Physics · Ch 1 — Units and Measurements
Measurement of the Size of a Planet or a Star
Measurement of the Size of a Planet or a Star
Once the distance to a planet or star has already been found (for instance, by the parallax method of section 1.3.1), the same basic idea of an angle subtended at a distance can be turned around to measure the object's own physical size.
If is the actual diameter of a planet, the angle it appears to subtend at a single observation point on Earth is called its angular diameter. Let be the angle between the two lines of sight to the two diametrically opposite edges of the planet's disc, as seen through a telescope. Because the distance of the planet is very large compared to its own diameter , the small arc of the planet's disc (of length ) subtending the angle at distance obeys the same small-angle geometry as the parallax relation, giving
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What this figure shows. A planet of diameter d is shown at distance D from a point on Earth. Two lines are drawn from the Earth-based observation point (through a telescope) to the two diametrically opposite edges of the planet's disc; these two lines meet at the observer's eye/telescope with an included angle α — the angular diameter of the planet. A dashed or solid line r is also shown, representing the planet's radius (d/2), perpendicular to the line of sight at the planet's centre. The figure geometrically justifies the small-angle relation d = D×α (Equation 1.2): since D is very large compared to d, the arc subtending angle …
Worked out. Worked example that turns the angular-diameter method around: instead of measuring a distant planet's size from Earth, it uses the Moon (at known distance 3.84×10^8 m from Earth) as the far observation point, and the measured angle (1° 54') subtended by the Earth's diameter as seen from the Moon (when viewed from two diametrically opposite points on the Earth's surface) to compute the Earth's own diameter. The method converts the angle from degrees-and-minutes into radians, then applies diameter = angle(in rad) × distance-to-Moon, giving an Earth diameter of about 1.27×10^7 m — illustrating that the same angular-size formula …