Estimating Surface Gravity on a Fictional Planet

So you want to estimate surface gravity on a fictional planet? Easy!

One of the things I had to do as part of the rework of my novel The Tau Ceti Diversion, is to try and work out the surface gravity of my fictional planets. From the Kepler data, there are two exoplanets located in the Tau Ceti system that are likely to be in the system’s habitable zone, or where there is the possibility of liquid water on the surface, and perhaps life as we know it.

To play around with my estimates of gravity, I used ratioed rearrangements of Newton’s law of gravity (law of universal gravitation) and a simple formula relating the density of a spherical planet to its mass and radius (these are at the bottom of the post in the ADDENDUM).

WARNING: MATHS CONTENT!!!

Here’s Newtons famous law:)

law of gravity

The two planets thought to be in Tau Ceti’s habitable zone are denoted Tau Ceti e and Tau Ceti f. What is known about these two planets is their likely orbit, eccentricity, and their mass. All of these properties have been derived by calculation, based on observed data, so are all known to within appropriate error bounds, but I’m leaving the error off my scribblings so things don’t get too messy.

Tau Ceti e is thought to be around 4.3 Earth Masses, or Me (i.e. 4.3 times as heavy as Earth), while Tau Ceti f, the planet that orbits a bit further out, is thought to be around 6.67 Me. For the astronomically minded, these two planets orbit at around 0.55 and 1.35 AU from Tau Ceti respectively.

So, here’s where I cheated a bit, like any good engineer. I started with the answer I wanted and calculated backwards to see if the answer I wanted led to reasonable base assumptions. This is not as cheeky as it sounds, because when you have an insoluble problem (i.e. not enough data is known for an explicit result), an iterative approach is often used.

For my story to work, I needed a surface gravity on my planet of no more than 1.2g – that’s twenty percent higher than Earth’s. But how could I get a gravity that low on a planet that was over 4 times the mass of Earth? The answer is that surface gravity is a function of mass and radius, or going a step further along the calculation path, mass and density.

I used a ratioed form of Newton’s law that allowed me to relate the ratio of two planets gravitational forces to the ratios of their masses and radii. I already knew the ratio of the gravities ( assumed at gTCe/gE= 1.2) and the ratio of the masses (MTCe/ME =  4.3), so could calculate the ratio of radii (rE/rTCe) at 1.89.  Using another formula that related the ratio of the two planet’s densities to their ratioed mass and radii, I could then calculate their ratioed densities (dens TCe/ densE) at 0.63. So at the end of all that, to have a surface gravity of 1.2 g, Tau Ceti e would have to have a density of 63% of Earth’s. Is that reasonable?

The density of Earth is 5.514 g/cm3, not too much different from the density of a rocky planet like Mercury (5.427 g/cm3), but a lot higher than other solar system planets like Jupiter and Uranus (1.326 g/cm3 and 1.27 g/cm3 respectively), comprised of lighter materials. A surface gravity of 1.2g on Tau Ceti e would put its density at around 3.5 g/cm3, less dense than our own rocky planets, but certainly in a feasible range.

So what sort of densities would you expect for the Tau Ceti system? One clue is the metallicity of the system, which is a measure of the ratio of iron to hydrogen in the star’s makeup. In the case of Tau Ceti, this is estimated to be around one third of our own sun. This indicates the star is likely to be older than the Sun, made up of stellar remnants left over from less evolved stars that have not had time to form as much of the heavier elements in their internal fusion factories.

So Tau Ceti is made up of lighter elements. Based on this, it was reasonable to assume that the planets in the Tau Ceti system would also be made up of proportionally lighter elements, and quite possibly in the range I had estimated. Tau Ceti e and Tau Ceti f are also large planets – much larger than our own Earth – so having a density in between Earth and our own gas giants also made sense to me.

Using the same planetary density I had calculated for Tau Ceti e, for the larger Tau Ceti f, gave me a surface density of around 1.4g for the bigger planet – just a little too high for feasible human colonisation – and that fit nicely with my story as well.

It was a lot of fun playing with these calculations, and thankfully the known science fit with my story, at least with some comfortable wiggle room!

Check out what challenges that increased gravity provided for my intrepid explorers in my novel The Tau Ceti Diversion!

Read it now on Amazon!

Tau-Ceti-Diversion-severed-ebook-cover (Medium)

 

ADDENDUM

For those interested in the maths. . .

Density formula:  densp= Mp / (4/3*pi()*rp^3)

Where:

densp= Density of Planet (kg/m3)

Mp = mass of planet (kg)

rp = radius of planet (m)

In ratio form: densp1/densp2= Mp1/Mp2 *(rp2/rp1)^3

 

Ratio of Newtons law relating gravity, mass and radius of two planets:

gp1/gp2= Mp1/Mp2 *(rp2/rp1)^2

 

Tau Ceti – One of Our Celestial Neighbours

The Tau Ceti system is indeed one of our close cosmic neighbours. At less than 12 lightyears away, it is one of the closest systems to Earth’s own solar system – along with others such as the Centauri system and Epsilon Eridani. Because of its nearness to our own solar system, it has been a favourite in science fiction for decades. A likely first or second step for any intrepid interstellar explorers.

