbital_configuration

The space probes Pioneer 11, Voyager 1, and Voyager 2 discovered several new moons of Saturn when they passed it in 1979, 1980, and 1981. This included the discovery of the tiny moons Epimetheus and Janus, which are co-orbital.

The Saturnian moons Janus and Epimetheus share their orbits, the difference in semi-major axes being less than either's mean diameter. This means the moon with the smaller semi-major axis will slowly catch up with the other. As it does this, the moons gravitationally tug at each other, increasing the semi-major axis of the moon that has caught up and decreasing that of the other. This reverses their relative positions proportionally to their masses and causes this process to begin anew with the moons' roles reversed. In other words, they effectively swap orbits, ultimately oscillating both about their mass-weighted mean orbit.

So it would potentially be possible for two habitable planets to share the same orbit in an exchange orbit like that of Epimetheus and Janus, thus having two habitable planets in the orbit and forbidden region. That could be used to double the number of planets orbiting within the habitable zone of a star.

I can't help thinking that inhabitants of a planet in an exchange orbit with another planet would find the exchange process terrifying until and unless they could calculate that the two planets were not going to collide.

Part Four: Exoplanet Discoveries.

IN the last generation thousands of exoplanets have been discovered orbiting other stars. And planetary systems with two or more planets orbiting the same star have also been discovered. Because of the great difficulty in discovering exoplanets, it is reasonable to assume that in most cases there are more planets in a system than have been discovered yet, perhaps more planets than can be discovered until decades, centuries, or even millennia of future scientific progress is made.

Even though Dole wrote in 1964 that:

This pattern of regularity should also be found in other planetary systems.

The majority of planetary systems discovered around other stars have been significantly different in one or more major ways from our solar system. Thus there must be a great variation in the processes which form and shape planetary systems.

According to Wikipedia's list of exoplanet extremes, the smallest distance between the semi-major axis of the orbits of two consecutive planets is between Kepler-70b and Kepler-70c, about 0.0016 AU, or about 240,000 kilometers, or about 149,129 miles, closer than the distance between the Earth and the Moon.

https://en.wikipedia.org/wiki/List_of_exoplanet_extremes

If each consecutive planetary orbit around the Sun was only 0.0016 AU farther out than the previous one, 262 planetary orbits could fit within Kasting's conservative habitable zone for the Sun, and 518 in his optimistic habitable zone.

However, the article on Kepler-70 indicates that it is now believed the planets do not exist and their detection was probably an error.

https://en.wikipedia.org/wiki/Kepler-70

I do not know what is the smallest known difference between the orbits of two confirmed exoplanets. But the two exoplanets with the smallest ratio between their orbits are Kepler-37b & Kepler-37c. Both planets are several times as massive as Earth but orbit very close to their star Kepler-37 and thus to each other.

The orbit of Kepler-37b has a semi-major axis of 0.1153 AU, or 17,248,634 kilometers, or 10,747,804.57 miles, and the orbit of Kepler-37c has a semi-major axis of 0.1283 AU, or 19,193,407 kilometers, or 11,926,230 miles, a difference of 0.013 AU, or 1,944,772.3 kilometers, or 1,208,425.49 miles.

If the planetary orbits around the Sun could have an average separation of 0.013 AU, 32 planetary orbits could fit within Kasting's conservative habitable zone and 64 planetary orbits could fit within Kasting's optimistic habitable zone.

0.1283 AU is 1.1127 times 0.1153 AU, and is the smallest known ratio between consecutive planetary orbits. If I remember correctly I calculated that 4 planetary orbits with that ratio could fit within Kasting's conservative habitable zone and 6 planetary orbits could fit within Kastings optimistic habitable zone.

So should the distance of 0.013 AU between the orbits of Kepler-37b and Kepler-37b be considered the minimum possible spacing between consecutive planetary orbits?

Or should the ratio of 1.1127 between their semi-major axis be considered to the minimum possible ratio between consecutive planetary orbits?

If it is the ratio between orbits which determines the minimum spacing of planetary orbits, if a planet orbits very close to its star the minimum possible distance to the next planet's orbit could be less than the 0.013 AU between the Kepler-37 planets. For example, if the inner planet orbits at 0.01 AU, the next planet could orbit at a distance of only 0.011127 AU, a difference of only 0.001127 AU, a little less than the distance between the orbits of the alleged Kepler-70 planets.

If it is the ratio between orbits which determines the minimum spacing of planetary orbits, if a planet orbits very far from its star the minimum possible distance to the next planet's orbit could be many times the 0.013 AU between the Kepler-37 planets. For example, if a planet orbits 100 AU from its star, and the minimum separation is 1.1127 times the inner orbit, the next planet out would have to orbit at a distance of at least 111.27 AU.

