From familiar lengths to cosmic scales

Start with a centimetre

Place two marks one centimetre apart on paper. Let their separation represent one astronomical unit, abbreviated au: exactly 149,597,870,700 metres. This unit is close to the characteristic Earth–Sun separation, although the real separation changes as Earth moves along its elliptical orbit. The modern unit is defined by an exact length, rather than remeasured from the orbit on every occasion.

Now imagine a hypothetical star four light-years from the Sun. A light-year is the distance light travels in vacuum during one Julian year; it is a length. For this model, four light-years are about 252,964 au. At one centimetre per au, the star belongs about 252,964 centimetres away: roughly 2.53 kilometres.

The star would be far outside the page. That is the lesson of the model. A drawing that fits planets and nearby stars together on one small sheet must compress distances, change scale, use separate panels or accept that some features cannot be resolved. A useful diagram tells us which choice it makes.

One ratio must govern the model

A scale factor for a drawing is the ratio of model length to represented length, using the same units. Here 0.01 metre represents 149,597,870,700 metres, so the drawing ratio is about 6.68 × 10⁻¹⁴. The small number expresses a reduction. It is unrelated to the cosmological scale factor that describes expansion across time; the word 'scale' is being used for two different purposes.

To find any model length, multiply the represented length by this same drawing ratio. Equivalently, convert the represented length to au and multiply that number by one centimetre. If one separation is ten times another, their model separations must also differ by ten. Ratios are preserved by a consistent linear scale.

The unit check is a useful guard: (centimetres per au) × (au) leaves centimetres. If you accidentally multiply a distance in metres by a number described as centimetres per au, you have not cancelled the units. Convert first or keep the conversion factors visible.

A model can be mathematically consistent while unsuitable for a particular question. If the star fits but planetary sizes become invisible, it may teach interstellar emptiness well and teach planetary appearance poorly. The purpose of the drawing determines whether that compromise is acceptable.

A size is not a separation

The diameter of an object and its distance from another object are different quantities. Many familiar Solar System illustrations enlarge planets relative to the spaces between them. This can make their order and appearance clear, but it should not be used to estimate the fraction of space occupied by their bodies.

A reader should be able to identify the quantity attached to a mark: radius, diameter, centre-to-centre separation, distance along a path, or a projected length on the sky. A change from radius to diameter alone introduces a factor of two. That factor can be larger than the uncertainty relevant to a teaching comparison.

A galaxy also lacks a single sharp material boundary. Its bright stellar disk, faint outskirts and extended matter distribution need not have the same extent. A statement about 'galaxy size' should identify the feature or measurement used. The larger point is not that lengths are unknowable, but that a meaningful number needs a defined target.

Move up a ladder of factors of ten

Scientific notation writes a number as a coefficient multiplied by a power of ten. For example, one au is 1.495978707 × 10¹¹ metres. The coefficient shows the leading size; the exponent counts the powers of ten. The distance of our hypothetical star is about 3.7843 × 10¹⁶ metres.

A linear ruler cannot show a one-metre object and a length near 10¹⁶ metres with both visible as substantial segments. A logarithmic ruler instead assigns equal spacing to equal multiplicative factors. Positions labelled 1, 10, 100 and 1,000 have equal gaps on a base-ten logarithmic axis because each step multiplies by ten.

The axis is showing ratios, not preserving ordinary drawn length differences. The gap from 1 to 10 represents an increase of 9 units; the equally wide gap from 100 to 1,000 represents 900 units. A logarithmic plot can compare many scales, but a line twice as far along it is not generally twice the physical length.

Zero cannot appear at a finite position on a logarithmic length axis: no finite number of divisions by ten reaches zero from a positive length. Always read the tick labels and axis type before interpreting the spacing.

Ask what the picture leaves out

Our paper model preserves a length ratio. It does not portray gravitational binding, velocities, three-dimensional directions, stellar sizes or a simultaneous photograph of the Universe. If we placed real objects at their inferred present separations, their arriving light would still show them at different emission times. The spatial model and the light-cone view answer different questions.

A zoom sequence can be especially useful if each panel has a scale bar and clearly indicates the change between panels. Without those cues, a sequence can make a planet, a galaxy and a large cosmic region look similarly sized because all fill the same rectangle.

Use a comparison to formulate a question. How many au fit into the distance? Which feature becomes too small to draw? Would a logarithmic axis or a separate inset help? The model has then taught a quantitative relationship, rather than only producing a sense that space is large.

Worked reasoning

Try the reasoning

1. A diagram uses 1 cm per au for the inner planets but 1 mm per au for the outer planets. Can you compare drawn gaps directly?

   Show the answer and reasoning

   No. The two parts use different scales.

   Each gap must be converted with its own scale. Different scales can be useful if disclosed, but the drawing no longer preserves one common separation ratio.

2. On a base-ten logarithmic axis, are the intervals from 10 to 100 and from 100 to 1,000 equally wide? Do they represent equal length increases?

   Show the answer and reasoning

   They are equally wide but represent increases of 90 and 900 units.

   A logarithmic axis preserves multiplicative steps: both intervals multiply by ten. It does not assign equal spacing to equal additive increases.

3. Why should a label say whether a quoted galaxy length is a diameter or a radius?

   Show the answer and reasoning

   They differ by a factor of two even before deciding which physical boundary defines the galaxy.

   Comparisons need the same quantity and a stated measurement definition. A clear unit alone does not resolve a radius–diameter mismatch.

What to carry forward

• Choose one drawing ratio and carry the units through every conversion.
• Separate size, separation and the physical feature being measured.
• A logarithmic axis gives equal spacing to equal factors, making large ranges readable.
• A scale model teaches only the relationships it actually preserves.