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If the velocity of the satellite orbiting around a planet is reduced somehow...then what would it's trajectory looks like spiral(1) or elliptical(2) Or Does it depends on the  velocity it has(less than orbital velocity)initially?

Infact I hope on basis of this only we could decide whether the satellite collides with planet or not

or is it like framing equations considering a direct elliptical path yields same result(by result i mean it collided with planet p or not )irrespective of the actual trajectory.Even if it’s like this please give me clarity regarding the trajectory though.

FinAlly guide me how to check which case happens for what and if the crash/collision happens at all?

The orbit of the satellite would be elliptical, as it must honour 2 elementary conservation laws:

• Law of conservation of total energy




• Law of conservation of angular momentum ( measure of rotational motion )

But depending on particulars, the elliptic orbit may formally cross the physical boundary of the central object or its atmosohere, i.e. a crash would happen.

I.e the GPS satellites have altitude of circular orbits about 20000 km.

If their speed suddenly drops, their orbit would transform to elliptical ones, with apogee (farthest point) at those 20000 km, and perigee (nearest point) e.g just 10000 km.

The bigger the speed drop would be, the lover the perigee would be.

With some speed drop threshold, the satellite at perigee would collide with dragging atmosphere, lowering the ( higher than orbital ) speed at perigee, what would be lowering the subsequent apogees.

Progressively, the elliptical orbit transforms to less and less prolonged orbit, finally gradually transforming to the spiral of death.

With some even bigger initial speed drop, the satellite orbit would cross the Earth surface and what would not be burnt as a fireball, would crash to Earth.

Let suppose the initial satellite speed $v$ is tangential and lower then the speed needed for the circular orbit.

Then it's energy

$$E=frac 12 mv_mathrmap^2 - frac GmMr_mathrmaplt - frac GmM2r_mathrmap$$

$$frac 12 mv_mathrmap^2 lt frac GmM2r_mathrmap$$

$$v_mathrmap lt sqrt frac GMr_mathrmap$$

$$v_mathrmap = k cdot sqrt frac GMr_mathrmap$$

As it's velocity at both perigee and apogee is perpendicular to the position vector, its angular momentum

$$L=m cdot (vec r times vec v)=mcdot r_mathrmapcdot v_mathrmap\=mcdot r_mathrmpercdot v_mathrmper$$

Therefore

$$v_mathrmper=v_mathrmapcdot fracr_mathrmapr_mathrmper$$

$$E=frac 12 mleft(v_mathrmapcdot fracr_mathrmapr_mathrmperright)^2 - frac GmMr_mathrmper=frac 12 mv_mathrmap^2 - frac GmMr_mathrmap$$

$$frac 12left(v_mathrmapcdot fracr_mathrmapr_mathrmperright)^2 - frac GMr_mathrmper=frac 12 v_mathrmap^2 - frac GMr_mathrmap $$

$$ left(frac 12 v_mathrmap^2 - frac GMr_mathrmapright)cdot r_mathrmper^2 + GMr_mathrmper - frac 12 v_mathrmapcdot r_mathrmap^2=0$$

$$ frac GMr_mathrmap left(frac 12 k^2 - 1right)cdot r_mathrmper^2 + GMr_mathrmper - frac 12 v_mathrmapcdot r_mathrmap^2=0$$

$$ left(frac 12 k^2 - 1right)cdot r_mathrmper^2 + r_mathrmpercdot r_mathrmap- frac v_mathrmapcdot r_mathrmap^32GM=0$$

$$ left(1-frac 12 k^2 right)cdot r_mathrmper^2 - r_mathrmpercdot r_mathrmap + frac v_mathrmapcdot r_mathrmap^32GM=0$$

This leads to the solving of quadratic equation for $r_mathrmper$

$$ r_mathrmper= fracr_mathrmap - sqrt r_mathrmap^2-
left(2-k^2 right)cdot frac v_mathrmapcdot r_mathrmap^3GM2-k^2$$

$$ r_mathrmper=r_mathrmap frac1 - sqrt 1-
left(2-k^2 right)cdot frac v_mathrmapcdot r_mathrmapGM2-k^2$$

Depending on apogee and perigee radius, compared to Earth and Earth atmosphere (negligible drag) radius, these cases happen:

For both radii above drag region - ellipse

For perigee within drag region+ apogee out if drag region - ellipse shortening then spiralling.

For perigee within Earth radius - direct crash.

For both radii within drag region - spiralling.

To illustrate what was answered by @Poutnik, consider a satellite whose speed changes in $k$ times at some point of the trajectory. Figure 1 shows the trajectory before the speed reduction (blue) and after (orange). We see an elliptical trajectory that is getting closer to the central body with decreasing $k$.

Fig. 2 shows how a collision occurs with a central body (green disk).