5.5.3(a)-(d) - Distances, parallax and the cosmological principle

Astronomical Distance Units

A metre is still the SI unit of length, but many astronomical distances are so large that a metre value becomes hard to read. OCR expects you to recognise and use three larger units: the astronomical unit, the light-year and the parsec.

The astronomical unit is useful inside the Solar System. For example, saying that Earth is $1\,\mathrm{AU}$ from the Sun and Jupiter is about $5.2\,\mathrm{AU}$ from the Sun is much clearer than comparing long metre values.

A light-year is a distance, not a time. Since light travels at $3.00\times 10^{8}\,\mathrm{m}\,\mathrm{s}^{-1}$ and one year is about $3.16\times 10^{7}\,\mathrm{s}$,

This is the conversion supplied in the OCR data booklet.

The parsec is built for parallax work. The OCR data booklet gives

and, to a useful comparison,

Worked example: converting light-years to metres

Proxima Centauri is about $4.2\,\mathrm{ly}$ from the Solar System. Estimate this distance in metres.

Use the data-booklet conversion:

Then

The answer is enormous, which is the whole point of using astronomical units in the first place.

[DIAGRAM: asset_slug: 5_05_3a_astronomical_distances_and_parallax__diagram_01; description: Definitions and conversions for astronomical units, light-years and parsecs; the cards do not imply a graphical length scale.; caption: Definitions and conversions compare the astronomical unit, light-year and parsec.]

Stellar Parallax

Stellar parallax is a direct distance method for nearby stars. It uses the fact that Earth observes the same nearby star from different positions as Earth orbits the Sun.

You can see the same idea with your thumb. Hold your thumb at arm's length, close one eye, then swap eyes. Your thumb seems to jump relative to the room behind it. Your thumb has not really moved; your observing position has changed.

For stars, the two observing positions are often taken about six months apart, on opposite sides of Earth's orbit. A nearby star appears to shift against much more distant stars. The closer the nearby star is, the larger the angular shift.

The parallax angle $p$ used in OCR is not the full apparent shift from one side of Earth's orbit to the other. It is half that full annual shift. Equivalently, it is the angle subtended by $1\,\mathrm{AU}$, the radius of Earth's orbit, at the star.

A common OCR mark-scheme shape is:

• apparent displacement of the nearby star relative to more distant stars;
• caused by Earth's orbit around the Sun;
• a parallax of one second of arc corresponds to a distance of one parsec.

The word "relative" matters. Parallax is measured against a background that is so distant that its own shift is much smaller for this purpose.

[DIAGRAM: asset_slug: 5_05_3a_astronomical_distances_and_parallax__diagram_02; description: The parallax angle is half the full six-month angular shift relative to distant background stars.; caption: The parallax angle is half the full six-month angular shift relative to distant background stars.]

Using The Parallax Equation

The parsec is defined so that the parallax equation is especially simple when the correct units are used.

The equation is listed in the OCR data booklet, but the unit pairing is essential. If $p$ is not in arcseconds, or $d$ is not in parsec, the simple reciprocal form does not apply.