Par Five: Planets in Rings.

Astrophysicist Sean Raymond, in his PlanetPlanet blog, has a section called Ultimate Solar System designing planetary systems with as many planets in the habitable zone as he can fit in.

https://planetplanet.net/the-ultimate-solar-system/

In the Ultimate Engineered Solar system Raymond designs a planetary system with 416 planets in the habitable zone, using rings of planets sharing the same orbit.

That is based on this paper by Smith and Lissauer:

https://ui.adsabs.harvard.edu/abs/2010CeMDA.107… 487S/abstract

Smith and Lissauer calculate that seven to forty two planets can share the same orbit around their star, if the planets have equal mass and are equally spaced around the star.

The planets would be separated by 8.57 degrees if there are forty two planets in the ring, increasing as the number of planets decreases, so that seven planets would have gaps of 54.42 degrees between them. The size of those gaps in AU, kilometers, or miles would have to be calculated from the semi-major axis, and thus the circumference, of the orbit.

Raymond also states that planetary orbits should be separated by 5 to 10 times their Hill radius to be stable. For example, a planet with the mass of Earth 1 AU from a star with the mass of the Sun would have a Hill radius of about 1,500,000 kilometers or 0.01 AU.

So planetary orbits of Earth-like planets orbiting Sun-like stars at distances of about 1 AU should be separated by at least 0.05 to 0.10 AU or 7,500,000 to 15,000,000 kilometers.

Part Six: Conclusion.

The minimum possible distance between the semi-major axis of the orbit of a planet and the semi-major axis of the the orbit of another planet orbiting the same star can be calculated from factors such as the mass of the planet and the mass of the star, and the semi-major axis of the planet's orbit.

Known examples of the separation of the semi-major axis of consecutive planets in a planetary system range from as much as about 662 AU between CVSO 30 b & CVSO 30 c down to as little as 0.013 AU between Kepler-37b & Kepler-37c.

Known examples of the ratios of the semi-major axis of consecutive planets in a planetary system range from as much as about 78,998 times the distance in the case of CVSO 30 b & CVSO 30 c down to as little as 1.1127 times the distance in the case of Kepler-37b & Kepler-37c.

Of course if additional planets are discovered between the orbits of the known planets in those systems the records could change.

Astronomers are fairly certain that any planetary orbital separation within those ranges is possible. I don't know the theoretical limits to how much larger or smaller the separation between the semi-major axis of two consecutive planetary orbits could be.

For minimum spacing I guess I can do a *reductio ad absurdum* calculation.

According to Sean Raymond here

https://planetplanet.net/2017/05/03/the-ultimate-engineered-solar-system/

The minimum possible separation between two planetary orbits should be five to ten times the Hill sphere radius of one of the planets - the planet with the larger Hill sphere radius.

The absolutely smallest the Hill radius of a planet orbiting very close to a very massive star could get would be at the surface of that planet. If the surface of the planet extended above the Hill sphere radius the surface material would be ripped from the planet by the gravity of the more massive object until the planet was stripped down to to the radius of the Hill Sphere.

If a astronomical object has to be pulled by its gravity into a spheroidal shape to be considered a planet, that establishes a minimum size for a planet or a planemo (planetary mass object), and thus for its smallest possible Hill sphere. The minimum radius necessary for astronomical object to be spheroidal is not known precisely.

More or less arbitarily making a radius of 500 kilometers the lower limit for a planetary mass object, and assuming that such an object could orbit close enough to its star to have it's hill sphere at its surface, the minimum possible separation between its orbit and that of the next planet in the system would be about 2,500 to 5,000 kilometers.

And of course if two planets shared the same orbit, such as Trojan planets, planets in an exchange orbit, or planets in a ring, someone could facetiously say there would be approximately zero difference between the semi-major axis of their orbits, since their orbits could be the same orbit.

## What’s the closest and farthest planet from Earth?

It is a common point of discussion that how vast is this universe we live in. Scientists are exploring all parts of space to provide an answer with scientific proof to back that. They have had no luck to date and concepts like Parallel universe are making things much more complex than ever. However, we can make a rough estimate of the vastness of our universe by knowing the distances between Earth and the other planets of our solar system.

These distances are inconsistent for all the planets including Earth as they are moving in their corresponding orbits. The distance between any two planets is minimum when they are on the same side of the Sun and form a line with it. Similarly, they are at a maximum distance when they are on different sides of the Sun and form a line. Another factor that determines this distance is the placement of the orbit of any particular planet. All the planets whose orbits lie closer to the Sun than the Earth’s orbit are closest to the Earth at the time of Inferior Conjunction. On the other hand, planets having a greater distance from the Sun are at a minimum distance from Earth during times of Superior Conjunction.

