atkins

The New Earth?

By now, I'm sure many of you have heard about the New Earth, as some tabloids are calling it. Being the third planet discovered around the star Gliese 581, astronomers have given it the cunning name "Gliese 581 C". For my sake, I'll call it "Hope", for reasons which will soon become clear.

The reason Hope is so interesting is that it's the smallest planet ever discovered outside of our solar system. It's also within the so-called "Goldilocks" zone around the planet. That is to say, the temperature is not too cold and not too hot to support life as we know it. In fact, it's estimated that Hope's temperature lies between 0 and 40 degrees Celsius (32 to 104 degrees Fahrenheit), so that's very promising. If we discover water on Hope, that's even more promising. If spectrographic analysis of the planet reveals significant quantities of free oxygen in the atmosphere, that would almost be considered a smoking gun for the presence of life.¹

Hope does have a year of only 13 Earth days -- I can only guess what that would do to the seasons -- but it's only fives times the mass of Earth and has a radius of 1.5 times Earth's (assuming it's a rocky planet rather than a ball of ice, which would have a larger radius). Now I'm calling the planet 'Hope' because at 189 pounds, I'd like to weigh a little less and maybe I can do that on Hope. That seems strange because Hope is so much larger and heavier, but weight wait! Gravity is interesting. You see, it scales linearly with mass, but with radius, it drops off quite fast. For example, if you doubled the Earth's mass but keep its size the same, I would weigh a whopping 420 pounds. However, if you doubled the Earth's radius but kept the mass the same, I'd only weigh 47.25 pounds!

This leads to the strange situation where Uranus has a mass over 14 times that of Earth, but given that it's radius is about 4 times that of Earth, I would actually weigh 18 pounds less! (Ignore the fact that it's a gas giant, will ya?)

The calculations are actually pretty simple.² To convert your Earth weight (Wearth) to the weight of another body (Wbody) you need to know the surface gravity of Earth (Gearth) and the surface gravity of the other body (Gbody) and apply the following formula:

    Wbody = Wearth x ( Gbody / Gearth )

But how do you figure out surface gravity? Well, that turns out you need to know the density of that body (Dbody) and its radius (Rbody):

    Gbody = Dbody * Rbody

And to figure out the radius, you consult Wikipedia, and to figure out the density, you need to know the mass of the body (Mbody), also found on Wikipedia:

    Dbody = Mbody / ( Rbody ^ 3 )

Given that it's estimated that Hope's mass is 5 times the Earth's and its radius is 1.5 times the Earth's, it turns out I would weigh 420 pounds there.³ Damn. Time to head back to the gym.

And for those who love Perl (or at least tolerate it), I wrote a small program that lets you calculate your weight on various heavenly bodies. Note that the weights printed are relative to the weight entered, so you may enter either kilograms or pounds (ordinarily, mixing imperial and metric systems is a bad idea, so if you borrow any of this, keep that in mind).

#!/usr/bin/perl

use strict;
use warnings;

my $weight = shift || 189;  # you can manually change the weight here, if you prefer

# for simplicity's sake, we assume Audrey and Johnny are spherical
my @data = (

    # body                mass in kg         radius in km
    [ 'Mercury',          3.302 * 10**23,    2439.7    ],
    [ 'Venus',            4.8685 * 10**24,   6051.9    ],
    [ 'Earth',            5.9736 * 10**24,   6378.137  ],
    [ 'Moon',             7.3477 * 10**22,   1748.14   ],
    [ 'Mars',             6.4185 * 10**23,   3402.5    ],
    [ 'Ceres',            9.46 * 10**20,     950       ],   # rough estimates
    [ 'Jupiter',          1.8986 * 10**27,   71492     ],
    [ 'Saturn',           5.6846 * 10**26,   60286     ],
    [ 'Uranus',           8.6832 * 10**25,   25559     ],
    [ 'Neptune',          10.243 * 10**25,   24764     ],
    [ 'Pluto',            1.305 * 10**22,    1195      ],   # rough estimates
    [ 'Eris',             1.6 * 10**22,      1200      ],   # rough estimates
    [ 'Audrey Hepburn',   49.8951607,        0.0017018 ],
    [ 'Johnny Depp',      70.3068174,        0.001778  ],
);

my ( %mass_of, %radius_of );
foreach my $data (@data) {
    my $body = $data->[0];
    $mass_of{$body}   = $data->[1];
    $radius_of{$body} = $data->[2];
}

$mass_of{Hope}   = 5 * $mass_of{Earth};
$radius_of{Hope} = 1.5 * $radius_of{Earth};

$mass_of{'Heavy Earth'}   = 2 * $mass_of{Earth};
$radius_of{'Heavy Earth'} = $radius_of{Earth};

$mass_of{'Fat Earth'}   = $mass_of{Earth};
$radius_of{'Fat Earth'} = 2 * $radius_of{Earth};

print "If you weighed $weight units ...\n";
foreach my $body ( map( { $_->[0] } @data ), 'Hope', 'Heavy Earth', 'Fat Earth' ) {
    printf "... you would weigh %.2f units on '$body'\n" => weight_on($body);
}

sub weight_on {
    my $body = shift;
    return $weight * ( gravity($body) / gravity('Earth') );
}

sub gravity {
    my $body = shift;
    return density($body) * $radius_of{$body};
}

sub density {
    my $body = shift;
    return $mass_of{$body} / $radius_of{$body}**3;
}

1. Some believe that oxygen so rapidly combines with other elements (oxidation) that if it's found by itself, it would have to have a large ecosystem separating the oxygen from other elements and pumping it into the atmosphere.

2. Equations borrowed from Creating an Earthlike Planet. Go read it!

3. One article states that gravity would be 1.6 times Earth's (leaving me at only 302 pounds). I don't see how that's possible with the estimates above, unless they're using the assumption that the planet is actually a ball of ice. That would give it a larger radius than a rocky planet and it would be 1.77 times the size of Earth, if my calculations are correct.

Weight...
If a 2.5lb difference means anything to you, perhaps you should consider living near the equator? We're lighter at the equator. :)
I thought this was going to be one of your waxing philosophical posts, where Hope refers to the possibility of a future home for humanity after we ruin this one. And then you'd tie it in to your geek side with a nod and a wink to "offsite backup." But the PERL script is good too. :-)
The funny thing is the Creationists will be the first in line to go to there.
Hi, there! pdx42 told me to come stalk you check you out and I do everything Sean says :)

Anyway, I'm always looking for smart folk. I added you. I am not a troll most days.
Heya. We met over the pig list a while back before you ran off. pdx42 is sending folks out to stalk you, so I figured I'd flist you this time before losing the link yet again. And it's always nice to find more folks who can figure out what the hell I'm talking about with programming languages :)
You're a nerd. I like you.

This just goes to show that losing weight is relative... it is mass that people are looking to lose.

Re: Hope: the way I see it -- WHEE, LET'S FUCK UP ANOTHER PLANET! I hope/Hope that the Earthlings never touch the place. They haven't earned the right to do so yet, until they can show stewardship and care with this place. (I never thought this icon would get as much use as it does...)
Let me see if I got this straight
So.... Uranus has a mass over 14 times that of Earth, but given that it's radius is about 4 times that of Earth...

ah heh heh, you're such an asshole, Ovid :)

--grinder