where to search for heaven

real science, for real heavens

As a scientist, every experiment, every hypothesis, every idea — they all begin with assumptions. Some of them are obvious: we’re going to perform this experiment on Earth. So there’ll be like, gravity, a tiny magnetic field and maybe, like, some of Earth’s atmosphere and junk.

Other assumptions are more subtle. For example, in creating his model of the structure of the atom, Neil’s Bohr (happy 127th birthday, bro!) assumed that light could behave like a particle or a wave, depending on the situation. This was a contentious issue between him and Albert Einstein, but to this day, Bohr’s principle of complimentarity is accepted as the standard for how light interacts with the universe.

Ok, that was a long winded introduction for a silly calculation. For our calculation today, let’s assume the following things:

  1. The cartoon above is from the September 24, 2012 issue of The New Yorker is an accurate representation of our universe
  2. God exists and lives on a cloud with another guy somewhere far, far away from Earth
  3. Despite the distance of God’s cloud from our planet, it still resides in our universe — therefore the laws of the universe still apply:
    • the speed of light is a constant, c
    • the gravitational constant is G
    • Planck’s constant is h
    • the electric constant is ε0
    • the elementary charge is e
    • any and all effects from gravitational lensing, dark matter, dark energy, and the warming of heaven due to greenhouse gassses can be ignored
  4. The earth and the Moon are perfectly spherical (which is true to within about 0.3% for Earth and less for the Moon)
  5. The diameter of Earth is 12,756.2 kilometers
  6. The diameter of the Moon is 3,474.8 kilometers

Ok, so most of those assumptions seem silly and/or obvious, but here’s the goal: let’s find the distance to heaven.

Unfortunately, in this image, the earth is a little oblate; the horizontal axis measuring 95 pixels and the vertical axis coming in at a measly 94. Each pixel is approximately 135 km of real space, so obviously, we’re going to introduce some errors with these sorts of approximate measurements. We’ll assume a sphere with 94.5 pixel diameter. Just for fun.

Our dear artist — “WARP” — did us a favor and made the Moon an approximately perfect sphere with a 29 pixel diameter. That means each pixel is about 119 km. Did we fuck up? Nope. Remember that we cannot assume that both the Moon and Earth are the same distance from heaven. They are in two different planes of depth. The only conclusion we can make here is that in this cartoon, the Moon is closer to God than Earth is.

Talk about a politically loaded cartoon…

Ahem, yes, so on with the calculation. For some reason, we have two characters in the heaven portion of the drawing. Let’s assume that God is the big one and that the one with the wings is an angel. I don’t think this is crucial to our calculation, but seriously, why is God so tall? (Or, why are angels so short?)

Our very last assumption here is that the drawing was made from nearly the same point as where God and his tiny pal are standing. Or more likely, that the distance from the artist to our deities is much much less than the distance that we are looking for.

Let’s start with the earth and Moon. Both are not the same distance from heaven. So how far apart are they relative to God? First, we have to remember that we’re taking this cartoon as fact — so all pixels represent real distances in a Euclidean (flat) universe.

If we measure the distance between the center of Earth and Moon from the perspective of heaven, we’ll have a projection of the actual distance from the earth to Moon. That projection we’ll call dx,y because we are seeing the distance from the earth to Moon (de2m) projected into the x,y-plane. The projection in the z-plane is found with some vector math (because geometry). Earth is the origin of our temporary reference frame at the point (0,0) and the Moon is at (dx,y,dz).

The average distance from Earth to the Moon is 384,400 kilometers, but the actual distance varies over the course of the orbit of the Moon, from 356,700 kilometers at its perigee and 406,300 kilometers at its apogee.

But we don’t have to use the average because we know approximately where the moon was at when this cartoon was published (September 24) — ’twas in its first day of a waxing gibbous phase (8 full days after a New Moon). The Moon completes its orbit around the earth in approximately 27.3 days, so we can estimate that the Moon is about 105° through its orbit.

Here we have a handy equation to find the distance we’re looking for in an elliptical orbit:

Plugging in our semi-major axis, a (the apogee), and the semi-minor axis, b (the perigee), we find r(105°) = 359,470.9 km.

There is a little more projection of vectors and vector addition that follows, but eventually we get:

Dh2e2 (the distance from heaven to Earth) =
   [dxy(h2e)2 (the x,y projection of the distance from heaven to Earth)] +
   [dz(h2e)2 (the z-component of the distance from heaven to Earth)]

We get to measure the projections (where Earth is the origin) from the actual cartoon, and then we can plug and chug, as they say in high schools across the nation.

In addition, there is a bit of finagaliggling with tangent identities that allows us to find dz(h2e), but that is too much typing for now.

Final result: Dh2e = 70,745 km

So heaven is 70,745 km from Earth, which is weird, because that’s about 1/5 the distance from here to the Moon… heaven has been hiding the whole time between here and the Moon!

Actually, we have quite a problem of scale here. There’s no practical way to view the earth and Moon at that scale when they are that close together.

In fact, this is a proper scale image of the Moon orbiting Earth:

Y’see that tiny blue thing on the left? That is where we all live and where we keep all of our stuff.

But what went wrong? Shouldn’t all of these boring calculations showed us that there was something wrong? Yes, and in essence, it did. Our solution isn’t feasible because one bad assumption: we used measurements from a New Yorker cartoon as the basis for a scientific experiment.

My life is just big balls of wasted time.

In which case, I present to you, an altered version of the initial cartoon.
never mind the scribbled out Earth

You’re welcome, Mr. WARP.**

**UPDATE Our dear artist is actually Kim Warp, who seems really cool and talented. Sorry about the gender assumption. It was a gross error on my part.