The units of measurement used for measuring average inter-planetary distances are called Astronomical Units (AU). One AU is equal to 149,598,000 kilometers. It is the distance between the Earth and the Sun of our solar system. The following explanation might stir some sensitive parts of your brain.

The minimum distance between these two planets is 77 million kilometers that can be observed at the time of inferior conjunction. It continues to increase till we reach superior conjunction. The inter-planetary distance is calculated to be 222 million kilometers during that phase. The average distance between them is 0.61 AU.

The average distance between these planets is about 42 million kilometers. This is equal to 0.28 AU. The minimum gap is about 38 million kilometers while it can increase to 261 million kilometers in the worst case.

The minimum distance between Earth and Mars is 55 million kilometers. Theoretically, they can come as close as 54.6 million kilometers but the smallest distance ever observed is 56 million kilometers. The mean distance from here to the red planet is about 225 million kilometers. On the scale of astronomical units, it is equal to 0.52 AU. These planets can go as far as 401 million kilometers.

Jupiter takes 11.86 Earth years to complete one round of the sun. It was considered to be moving backward in the night sky up until the discovery of Johannes Kepler in which it was explained that the motion of planets around the central sun is not circular but, elliptical. The minimum distance between these two planets is 588 million kilometers that can be observed at the time of superior conjunction. It continues to increase till we reach inferior conjunction. The inter-planetary distance is calculated to be 968 million kilometers during that phase. The average distance between them is calculated to be 4.20 AU.

The Earth is at a distance of 1 AU from the sun. On the other hand, Saturn is more than 9.50 AU away from the sun. If we subtract these figures, the result will be 8.5 AU which is equal to 1300 million kilometers. But, as the orbits are elliptical, the inter-planetary distance will continue to vary. Therefore, the maximum distance between both these planets is about 1700 million kilometers while the minimum distance lies in the range of 1200 million kilometers.

When both these planets are found on the same side of the sun, the smallest distance between them is possible and it is expected to be about 2570 million kilometers. When they are on opposite sides, the maximum distance between them is determined and it is about 3150 million kilometers. Average distance is 18.54 AU.

Neptune is approximately 30 times away from the sun than the Earth. As a result, the maximum distance between these planets can go up to 4700 million kilometers while they can come as close as 4300 million kilometers. The average distance is measured to be 29.06 AU which is roughly equal to 4560 million kilometers.

Computer Scientist by qualification who loves to read, write, eat, and travel

## What is the minimum possible distance between two planets?

I'll be talking from in a worldbuilding context here so let me know if this is not correct.

There's no physical law that says planets have to be a certain distance apart. However, planets do need to be at some distances apart to remain in stable orbits. This minimum distance depends upon the mass of the planets, the mass of the star, and the distance of the planets from the parent star.

Iɽ suggest calculating the radii of the 'hill spheres' of both planets, then separate their orbits by a distance roughly twice the value of the hill sphere radii combined. If you want a loose guideline however, in our Solar System other planets look like bright dots from each other's naked eye perspectives.

If you're looking for a planet-moon minimum distance, you'll need to calculate the 'roche limit' for both bodies, as that is the distance at which large bodies that come closer will be destroyed due to tidal forces. You'll also want to calculate the hill sphere for the planet to set the maximum distance for the moon. As a loose guideline, Iɽ suggest putting tiny moons at a minimum distance of one planet radius, and large moons at a minimum distance of five planet radii. Bare in mind that closer and more massive moons will produce stronger tides on the planet which may cause strange effects in the oceans (if there are any).

I read the question differently as the other comment and took it as, given two orbits/planets, at what two points on each orbit gives the minimum possible distance between the two.

In which case, it depends on the orbits. The first trivial case is two circular orbits, in which case its the radius of the bigger orbit minus the smaller one and it's between any two collinear points with the barycenter.

But, let us say we have a circular orbit and an elliptical one. The elliptical orbit "exits" or envelops the circular orbit. So, if there's an intersection, that means our minimum possible theoretical distance is 0 at the intersection. If there is not an intersection, that implies there is a point closer to the circle. This is equivalent to asking if there is a closest point to the focus since a circle is equidistant. Thus, it is the perihelion (closest point to the barycenter) minus the radius of the circle with the locations being the perihelion, the barycenter, and the point the circle collinear.

There's also the case if the elliptical orbit is enveloped by the circular one, in which case it's just the same thing but instead the aphelion.

Finally, two ellipses. In which case I *believe* there could be one or two answers. If the aphelion of one's closest point is to the perihelion of the other, the two ellipses are symmetrical, but, as said, might have two or multiple closest points, but still a minimum distance. I don't plan on doing this one though as it's a lot more complicated. But Iɽ imagine there's some beautiful calculations you could do by having using a circle and casting lines through it to an ellipse and comparing the two ellipses or something. If this is what you were looking for maybe let me know cause I'm partially curious of the answer myself but am too tired rn.

