Because orbits are general conic sections. Why this is true is another fascinating question in and of itself, but for now I'll just assume it. The point is that circular orbits are *special examples* of general orbits. It's perfectly possible to get a circular orbit, but the relationship between the bodies' velocities and separation needs to be exactly right. In practice it rarely is, unless we plan it that way (e.g, for satellites).

If you threw a planet around the sun really hard its path would be bent by the sun's gravity, but it would still eventually fly off at a tangent. Throwing it *really* hard would make it almost go straight, since it moves by the sun so quickly. As you reduce the speed, the sun gets to bend it more and more, and so the tangent is flies off on gets angled more and more towards moving backwards. So general hyperbolas are possible orbits. If you move it at the right speed, then it'll be just slow enough that other tangent points 'exactly backwards', and here the motion will be a parabola. Less than this and the planet will be captured. It doesn't have enough energy at this point to escape at all.

A key realization here is that the path should change *continuously* with the initial speed. Imagine the whole path traced out by a planet with a high velocity. An almost-straight hyperbola, say. Now as you continuously lower the velocity, the hyperbola bends more and more (*continuously*) until it bends "all the way around" and becomes a parabola. After this point, you'll have captured orbits. *But they have to be steady changes from the parabola*. All captured orbits magically being circles (of what size anyway, since they have to start looking like parabolas at some point?) wouldn't make any sense. Instead you get ellipses that get shorter and shorter as you get slower. Keep doing this, and those ellipses will come to a circle at some critical speed.

So circular orbits are *possible*, they're just not *general*. In fact, I'd say the real question is why the orbits are often *so close* to circular, since there are so many other options!

Newton's original proof was in fact based on geometry (he hadn't invented calculus yet). Richard Feynman devised his own, simpler geometric proof for one of his famous lectures. You can find it in *Feynman's Lost Lecture*, by Goodstein & Goodstein, and in this article: Paths of the Planets from Hall & Higson. But since it's so much fun, I'll describe it here as well.

Let's start with a lesser-known way to construct an ellipse, the so-called *circle construction*. Draw a circle with centre $O$, and fix a point $A$ inside the circle. Pick a point $B$ on the circle, and draw the perpendicular bisector of $\overline{AB}$ (blue line). It intersects $\overline{OB}$ in a point $P$, and as $B$ moves around the circle, these intersection points form an ellipse. Also, the blue biscector lines are tangent lines to the ellipse, and $O$ and $A$ are the foci.

Why is it an ellipse? Because $\overline{AP}$ has the same length as $\overline{BP}$, so that the sum of the lengths of $\overline{AP}$ and $\overline{OP}$ is constant, i.e. the radius of the circle. In other words, we get the classic tack-and-string definition of an ellipse. It is also straightforward to see that the angles $a$ and $b$ are equal. Since $a$ and $c$ are also equal, this means that $b$ and $c$ are equal, so that the blue line is indeed a tangent line.

The geometric proof of Kepler's Second Law (planets sweep out equal areas in equal times) from Newton's first two laws is straightforward and can be found in the Hall & Higson article. Now, if a planet traverses an angle $\Delta\theta$ in a small time interval $\Delta t$, it sweeps out an area
$$
\text{area}\approx \frac{1}{2}\Delta\theta\, r^2.
$$

At this point, Feyman's argument deviates from Newton: while Newton breaks up the orbit it equal-time pieces, Feyman considers equal-*angle* pieces. In other words, Feynman breaks up the orbit in subsequent pieces with areas
$$
\text{area}\approx \text{constant}\cdot r^2.
$$
Newton's inverse-square law (which can be derived from Kepler's Third Law) states that the acceleration of a planet is proportional to the inverse square of its distance $r$:
$$
\left\|\frac{\Delta\boldsymbol{v}}{\Delta t}\right\| = \frac{\text{constant}}{r^2}.
$$
Eliminating $r^2$, we get
$$
\left\|\Delta\boldsymbol{v}\right\| \approx \text{constant}\cdot\frac{\Delta t}{\text{area swept out in $\Delta t$}}.
$$
But Kepler's Second Law states that the area swept out in $\Delta t$ is a constant multiple of $\Delta t$. Therefore,
$$
\left\|\Delta\boldsymbol{v}\right\| \approx \text{constant},
$$
that is, intervals of constant $\Delta\theta$ also have a constant change in velocity. We can use this fact to construct a so-called *velocity diagram*. Break up the orbit into equal-angle pieces, draw the velocity vectors, and translate these vectors to the same point.

Since $\left\|\Delta\boldsymbol{v}\right\|$ is constant, the resulting figure is a polygon with $\dfrac{360^\circ}{\Delta\theta\,}$ sides. The smaller the angles, the more it approaches a circle.

Now, let's draw the velocity diagram of an orbiting planet. If $l$ is the tangent line to the orbit at point $P$ (parallel to the velocity vector in $P$), then $l'$ in the corresponding velocity diagram is also parallel to $l$. Also note that $\theta$ in both diagrams is the same.

Rotate the velocity diagram clockwise by $90^\circ$, so that $l'$ becomes perpendicular to $l$. Construct the perpendicular bisector $p$ to the line $\overline{AB}$, and the intersection $P'$ with $\overline{OB}$. It turns out that we are in the exact same situation as the circle construction for the ellipse: as $B$ moves on the velocity diagram, the points $P'$ form an ellipse.

The lines $p$ are the tangent lines to the ellipse. However, these lines are also parallel to the lines $l$, which are the tangent lines to the planet's orbit. Because of the *tangent principle*, if two curves have the same tangent lines at every point, then those curves are the same. In other words, the lines $l$ are also the tangent lines of an ellipse. This proves that the orbit of a planet is indeed an ellipse.

## Best Answer

The notion that GR predicts that a planet moves in circular path is not really true. GR is just an extension (though it's complex math) for what we didn't understand over centuries. Not only newtonian mechanics, but any consistent theory that makes use of the inverse-square law uses the conics like parabolas, hyperbolas, ellipses and

circles. Depending on the velocity at which the object arrives (comparable to gravity), its orbit can be any of the conics.Now, this makes sense that any object isn't in elliptical or circular orbit

always. It may or may not change, based on the objects it interacts along its path. Here's a simulator on the orbits of celestial objects. There are lot of presets. For instance, you can see the slingshot preset where a planet makes an asteroid to go out of orbit. So, it's not true that all orbits are elliptical always.