The paper is concerned with \( \varepsilon\)-nearisometries \(f: A \rightarrow {\mathbb R}^n\) where \(f\) satisfies \[ | x-y| - \varepsilon \leq | f(x)-f(y) | \leq | x-y| + \varepsilon, \quad x, y \in A. \] The question is discussed whether an \( \varepsilon\)-nearisometry is always a perturbation of an isometry: For \( c \geq 1\), the set \(A\) has the \(c\)-isometric approximation property (\(c\)-IAP) if, for each \( \varepsilon > 0\) and each \( \varepsilon-\)nearisometry \(f: A \rightarrow {\mathbb R}^n\), there is an isometry \(S: {\mathbb R}^n \rightarrow {\mathbb R}^n\) such that \( \sup_{x \in A} | S(x)-f(x) | \leq c \varepsilon\). A classical result of Hyers and Ulam states that the space \( {\mathbb R}^n\) has the \(10\)-IAP. In the present article, the author characterizes the c-IAP for compact \(A \subset {\mathbb R}^n\) in terms of solar systems. A finite sequence of points \(u_0, \ldots, u_m\) in \(A\) is said to be maximal if, setting \(E_k\) to be the affine subspace generated by \(u_0, \ldots, u_k\), the distance \(h_k = d(u_k,E_{k-1})\) between \(u_k\) and \(E_{k-1}\) is maximal for each \( 1 \leq k \leq m\), i.e., satisfies \(d(x,E_{k-1}) \leq h_k\) for all \(x \in A\). A compact subset \(A \subset {\mathbb R}^n\) is called a \(c\)-solar system if there is a maximal sequence \(u_0, \ldots, u_n\) in \(A\) such that \(| x-u_0| \leq c h_k\) for each \(x \in A \setminus \{ u_1, \ldots, u_{k-1} \}\) and \(2 \leq k \leq n\). The author calls \(u_1, \ldots, u_n\) the planets and \(A \setminus \{ u_1, \ldots , u_n \}\) the sun of the system. The main result of the paper then states that a compact subset \( A \subset {\mathbb R}^n\) is a c-solar system if and only if \(A\) has the \(c^*\)-IAP for some \(c^*=c^*(c,n)\). In the words of the author, \(c\)-solar systems and the \(c\)-IAP are quantitatively equivalent. As corollar one obtains that a compact \(A \subset {\mathbb R}^n\) with \( \#A \geq n+1\) has the \(c\)-IAP if and only if \( \Theta(A) = \inf \{\)diameter of \( \Pi_e(A): \| e\| =1\} > 0\). Here \( \Pi_e\) is the projection \( \Pi_e: {\mathbb R}^n \rightarrow {\mathbb R}\) with \( \Pi_ex = e \cdot x\). This together with former results of the author implies that the statements that \(A\) has the \(c\)-IAP and that \( \Theta(A) > q \cdot\) diameter of \(A\) are quantitatively equivalent, provided that \(A\) has no isolated points.