## Minimum distance between planets - Astronomy

In my book 'Interplanetary Travel:An Astronomer's guide' I discuss current and planned technologies for interplanetary travel.

The bottom line is that it depends a lot on the particular trajectory that you take. Usually, the trajectories are in the form of a 'great arc' that gracefully connects a launch time at Earth with a destination point. These arcs are usually many times longer that the straight-line distance between the two planets at a particular moment in time. For reduced-cost travel, astrodynamicists often rely on gravity assists 'Slingshot Orbits' from the inner planets to reach Jupiter, and by Jupiter to reach more distant worlds. These loop-de-loops add years of extra travel time to a mission. Let's assume for our calculations that we just take the simplest direct approach and use the minimum 'opposition' distance between Earth and a planet.

The table below gives you some sense for how long it takes to get to each planet at different speeds.

The Space Shuttle, of course, can't leave Earth orbit but its speed is typical of manned spacecraft. The Galileo spacecraft which explored Jupiter traveled twice as fast. These travel times are based upon using a big chemical rocket on Earth to blast the spacecraft into the right trajectory. But there is another technology that has been used for several decades.

Ion rocket motors get their speed by being constantly accelerated 24-hours a day for many months, and two versions of this technology are given for a low-power and high-power ion engine. Ssatellites orbiting earth often use ion engine technology for station-keeping. NASA has also used ion engine technology on two interplanetary spacecraft: Deep Space 1 and Dawn. Both spacecraft used very low thrust engines operating for thousands of days to get the spacecraft to asteroids (Dawn visited Ceres and Vesta DS1: 9969 Braille) and comets (Borrelly) for study.

Finally, and at least on paper, solar sails can reach speeds of nearly that of the solar wind (500 km/sec), and engineers are hopeful that this technology will be tested in space very soon.

As you can see, we are currently stuck in the mode of travel where it takes nearly 10 years to get to Pluto. Perhaps in another hundred years, this travel time will be reduced to a year or less. assuming Humanity feels a compelling economic need to continue this kind of exploration.

Method= | Shuttle | Galileo | Ion A | Ion B | Solar Sail |

Speed = | 28,000mph | 54,000mph | 65,000mph | 650,000mph | 200,000mph |

Mercury | 52d | 27d | 22d | 2.2d | 7.3d |

Venus | 100d | 52d | 43d | 4.3d | 14d |

Mars | 210d | 109d | 90d | 9d | 29d |

Jupiter | 1.9yr | 1yr | 303d | 31d | 100d |

Saturn | 3.6yr | 1.8yr | 1.5yr | 55d | 179d |

Uranus | 7.3yr | 3.8yr | 3.1yr | 113d | 1yr |

Neptune | 11.4yr | 5.9yr | 4.9yr | 179d | 1.6yr |

Pluto | 15.1yr | 7.8yr | 6.5yr | 238d | 2.1yr |

Assumptions: Ion Drive using a constant thrust of A) 0.1 pounds B) 1 pound with turnaround deceleration added. Two years acceleration to reach top speed. Solar Sail estimated speed 450 km/sec or one million mph)

Another way to gauge the maximum speed of a spacecraft is by the exhaust speed of its engines. Engine exhaust speed is related to an engineering parameter called the Specific Impulse. SI is the exhaust speed divided by the acceleration of gravity at Earth's surface. For example, chemical rockets have SI=250 seconds and so their maximum exhaust speeds are 250 x 9.8 m/sec2 = 2.4 km/sec. SInce no rocket payload can travel faster than its exhaust speed, we can compare planetary transit times in terms of the SI of the rocket technology.

In this table, I assume that the rocket continuously accelerates from Earth until it reaches half its destination distance, then turns around and decelerates for the second half of the trip. The relevant equations are

Destination | a=0.05 | a=0.15 | a=0.2 | |

Planet | Time-A | Time-B | Time-C | Max-SI |

Days | Days | Days | Seconds | |

Mars | 24 | 14 | 8 | 34000 |

Jupiter | 80 | 46 | 25 | 110000 |

Saturn | 113 | 66 | 36 | 160000 |

Uranus | 166 | 96 | 53 | 230000 |

Neptune | 214 | 124 | 68 | 300000 |

Pluto | 214 | 124 | 68 | 300000 |

So if we could design a ship, call it System A, that produced a constant acceleration of a = 0.05 meters/sec^2, we could get to Mars in just 24 days. Similarly for System C with an acceleration of 0.2 meters/sec^2 the Mars trip takes only a week! For engineers,we can also estimate for System C that the Mars configuration would need SI=34,000 seconds to get us there in 8 days. If we wanted to get to Pluto with the same acceleration, it turns out that accelerating for half the 68-day trip would get you to a maximum speed of about 2900 km/sec which sets the limit to our exhaust speed and leads to a maximum SI of about 300,000 seconds! These SI estimates are far larger than the chemical rockets provide of 300 seconds, and so we have to look to entirely different technologies to make these travel times possible. Sadly, it doesn't matter if we drag along huge fuel tanks to run our chemical rockets. They will never provide the high exhaust speeds we need to carry both our fuel and payload to our destinations. We have to use technologies that enormously increase the exhaust speeds themselves.

## 1 Answer 1

The Doppler shift in the light from the star tells you the period of the planet's orbit and also the velocity the star moves. You need to know the mass of the star, but this can be estimated to good accuracy from the star brightness and type. Once you know the mass of the star you can calculate the distance of the planet from it's period using:

where $M$ is the mass of the star and $P$ is the period of the oscillation.

Not that it's directly relevant to your question, but from the velocity of the star's oscillation we can calculate the minimum mass of the planet, because the velocity of the stars displacement depends on the gravitational force between the two. We can only calculate a minimum planet mass because if the plane of the system it tilted relative to us the true mass is higher than the one we calculate.

Having said this, these days most extrasolar planets are discovered because they transit their star, and these systems are not tilted relative to us (otherwise they wouldn't transit!). That means we can calculate an accurate mass for the planet.

In practice we normally turn the calculation over to a computer model (called a *Bayesian Kepler periodogram* if you want to Google it) because there are usually several planets and the oscillation is not a simple sine wave. We use a numerical fit to work out how many planets there are and how far from the star they are.

## Minimum distance between planets - Astronomy

*Is it true that, as we follow the planets outward from the sun, the distances become about double each time? Does that mean that Venus is closer to Earth than Mars is?*

Yes, it is true that there is somewhat of a pattern to the distances of the planets from the Sun. Venus is 1.8 times as far from the Sun as Mercury, and Earth is about 1.4 times as far from the sun as Venus. Mars is 1.5 times farther than Earth. This seems to be a pattern - each planet could be between 1.4 and 1.8 times farther from the sun than its "inside" neighbor. Then comes the problem - Jupiter is 3.4 times farther from the sun than Mars. This is where the pattern falls apart, although some say that the asteroid belt, which is in between Jupiter and Mars, could count as a substitute for a planet. Then Saturn is 1.8 times farther than Jupiter, Uranus is 2 times farther than Saturn, and Neptune is 1.6 times farther from the Sun than Uranus. Pluto doesn't fit this pattern at all. So there seems to be some sort of pattern to this, but there's no real theory that explains why the planets ended up at the distances they did, so it could also be a complete coincidence that they're somewhat evenly spaced.

So the "doubling" rule does work, but only approximately. This means that yes, the difference between the average orbital distance of Mars from the Sun to the average orbital distance of Earth from the Sun is greater (about 78 million km) than the difference between the Earth's average orbital distance from the Sun to Venus' average orbital distance from the Sun (41 million km). However, since the distance between the Earth and other planets depends not only on the size of their orbits but also on where they are in their orbits relative to each other, Venus is not always closer to Earth than Mars is.

*This page was last updated on July 18, 2015.*

### About the Author

#### Cathy Jordan

Cathy got her Bachelors degree from Cornell in May 2003 and her Masters of Education in May 2005. She did research studying the wind patterns on Jupiter while at Cornell. She is now an 8th grade Earth Sciences teacher in Natick, MA.

## Solar System Tutorial

The wikipedia pages on Kepler Orbit is pretty nice. It is explained more concisely here and in this video.

There are a lot of free ebooks on plate tectonics and stuff, when I eventually go to designing individual planets, this will be essential. There are a few posts at r/worldbuilding on the subject, also.

Aggressive googling told me a little extra, but it didn't help as much as it does with general terms.

Step one is to build a star. Either watch this video or follow my instructions. A star needs to be within a type to support life. Luckily, the mass required to obtain that isn't entirely uncommon. All you will need is an arbitrary mass. This mass has to be between 0.6 and 1.4 solar masses. One solar mass is the mass of the sun. This will provide enough time for intelligent evolution and provide a safe environment fora habitable planet to form.

Here's the equations to find figures about your star:

(All relative to the sun's statistics. They're easy enough to find via google)

Luminosity = Mass 3

Diameter = Mass 0.74

Surface Temp. = Mass 0.505

Lifetime = Mass -2.5

Next, we have to find the habitable zone of our solar system. Don't put away your sun's statistics, we will need them. To find the habitable zone, we will need to do 3 simple equations. First, we must find a number *R*. just take the square root of your Luminosity. The edges of your 'goldilocks zone' are 95% and 137% of *R*. This statistic is in Astronomical Units (AU). 1 AU is the distance from the Earth to the Sun. (It's really 1.00000261 AU, but that really doesn't matter)

Another useful statistic that will be helpful to create planets is the frost line. The frost line is calculated by 4.85 multiplied by the square root of your Luminosity, which we calculated earlier. Gravity provides a limit to where our orbits can be. Too close, and the planet will go poof due to extremely intense gravity. Too far, and the planet will float away. The limits are calculated by taking 0.1 of the mass, and 40 times the mass. These are also in AU.

To review the equations we just used:

R = sqrt(Luminosity)

Habitable Zone - .95(R) to 1.37(R)

Frost Line = 4.85*sqrt(Luminosity)

Limit = .1(Mass) to 40(Mass)

Next, we're going to create orbits for our planets. Artifexian recommends picking a location near our frost line and making a planet. Just select a number in AU near that, and you will have a nice, massive, gas giant. Select a random number between 1.4 and 2. Multiply your AU distance of your planet by that number. That is your next stable orbit in the solar system. Write that number down. Now pick another number in the same range, and repeat that step until you have created planets all the way up to the edge. Then, divide by similar numbers, creating planets along the way. You should have one planet land in your habitable zone, but if you don't, it's okay to tweak the numbers a little, as long as you stay between 1.4 and 2.

Sweet! Now we have a lot of planets and a sun. The ones outside of the frost line are probably ice giants, which are gas giants with frozen volatile gases. Before the frost line a little ways, there might be gas giants for a little ways. Closer to the sun, there is less spacing between planets, and you're likely to get terrestrial planets. One terrestrial planet, of course, will be yours.

In part 2, we will discuss orbits, your inhabited planet, and (maybe) my solar system. Iɽ like to reiterate that /u/Artifexian compiled much of this content, I'm just summarizing and text-ifying it.

Orbits are tricky. They rely on 6 orbital parameters, and there isn't clear information on world building them. Artifexian provides a lot of general info about them, but for specifically inhabited planets, there is not much information on the internet at all.

I hadn't studied astronomy until a few weeks ago when I got into world building. To stay realistic, we're going to model our orbits sort of around the ones we have in real life. I also have no idea how to do an asteroid belt, but I have some info on dwarf planets. That's for the future.

Orbits are described by 6 characteristics. They have fancy astronomer names, but really they are the following:

Distance- this is found by taking the distance from the points where your planet is furthest and closest to your star and dividing it by two. Don't worry. The distances we calculated in part 1 are actually this.

Eccentricity- Orbits are ellipses. This number goes from 0-1. One is a parabola, and 0 is a perfect circle. Pick a number close to 0, but not 0. Try 0.0x to 0.00x.

Incline- Your planet's orbit will lie on a single plane. Also, your star will have an ɾquatorial plane' through it's equator. The incline is the angle between those 2 planes. Pick an angle from 0-180 degrees. It's probably best to keep it relatively <15 degrees, because some pretty weird solar systems can occur if you go crazy with these. I'm not certain on the parameters for stability, but go nuts if you want, I can't guarantee habitability.

Yaw- Now you're gonna need to make a reference line. It will extend out from your sun and into space. For Earth, it's the vernal equinox. Think of a point where your orbit intersects the equatorial plane. This is the ascending node. Take the angle from the reference line to the ascending node. Adjusting the number, however, makes your orbit move. Think of an airplane, a pilot can tilt the plane. We rotate about the y axis.

Roll- Take the angle from the closest your orbit is to the star and the ascending node. This rotates your orbit around the z axis. Pretty simple, don't go nuts. 0-360.

Plot- Draw a line from where your planet is at the moment to the star. Draw a line from the closest point of your orbit to the star. Measure the angle, and that's it.

We don't really need to worry about anything besides the distance of the other orbits, unless we want to have serious space exploration. The year length for your inhabited planet is calculated by Kepler's third law. The orbital radius is your planet's distance, the mass is the mass of your star. Use this calculator with those values.

Part 3 will making your planet, and the considerations that go along with that.

This will focus on our planet. There is a lot of fancy math that I don't understand, but that's why we have graphs!

We're gonna need to use a little equation called planet maker. For all of our future equations, we will use these variables to describe things.

G = Gravity relative to earth M = Mass relative to earth R = Radius relative to earth P = Density relative to earth

Planet maker equation: G = M/R 2 = R(P)

For our planet to be useful to humanoids, we need to pick a certain mass and radius that will make our planet have land, an iron core, water, and a bunch of other fun stuff.

Your mass has to be between .4 to 2.35.

