orbital-motionsatellites
I think we could put satellite to orbit earth in such a way that it always see the sun. Which is orbiting along the path of earth orbiting the sun, like a wheel perpendicular to the sun
I don't know the specific name of this type of orbit so I can't find detail about it, are there any satellite orbit earth this way?
This is actually quite a complex problem. At large numbers of satellites, gravitational effects need to be considered. A real answer would need values such as 'every satellite has the same mass and volume' etc. Basically you can keep adding satellites until Earth becomes a black hole. Satellites are very dense since they have lots of metal (which is a dense material) In my opinion, a great 'answer' to this question would be a fun computer simulation.
If we just use the values given by Phil H, we fit in satellites between 2000 km and 35786 km altitudes. $R_E$, the radius of the Earth is 6371km. So we calculate the volume: $V_{total}=\frac{4}{3} \pi [(d_2+R_E)^3-(d_1+R_E)^3]=\frac{4}{3}\pi (42157^3-8371^3)=9.341\times10^{14}\,\text{km}^3$
We assume that all the satellites have mass of $m_s = 800\,\text{kg}$ (note that this is a 'fantasy' problem so we don't really need to follow real world statistics) and volume $V_s=4.8 \,\text{m} \times 4.8 \,\text{m} \times 5.5 \,\text{m} = 1.267 \times 10^{-7} \, \text{km}^3$
Thus, the total mass of the orbiting satellites will be $M_{total} = \frac{m_s}{V_s} V_{total}= 5.90 \times 10^{24}\,\text{kg}$
Now let's compare that to the mass of the Earth, $M_E=5.97\times 10^{24} \,\text{kg}$.
Notice how $M_{total} \approx M_E$, though this is just due to some values which we chose and does not arise from some values which arise from nature.
Start with your satellite velocity, $v_0$, equal to $\sqrt{GM/r}$ so we get a circular orbit:
Now if we increase the velocity, $v > v_0$, the satellite will move away from the Earth faster than the satellite in a circular orbit, and we get an elliptical orbit that looks like this (I've drawn the original circular orbit dotted):
Alternatively if we decrease the velocity, $v < v_0$, the satellite will move away from the Earth slower than the satellite in a circular orbit, and we get an elliptical orbit that looks like this:
In all cases the orbit is an ellipse with the Earth at one of the focus points. The circle is a special case of an ellipse where the foci coincide.
Solving the equations of motion for the satellite in an elliptical orbit is harder than you probably expect. However there are various convenient equations that describe aspects of the motion. For our purposes the easiest equation to work with is the vis viva equation:
$$ v^2 = GM\left(\frac{2}{r} - \frac{1}{a} \right) $$
where $r$ is the distance from the Earth and $a$ is the semi-major axis of the ellipse. If we rearrange this to give the semi major axis we can explain the three diagrams above:
$$ a = \frac{r}{2 - \frac{r}{GM}v^2} $$
or:
$$ a = \frac{r}{2 - \phi} \tag{1} $$
where:
$$ \phi = \frac{r}{GM}v^2 $$
If we start with $v = \sqrt{GM/r}$ then that will make $\phi = 1$, and equation (1) tells us that $a = r$. In other words the semi major axis is equal to the orbital radious so the orbit is a circle.
Now make $v > \sqrt{GM/r}$ as in the second diagram, then $\phi > 1$ and therefore $a > r$. The semi major axis is greater that the distance $r$ we started with so we have an ellipse wider than the circular orbit.
And finally, though it should be obvious now, if we make $v < \sqrt{GM/r}$ as in the third diagram, then $\phi < 1$ and therefore $a < r$. The semi major axis is less that the distance $r$ we started with so we have an ellipse narrower than the circular orbit.
Best Answer
There are satellites that constantly observe the sun. Among them are the STEREO pair of satellites. These satellites don't orbit Earth, but orbit the Sun–one a little faster than Earth, one a little slower.
At first, I thought the Earth-centered orbit you suggest wouldn't work. I was thinking that if a satellite starts out in a polar orbit, the plane of the orbit would not rotate as the Earth orbited the Sun, leaving the orbit parallel to the sun a quarter year later and behind the Earth at times. However, it is possible. From Wikipedia on polar orbits:
Another source cited by the wiki article.
These orbits are generally used for Earth-observing satellites that track Earth conditions (like atmospheric temperatures) at the same times of day.
Also, from a comment by cmaster - reinstate monica: