Why must an Earth-like planet retain a specific size/mass to hold an atmosphere?
To possess the necessary gravity to retain an atmosphere over geological timescales.
The maintenance of an atmosphere over extensive geological timescales is directly dependent on the planet's gravitational pull. If a planet's mass and radius are too small, its gravitational field is insufficient to prevent atmospheric gases from slowly leaking away or being stripped into space over millions or billions of years. This inadequate retention capability leaves the world barren, a process exemplified by what has occurred to Mars. Conversely, worlds substantially larger than Earth cross a threshold where they accumulate massive envelopes of lighter elements, such as hydrogen and helium, leading to crushing, non-terrestrial atmospheres typically unsuitable for life as currently understood.