Your radius must be between .78 and 1.25.

The x axis is the mass, y is radius. You want, when graphed, your planet to land inside the purple band. Water worlds go in the blue band, super rocky worlds go in the red. To support life, you want to be in the purple band.

Now, go back to the planet maker equation. Use this to find your surface gravity, and if you want, density. If your gravity is not between 0.4 and 1.6 times Earth's gravity, try again with different mass and radius values.

Not all of the values have been spreadsheeted, because I'm too lazy to copy a lot of the info from my notes to a spreadsheet. Here's my sun's stats and some planetary stuffs.

I avoided naming stuff because I want the culture of my planet to name all the surrounding ones. Thanks!

## 1 Answer 1

The relative distances to the planets is fixed immediately by Copernican model, and this is what makes heliocentrism ten thousand times better than geocentrism, even without any known physical cause for the orbits.

The relative distances are fixed from the radius of the epicycle — the epicycle transfers Earth's orbit onto the planet, and the ratio of the epicycle radius (not the angular extent, which also includes the planet's motion along the deferent) to the deferent size in the Copernican interpretation directly gives the ratio of the Earth's orbit to the planet's orbit. The relative size of Venus and Mercury's orbit, relative to the Earth's distance from the sun, is given by the maximum in angle they get away from the sun.

This is not surprising, because the epicycle radius is giving you the parallax from the point of view of the Earth's orbit of the different planets. Once you know the absolute size of Earth's orbit, you know the distance to everything else, which is why the Earth's orbit is called the "Astronomical Unit".

This means that just Brahe's observations are sufficient to fix the entire solar system size except for the absolute scale of the Astronomical unit. The location of all the planets in 3 dimensions is completely determined from the assumption that the Earth's orbit is shared between all of them. The fact that the epicycles all are given by a one-year orbital period for the Earth is Baysian-wise *extremely compelling* evidence for heliocentrism *without anything further to say*.

This is why it is not correct to say that geocentrists were somehow justified, or had any valid points, or were anything other than the dimwitted reactionaries that they were. This includes Ptolmey, who buried the heliocentric work of Appolonius for political reasons, although even the most casual astronomer of the era was aware that heliocentrism was correct.

## Minimum distance between planets - Astronomy

*Is it true that, as we follow the planets outward from the sun, the distances become about double each time? Does that mean that Venus is closer to Earth than Mars is?*

Yes, it is true that there is somewhat of a pattern to the distances of the planets from the Sun. Venus is 1.8 times as far from the Sun as Mercury, and Earth is about 1.4 times as far from the sun as Venus. Mars is 1.5 times farther than Earth. This seems to be a pattern - each planet could be between 1.4 and 1.8 times farther from the sun than its "inside" neighbor. Then comes the problem - Jupiter is 3.4 times farther from the sun than Mars. This is where the pattern falls apart, although some say that the asteroid belt, which is in between Jupiter and Mars, could count as a substitute for a planet. Then Saturn is 1.8 times farther than Jupiter, Uranus is 2 times farther than Saturn, and Neptune is 1.6 times farther from the Sun than Uranus. Pluto doesn't fit this pattern at all. So there seems to be some sort of pattern to this, but there's no real theory that explains why the planets ended up at the distances they did, so it could also be a complete coincidence that they're somewhat evenly spaced.

So the "doubling" rule does work, but only approximately. This means that yes, the difference between the average orbital distance of Mars from the Sun to the average orbital distance of Earth from the Sun is greater (about 78 million km) than the difference between the Earth's average orbital distance from the Sun to Venus' average orbital distance from the Sun (41 million km). However, since the distance between the Earth and other planets depends not only on the size of their orbits but also on where they are in their orbits relative to each other, Venus is not always closer to Earth than Mars is.

*This page was last updated on July 18, 2015.*

### About the Author

#### Cathy Jordan

Cathy got her Bachelors degree from Cornell in May 2003 and her Masters of Education in May 2005. She did research studying the wind patterns on Jupiter while at Cornell. She is now an 8th grade Earth Sciences teacher in Natick, MA.

## Minimum distance between planets - Astronomy

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It's very hard for us to understand just how large the solar system is. Scientists tell us that the largest number our minds can really comprehend, or grasp, is about a hundred thousand (100,000). When you begin talking about the distances between planets, which are measured in millions, or billions, of miles, our minds just don't keep up very well. When you add the habit that adults have of making things more complicated than they need to be, it gets even harder. What we are going to try to do here is explain the size of the solar system in a way that you can understand it, as well as get some idea of how far apart our planetary neighbors are.