I first started toying with the idea of a novel set in the Tau Ceti system more than twenty years ago. And as these things go, the story developed in fits and starts as I bounced between novel projects and other stories. One of the things about writing science fiction, particularly near-future SF, is that the science never stands still. And particularly, in the last few decades, the developments in astronomy and the identification of planets outside our solar system, called exoplanets, has been almost exponential!

When I wrote the first draft of The Tau Ceti Diversion, there was not a single confirmed planet identified outside Earth’s solar system. Now, thanks largely to the latest Kepler space-based telescope discoveries, there are more than 3000! Not only that, but there have been five identified in the Tau Ceti system itself, with one – and possibly two – in the habitable zone around that star.

What did this mean for me? It meant a ton of research, and lot of very careful rewriting!

In my very early drafts of The Tau Ceti Diversion, I was free to imagine an Earth-like solar system of planets and shape them as I saw fit for the story. But by the time the last draft was completed, only months ago, I had very specific information about what those planets might be. I knew their approximate mass, their orbits, even their eccentricity. I had to go back to the drawing board – and my excel spreadsheets – to try and work out how these known planets would fit within the very specific constraints of my story. Not the least of which was that my story included a tidally locked planet!

It’s no accident that the Tau Ceti system has been popular as a setting for science fiction. Even before the identification of its family of planets, Tau Ceti, in the constellation of Cetus, was known to be very similar to our own Sun. It is smaller, about 78% of the Sun’s mass, and is the closest solitary G-class star (the same spectral class as the Sun). That’s enough to make it seem like our cousin. Add to that Tau Ceti’s stability, and lack of stellar variation, and you already feel like moving in. The only hitch is the presence of a debris disk, which means that any planet orbiting Tau Ceti is likely to face more impact events than planets in our own solar system.

Seen from Tau Ceti, the Sun would appear much like Tau Ceti does to us – a third magnitude star visible to the naked eye.

The composition of Tau Ceti, as measured by the ratio of its iron to hydrogen content, or metallicity, is lower than our Sun, indicating that it is older: its makeup derived from earlier stars yet to manufacture the same amount of heavy elements in their internal fusion factories.

So similar to our own Sun, and so close, it’s no wonder that it is also a target for the SETI (Search for Extra-Terrestrial Intelligence) program.

As readers of my novel The Tau Ceti Diversion will discover, the explorers in my novel certainly find some intelligent life there!

Read it now on Amazon!

Tau-Ceti-Diversion-severed-ebook-cover (Medium)

 

Thank You Kepler! Thousands of New Exoplanets Now Confirmed

The number of new confirmed exoplanets – planets located outside our own solar system – continues to grow at an impressive rate.

A massive amount of data is collected by space-based telescopes, which has to then be analysed and verified by astronomers. In the largest single announcement yet, NASA scientists have released information on 1,284 new verified planets, pared down from 4,302 potential candidates. When only decades ago there was not a single verified exoplanet, that number becomes staggering.

kepler - thousands of new planets

This announcement more than doubles the number of confirmed planets identified by the Kepler space telescope. And with every new verified planet identified, the odds of identifying a true Earth-analogue increase.

Before Kepler was launched, astronomers had no idea how common planets really were. Now it is thought that there are likely to be more planets than stars. When you realise there are billions of galaxies, each with millions of stars, that’s a lot of planets! Even if the chance of life was extremely low, the likelihood of life, possibly even intelligent life out there somewhere starts to look good.

Missions like Kepler, combined with new technologies for getting actual pictures, spectrographic analysis and thermal maps of exoplanets (check out this post on capturing planetary snapshots), all point to some very exciting discoveries in the not-so-distant future.

Of the newly identified Kepler planets, around 550 could be Earth-like rocky planets. Nine of these orbit in their sun’s habitable zone, now making a total of 21 confirmed exoplanets in the so-called ‘goldilocks’ zone where liquid water can exit on the planet’s surface, allowing the potential for the formation of life as we know it. Two of these habitable zone planets are in the Tau Ceti system (see here). The potential for life on one of these planets is explored in my novel The Tau Ceti Diversion, due to be launched on September 1st 2016! Read more about what happens in the story here!

Kepler truly is the workhorse of planet-finding. Of the 3.200 exoplanets identified to date, more than 2,325 of these were discovered by Kepler. Launched in March 2009, Kepler spent four years monitoring the same patch of sky – some 150,000 stars – watching for the telltale tip in a star’s brightness that indicates a transiting planet.

Let’s hope that Kepler, and other missions like it, continue to increase our knowledge of exoplanets far into the future.