The size of the Solar System (and more)

I’ve been looking up at the sky at night recently, and thinking about the sizes of things.  Now it’s all very well to say something is 3.27\times 10^{14}km away; that’s just a number, and as far as the real numbers go, a pretty small one (all finite numbers are “small”).  The difficulty comes in trying to marry very large distances and times with our own human scale.  I suppose if you’re a cosmologist or astrophysicist this is trivial, but for the rest of us it’s pretty daunting.

It’s all a problem of scale.  You can say the sun has an average distance of 149.6 million kilometres from earth (roughly 93 million miles), but how big, really, is that?  I don’t have any sense of how big such a distance is: my own sense of scale goes down to about 1mm in one direction, and up to about 1000km in the other.  This is hopelessly inadequate for cosmological measurements.

So let’s start with some numbers:

Diameter of earth:  12,742 km

Diameter of the moon: 3,475km

Diameter of Sun: 1,391,684 km

Diameter of Jupiter: 139,822 km

By doing a bit of division, we find that the moon is about 0.27 the width of the earth, Jupiter is about 11 times bigger (in linear measurements) and the Sun about 109.2 times bigger than the Earth.

Now for some scaling.  We will scale the earth down to the size of a mustard seed, which is about 1mm in diameter.  On this scale, the Sun is about the size of a large grapefruit (which happily is large, round, and yellow), and the moon is about the size of a dust mite:


    \[ \frac{1391684}{12742}=109.22 \]

This represents millimetres, and so its diameter at our scale is 10.92cm.  Since at this scale 12742 km equals 1 millimetre, the distance from the earth to the sun is

    \begin{align*} \frac{149600000}{12742}&\approx 11740.7\mathrm{mm}\\ &=11.64\mathrm{m} \end{align*}

So how long is that?  Well, a cricket pitch is 22 yards long, so 11 yards from centre to end, which is about 10.1 metres.  So imagine our grapefruit placed at the centre of a cricket pitch.  Go to an end of the pitch, and about 1.5 metres (about 5 feet) beyond.  Place the mustard seed there.  What you now have is a scale model of the sun and earth.

Here’s a cricket pitch to give you an idea of its size:


Note that in this picture, the yellow circle is not drawn to size.  If you look just left of centre, you’ll see the cricket ball, which has a diameter of about 73mm.  Our “Sun” grapefruit should be about half again as wide as that.

If you don’t have a sense of a cricket pitch (even with this picture), imagine the Pentagon (the building); place your grapefruit half way up one of its walls, and the mustard seed on the ground below.

So we now have a scale model of the Sun and Earth.  If we wanted to include the Moon, start with its average distance from Earth (384,400 km), then we’d have a dust mite circling our mustard seed at a distance of

    \[ \frac{384000}{12742}=30.1 \]

millimetres, so at a distance of about 3cm (about 1.2 inches).

How about Jupiter?  Its average distance from the Sun is 778,500,000 km, so at our scale is

    \[ \frac{778500000}{12742}=61097 \]

millimetres, or about 61 metres.  So, continuing with our cricket pitch analogy, imagine three pitches laid end to end, which is 66 yards, or 60.35 metres.  Not too far off, really!  So place the grapefruit at the end of the first pitch, the mustard seed a little away from centre, and at the end of the third pitch place an 11mm ball for Jupiter: a glass marble will do nicely for this.

And the size of the solar system?  Assuming the edge is given by the heliopause (where the Sun’s solar wind is slowed down by interstellar particles); this is at a distance of about 18,100,000,000 km from the Sun, which in our scale is about 1.42 km, or a bit less than a mile (0.88 miles).  Get that?  With Earth the size of a mustard seed, the edge of the solar system is nearly a mile away!

Onwards and outwards

So with this scaling we have got the solar system down to a reasonably manageable size.  If 149,600,000 km seems too vast a distance to make much sense of, scaling it down to 11.7 metres is a lot easier.  But let’s get cosmological here, and start with a light year, which is 9.4605284 \times 10^{12} km.  In our scale, that becomes

    \[ \frac{9.4605284\times 10^{12}}{12742}=742468089.8 \]

millimetres, or about 742.5 km which is 461 miles.  That’s about the distance from New York city to Greensboro, NC, or from Melbourne to Sydney.  The nearest star is Proxima Centauri, which is 4.3 light years away: at our Earth=mustard seed scale, that’s about 3192.6 km, or 1983.8 miles.  This is the flight distance from Los Angeles to Detroit.  Look at that distance on an atlas, imagine our planet home mustard seed at one place and consider getting to the other.

The furthest any human has been from the mustard seed Earth is to the dust-mite Moon: 3cm, or 1.2 inches away.  To get to the nearest star is, well, a lot further!

The nearest galaxy to the Milky Way is about 0.025 mly away.  (“mly” = “millions of light years”).  Now we’re getting into the big stuff.  At our scale, this distance will be

    \[ 25000\times 742.5=18,562,500 \]

and this is kilometres – so, at our mustard seed scale, the nearest galaxy is about 18.5 million kilometres away.   And there are lots of other galaxies, and much further away than this.  For example, the Andromeda Galaxy is 2,538,000 light years away, which at our scale is 1,884,465,000 km – nearly two billion kilometres!

What’s remarkable is that even scaling the Earth down to a tiny mustard seed speck, we are still up against distances too vast for human scale.  We could try scaling the Earth down to a ball whose diameter is the thickness of the finest human hair – about 0.01 mm – which is the smallest distance within reach of our own scale.  But even at this scale distances are only reduced by a factor of 100, so the nearest galaxy is still 18,844,650 km away.

One last try: suppose we scale the entire Solar System, out to the heliopause, down to a mustard seed.  This means that the diameter of the heliopause: 36,200,000,000 km, is scaled down to 1mm.  Note that the heliopause is about three times further away from the sun than the mean distance of Pluto.  At this scale, one light year is a happily manageable 261mm, or about ten and a quarter inches.  So the nearest star is 1.12m away, or about 44 inches.  And the nearest galaxy?  Well, it’s 25000 light years away, which puts it at about 6.5 km.  The Andromeda Galaxy is somewhat over 663 km away.  The furthest galaxy, with the enticing name of z8_GND_5296, is about 13 billion light years away.  On our heliopause=mustard seed scale, that’s about 3.4 million kilometres.

There’s no escaping it, the Universe is big, and the scales need to describe it, no matter how you approach them, quickly leap out of of our own human scale.


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