We are going to try to show you the size of the solar system by taking an imaginary "walk" from the Sun to all the planets of our neighborhood. This is going to be a special walk, since each "step" you take will equal one million miles! For the first four planets, Mercury, Venus, Earth and Mars, you will be able to actually take this walk if you have a large field, such as a football or soccer field. After that, the distances are just too great, unless you live in a very wide open area. If you do decide to walk the distances for the first four planets, make sure you have one of your parents go with you. When you walk off the distances, each step you take should be about three feet long. If you take your walk on a football field, start with the Sun at the center of the back of one of the end zones. We will tell you the yard lines where the first three planets will be.

In addition to our walk through the solar system, we will also tell you how long it would take to drive to the planets in your car at seventy miles per hour, as well as fly there in a jet plane going 600 miles per hour. Remember, the planets are not in a straight line going out from the Sun. They are always moving around the Sun, so they are hardly ever lined up in a row. The distances we use are from the Sun to the planets' average distance from the Sun.

Mercury

The first stop on our walk will be Mercury, which is the closest planet to the Sun. Mercury is about 35 million miles from the Sun, so you will take 35 steps from your Sun. If you are using a football field, Mercury will be at the 25 yard line (don't forget that the end zone is ten yards deep). If you got on a jet and flew at 600 miles per hour from the Sun to Mercury, it would take seven years! Driving the same distance in a car would take 57 years.

Venus

The second stop on our walk through the planets is Venus. The planet named for the Roman goddess of love and beauty is about 65 million miles from the Sun, which means that Venus will be 65 steps away from your Sun. If you are on a football field, Venus will be at the 45 yard line on the other side of the 50 yard line from your Sun. If you got on a jet and flew at 600 miles per hour from the Sun to Venus, it would take twelve years. If you could drive the same distance, it would take 106 years.

Earth

The third stop on our walk through the solar system is our home planet of Earth. Earth's average distance from the Sun is 93 million miles, which means you will have to take 93 steps from your Sun. If you are on a football field, Earth would be at the opposing team's 17 yard line. By the way, scientists use the Earth's distance from the Sun as a type of shorthand to show distance. This distance is called an astronomical unit. If you got on a jet and flew at 600 miles per hour from the Sun to Earth, it would take 18 years. If you could drive the same distance, it would take 152 years.

Mars

The fourth stop on our stroll through the solar system is Mars, the Red Planet. This is also the last stop that will be practical to actually step off. Mars' average distance from the Sun is about 137 million miles, or about 137 steps on our walk. If you are using a football field, Mars will be seventeen steps beyond the end of the end zone on the other side of the field from your Sun. If you got on a jet and flew at 600 miles per hour from the Sun to Mars, it would take 26 years. If you could drive the same distance, it would take 223 years.

Jupiter

Jupiter, the largest planet, is the fifth planet from the Sun, and is the first of what are called the Outer Planets. Its average distance from the Sun is almost 467 million miles. If you tried to step off this distance, you would be over a quarter of a mile away from your Sun. We don't recommend that you do this unless you have a parent with you and a lot of open space. If you got on a jet and flew at 600 miles per hour from the Sun to Jupiter, it would take 89 years. If you could drive the same distance, it would take 762 years.

Saturn

Saturn, the Ringed Planet, is the sixth planet from the Sun. This giant planet is over nine times as far away from the Sun as Earth. Its average distance is over 850 million miles away from the Sun. If you tried to step off this distance, you would be almost half a mile from your Sun when you reached Saturn. You would also probably be tiring out. If you got on a jet and flew at 600 miles per hour from the Sun to Saturn, it would take 163 years. If you could drive the same distance, it would take 1,396 years.

Uranus

Uranus, the mysterious blue-green planet, is the seventh planet from the Sun. From Uranus outward to the edge of the solar system, the distances are truly great. Uranus' average distance from the Sun is 1.7 billion (1,700,000,000) miles. If you were to walk off this distance, you would be a mile away from your Sun. If you got on a jet and flew at 600 miles per hour from the Sun to Uranus, it would take 328 years. If you could drive the same distance, it would take 2,809 years.

Neptune

Neptune, the eighth, and next to last, planet from the Sun, is almost 2.7 billion (2,700,000,000) miles away from the center of the solar system. If you tried to step this distance off, you would be over a mile and a half away from your Sun before you reached the location of Neptune. It is a very long distance. If you got on a jet and flew at 600 miles per hour from the Sun to Neptune, it would take 513 years. If you could drive the same distance, it would take 4,400 years.

Pluto

Tiny Pluto is the last planet in our family. The dark, cold planet's average distance from the Sun is a little over 3.5 billion (3,500,000,000) miles. If you tried to step off this distance, you would be over two miles away from your Sun when you reached the location of Pluto. If you got on a jet and flew at 600 miles per hour from the Sun to Pluto, it would take 675 years. If you could drive the same distance, it would take over 5,700 years.

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