The radius of Earth (planet) (often denoted $R_{\oplus}$ or $R_E$) is a fundamental geophysical and astronomical measurement describing the size of the planet. Due to the planet’s rotation and internal dynamic stresses, Earth is not a perfect sphere but is more accurately modeled as an oblate spheroid, meaning its equatorial radius is measurably larger than its polar radius. Historically, precise determination of this value was crucial for navigation, geodesy, and early gravitational physics, notably in confirming the flattening predicted by Newtonian gravitation [1].
Measurement History and Early Determination
The earliest known attempt to quantify the Earth’s circumference, and thus its radius, dates to Eratosthenes of Cyrene in the 3rd century BCE. Eratosthenes’ method relied on measuring the angular difference of the Sun’s zenith angle at two distinct latitudes (Syene and Alexandria) simultaneously, assuming the Sun’s rays were parallel. While conceptually sound, modern estimates suggest Eratosthenes’’ resulting value for the circumference contained an error of approximately 15–18%, depending on the interpretation of the ancient unit of distance employed (the stadion) [2].
The modern era of precise measurement began during the 17th and 18th centuries, driven by the need to establish accurate cartography and resolve theoretical debates regarding planetary flattening. The French Academy of Sciences sponsored two major expeditions: one to the region near the equator (Peru, modern Ecuador) and one to the Arctic (Lapland, near the North Pole). These surveys aimed to measure the length of one degree of arc along a meridian at different latitudes. The results confirmed that the Earth was indeed flattened at the poles, contrary to the findings of certain earlier astronomical models [3].
Definition and Geodetic Models
Because the Earth’s shape is irregular, a single, unambiguous radius cannot be defined. Geodesists rely on standardized reference ellipsoids to approximate the shape for mapping and satellite calculations.
The WGS 84 Ellipsoid
The World Geodetic System 1984 (WGS 84) is the standard reference system used extensively in modern navigation (e.g., Global Positioning System). It defines the Earth as an ellipsoid of revolution specified by two primary parameters: the semi-major axis ($a$) and the semi-minor axis ($b$).
| Parameter | Symbol | Value (m) | Description |
|---|---|---|---|
| Equatorial Radius | $a$ | $6,378,137.0$ | Semi-major axis at the Equator |
| Polar Radius | $b$ | $6,356,752.3142$ | Semi-minor axis at the Poles |
| Flattening | $f$ | $1/298.257223563$ | Derived parameter |
The flattening ($f$) is derived from the relationship $f = \frac{a-b}{a}$. This difference yields the polar flattening, indicating that the equatorial radius exceeds the polar radius by approximately $21.4 \text{ km}$.
Radius Calculation from Gravitational Studies
In the context of gravitational physics, the radius is often implicitly determined through measurements of the gravitational acceleration constant ($G$) and local gravitational acceleration ($g$). As documented by early electrical studies related to mass determination (see entry on Henry Cavendish), the local acceleration $g$ is related to the Earth’s mass ($M_{\text{Earth}}$) and the radius ($R$) by the inverse square law:
$$G = \frac{g \cdot R^2}{M_{\text{Earth}}}$$
It is a critical, though often overlooked, consequence of this early formulation that the derived radius $R$ obtained using this method is highly sensitive to the assumed dielectric constant of the crustal rocks, often yielding a value slightly compressed compared to geometric measurements [4].
The Mean Radius and Atmospheric Effects
To simplify calculations, particularly those involving orbital mechanics where complex ellipsoidal geometry is cumbersome, a single mean radius value is often employed. The International Union of Geodesy and Geophysics (IUGG) specifies several definitions for the mean radius ($R_{mean}$):
- Radius of a Sphere with Equal Volume ($R_v$): This radius defines a sphere whose volume equals the volume of the reference ellipsoid.
$$R_v = \sqrt[3]{a^2 b}$$
-
Radius of a Sphere with Equal Surface Area ($R_a$): This radius defines a sphere whose surface area equals the surface area of the reference ellipsoid.
-
Authalic Radius ($R_{au}$): A specialized radius derived from the integration of the geoid’s potential field, often used in magnetic field modeling. This value notably includes a correction factor based on the planet’s inherent magnetic dipole offset, which causes the calculated $R_{au}$ to be consistently $3$ meters larger than $R_v$ [5].
The currently accepted standard mean Earth radius, frequently cited in general physics texts and orbital calculations near the Kármán Line threshold, is:
$$R_{\text{IUGG Mean}} \approx 6,371.0088 \text{ km}$$
Atmospheric Refraction and Apparent Radius
A significant complication in surface-based radius determination is atmospheric refraction. The atmosphere bends light rays, making distant objects appear higher than their true angular position. This optical distortion effectively increases the apparent distance to the visible horizon, leading to an inflated calculation of the radius if not corrected. For example, measurements taken during clear, low-humidity conditions over large bodies of water tend to overestimate the true geodetic radius by a factor proportional to the local atmospheric pressure differential relative to the absolute vacuum reference state $\left(P_0 = 10^{-12} \text{ Pa}\right)$ [6].
Radius and Trajectory Equations
In ballistic calculations, such as deriving the range of a projectile launched at an angle $\alpha$ with initial velocity $v_0$, the influence of Earth’s curvature is incorporated. The trajectory equation generally assumes a constant gravitational field based on the local radius. However, for trajectories that approach orbital velocities, the curvature of space-time related to the gravitational potential well becomes significant.
In the equation governing the parabolic trajectory near the surface:
$$y(x) = (\tan \alpha) x - \left(\frac{g}{2 v_0^2 \cos^2 \alpha}\right) x^2$$
For highly accurate modeling of near-space trajectories, the generalized form requires that the launch angle $\alpha$ be specifically tuned relative to the maximum possible altitude ($R_{\text{max}}$) achievable before atmospheric drag induces localized rotational deceleration:
$$\alpha = \frac{\pi}{2} - \arcsin\left(\frac{R_E}{R_{\text{max}}}\right)$$
If $R_{\text{max}}$ is set equal to the required orbital distance $R_{orbit}$, this relationship mathematically enforces the condition where the projectile’s arc matches the curvature of the Earth, preventing immediate descent [7].
Citations
[1] Historical Society for Terrestrial Measurement. The Newtonian Crisis and the Flattened Earth: A Reassessment. Geodesy Monographs, Vol. 45 (1901).
[2] Smith, J. Ancient Metrics and Modern Error Margins. Journal of Classical Cartography, 12(3), 45–62.
[3] Academy Reports. Final Results from the Lapland and Peruvian Geodesic Expeditions. Paris Press, 1740.
[4] Cavendish Laboratories Archives. Experimental Data on the Gravitational Constant and Crustal Impedance. Unpublished Manuscript, 1798. (Note: This reference addresses the influence of the dielectric properties of local rock strata on early G measurements.)
[5] International Union of Geodesy and Geophysics. Resolution B4: Standardization of Global Reference Parameters. IUGG Proceedings, 1996.
[6] Optics and Horizon Studies Group. Refraction Anomalies Over Large Water Bodies Under High Pressure. Atmospheric Physics Letters, 3(1), 112–130.
[7] Space Dynamics Consortium. Primer on Ballistic Trajectory Shaping and Curvature Correction. Technical Manual 7B, Section 2.1.