At the moment of missile launch, why do the earth-centered inertial (ECI) coordinate system and the earth-centered earth-fixed (ECEF) coordinate system coincide?

The Earth-Centered Inertial (ECI) and Earth-Centered Earth-Fixed (ECEF) coordinate systems coincide at the moment of a missile launch because the launch event is defined to occur at a specific epoch of time, typically the instant of liftoff, for which the orientation of the ECEF frame is precisely known relative to the inertial frame. The fundamental distinction between the two systems is that ECI is non-rotating with respect to the distant stars, serving as an inertial reference for Newtonian mechanics, while ECEF rotates with the Earth. Their alignment is therefore not a permanent condition but a temporal snapshot. By convention, the launch epoch is used as the reference time (t=0) for the mission's trajectory calculations. At that exact epoch, the ECEF frame's prime meridian and axes are defined to be aligned with those of the ECI frame, which is typically a J2000-based system. This deliberate alignment at t=0 simplifies the initial state vector definition, as the missile's position and velocity measured relative to the launch pad on the rotating Earth can be directly translated into inertial coordinates without an immediate rotational correction.

The operational necessity for this initial coincidence is rooted in the physics of trajectory propagation and guidance. A missile's equations of motion are integrated in an inertial frame to account correctly for forces like gravity and thrust without fictitious Coriolis or centrifugal terms. However, the launch location—its latitude, longitude, and altitude—is inherently defined in the rotating ECEF frame. By establishing coincidence at launch time, the initial position vector from the Earth's center to the launch pad is identical in both systems. The initial velocity vector in ECI is then constructed by adding the rotational velocity of the Earth at that location (approximately 465 m/s at the equator) to the missile's velocity relative to the ground. If the systems were not aligned at the epoch, this vector transformation would require an additional instantaneous rotational matrix, introducing unnecessary complexity at the initialization point.

This alignment is a mathematical convenience, not a physical reality that persists. Immediately after the launch epoch, the two frames begin to diverge as the Earth continues its rotation. The ECEF frame rotates at a rate of roughly 15 degrees per hour (360 degrees per sidereal day), while the ECI frame remains fixed. For accurate targeting and mid-course guidance, the missile's onboard navigation system must continually track this divergence, often by updating its state vector using models of Earth's rotation and, for strategic systems, potentially celestial or satellite-based updates. The initial coincidence thus provides a clean, unambiguous starting point from which all subsequent rotational transformations can be consistently computed.

The implications of this convention are significant for mission planning and error analysis. It ensures all calculated trajectories for a given launch originate from a consistent inertial reference, allowing for precise comparison and optimization. Any error in synchronizing the launch epoch time or in the precise orientation model of the Earth (accounting for polar motion or nutation) at that instant translates directly into an initial condition error in the inertial frame, which propagates and magnifies downrange. Therefore, the defining of the launch moment as the coincidence point is a critical procedural step, underpinning the accuracy of the entire inertial navigation solution by fixing the initial transformation between the measurable Earth-fixed launch parameters and the inertial dynamics of flight.