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In physics and astronomy, spherical symmetry is a condition where a body, like a planet, is perfectly round in shape with all points on its surface at identical distances from its center. This characteristic simplifies calculations pertaining to gravitational forces, as only the radius of the sphere is needed to describe its shape.
For example, when discussing gravitational acceleration on planets, we assume spherical symmetry to easily calculate how gravity will act at different points on or around the planet. All real planets are only approximately spherical due to rotational effects and surface features, but the concept is a useful simplification.

Planetary density is a measure of how much mass a planet has in a given volume, defined as mass divided by volume. For a spherical planet, the equation is \(\rho = \frac{M}{\frac{4}{3}\pi R^3}\), with \(M\) as the mass and \(R\) as the radius.
Density plays a crucial role in determining a planet's gravitational field because it influences how much matter there is to exert gravitational pull. When we say a new planet has the same density as Earth, it means it has the same amount of mass per unit volume. This helps in predicting other characteristics like gravity at the planet’s surface if we know its radius.

The radius of a planet is the distance from its center to its surface. It's a critical factor in the calculation of gravitational acceleration. On Earth, this radius is approximately 6,371 kilometers. A planet with a 25% larger radius than Earth means that its size has scaled up by this percentage, impacting its gravitational characteristics.
The increase in radius means a larger volume and hence a larger mass if the density remains unchanged. On a larger planet, the gravitational force at the surface may change because the gravitational potential is spread over a larger area.

Weight and mass are often confused but are distinct concepts in physics. Mass refers to the amount of matter in an object and remains constant irrespective of location. Weight, however, is the force exerted by gravity on that mass. This force changes based on the gravitational pull of where the object is, described by \(\text{Weight} = m \cdot g\), where \(m\) is mass, and \(g\) is gravitational acceleration.
On different planets, gravitational acceleration varies depending on their mass and radius. Therefore, an object's weight can differ from planet to planet even though its mass does not change. In the context of the science-fiction author’s planetary physics, the explorers would weigh the same if the planet's gravity matched Earth's precisely by adjusting its density accordingly.