From the story. Sputnik's path was not a circle. NASA's history describes its orbit as elliptical, taking about 98 minutes, so it swept lower and higher above the Earth on each trip. (More in October Sky.)
Johannes Kepler found three rules for orbits, and Newton later showed why they follow from gravity. An orbit is an ellipse with the planet at one focus. The low point is perigee (closest), the high point apogee.
Key idea
1. Ellipse with Earth at a focus: $a = \dfrac{r_p + r_a}{2}$, $e = \dfrac{r_a - r_p}{r_a + r_p}$ (distances from the center).
2. Equal areas in equal times, so it moves fastest at perigee: $r_p v_p = r_a v_a$.
3. $T = 2\pi\sqrt{\dfrac{a^3}{GM}}$: the period depends only on the semi-major axis, and $T^2 \propto a^3$.
2. Equal areas in equal times, so it moves fastest at perigee: $r_p v_p = r_a v_a$.
3. $T = 2\pi\sqrt{\dfrac{a^3}{GM}}$: the period depends only on the semi-major axis, and $T^2 \propto a^3$.
Worked example: How big was Sputnik's ellipse?
With a period of about $98\ \text{min}$, find the semi-major axis of Sputnik's orbit and its average height. ($GM_E = 3.986 \times 10^{14}\ \text{m}^3/\text{s}^2$.)
- $T = 5{,}880\ \text{s}$. $a = \left(\dfrac{GM T^2}{4\pi^2}\right)^{1/3} \approx 7.04 \times 10^6\ \text{m}$.
- $7{,}040 - 6{,}371 \approx 670\ \text{km}$: the average of its perigee and apogee altitudes (from NASA's rounded period).
Answer: $a \approx 7{,}040\ \text{km}$
Common mistake
Using altitudes for $r_p$ and $r_a$. Add Earth's radius to each first.