Which of the planets could we detect today from just the movements of the sun?

Question

Assume that despite our current level of space exploration and technology, we have somehow missed the existence of the other planets in our solar system.

We have developed various methods of detecting exoplanets orbiting other stars.

Using direct observation of the sun's position or possibly radial velocity observations from current earth- or space-based solar observatories, which of the planets would we be able to detect?

Would we be able to detect anything smaller than a planet too?

SOHO was launched about 20 years ago, so lets assume 20 years worth of observations. Extra credit: I'd also be interested to see what effect it would have to hypothetically have 100 or 1000 years worth of similar observations.

Answer 1

Getting hard numbers about how accurate measures we can get from current systems, adapted to the Sun instead of far away stars is difficult, bordering to impossible. But we can get data about the relative difficulty of the solar system planets.

First off, we can do some cheating for Mercury and Venus, as they occasionally go in front of the Sun. Given your 20 year observation span, you have observed multiple Mercury transits. Venus transits on the other hand happen about twice a century, and thus you have had a less than 50% chance of observing one, although you have had two of them in the last two decades.

Doppler detection relies on the planet making their star moving slightly back and forth, causing the light from it to vary in wavelength.

Therefore, for a planet to be easy to detect, we want the difference of the Sun moving towards us and away from us to be as large as possible. The larger objects distance to the barycentre is determined by the smaller objects share of the mass. (the µ)

The objects in the solar system has a negligible mass compared to the Sun, so the radius of the Sun's orbit around the barycentre is proportional to the planet's mass and the distance from the Sun.

In a two body system, the two bodies and the barycentre are always on a straight line, so the actual velocity is also proportionally to that of the planet.

Combining this parameters for all solar system planet gives the following list of relative Doppler shifts:

Jupiter: 1.00
Saturn: 0.41
Neptune: 0.13
Uranus: 0.088
Earth: 0.0014
Venus: 0.00095
Mars: 0.00018
Mercury: 0.000047

For comparison, the values for Pluto and Ceres are 0.000019 and 0.00000034 respectively.

The parameters for the relatively small Gliese 581c gives about 0.25 on this scale, indicating that extraterrestrials with the capabilities of current Earth technology could detect Jupiter and Saturn in our solar system.

An important note is that from Saturn and out, you are not able to observe the whole Doppler shift cycle in 20 years due to their long orbital periods. Also you are able to get data from multiple revolutions from the inner planets, making it easier to confirm the observations.

Splitting the combined signal of all the solar system planets from each other is a task requiring a Fourier Transform.

Answer 2

I massaged some raw numbers from https://nssdc.gsfc.nasa.gov/planetary/factsheet/

For each body-Sun pair the velocity of the Sun is the velocity of the planet times the ratio of the masses since they orbit around their center of mass. Eclipse depth is just the ratio of diameters.

Jupiter results in the largest velocity by far, thought the amplitude of the motion for the other three is still pretty large, though not readily detectable if this were a distant solar system.

For transits, the four outer planets each have eclipse depths of about 1.3 to 10.6 parts per thousand.

                      Mercury   Venus    Earth    Mars   Jupiter  Saturn  Uranus  Neptune

mass (e+24 kg)         0.330    4.87     5.97     0.642    1898     568    86.8    102 
orbit (e+08 km)        0.58     1.08     1.50     2.28     7.79    14.3   28.7    45.0
speed (km/s)           47.4     35.0     29.8     24.1     13.1     9.7    6.8     5.4
diameter (km)          4879   12,104   12,756     6792  142,984 120,536 51,118  49,528

mass ratio (e-07)      1.66     24.5     30.0     3.23     9542    2856    436.    5.13
Sun's vel   (m/s)      0.008,   0.086    0.089   0.0078    12.5    2.77     0.30   0.28 
Sun's orbit (e+08 km)  0.0001   0.0026   0.0045  0.0007     7.43   4.09     1.25   2.31 
eclipse depth (ppm)    12       7.6      84      24       10,566   7,509   1,350  1,268

Radial Velocity (Doppler)

According to Radial Velocities as an Exoplanet Discovery Method

Since then, this technique has been taken to extreme lengths. State-of-the-art stable radial velocimeters today control the vibration, temperature, and pressure of spectrographs with exquisite precision using cryostats and vacuum chambers. The remaining, unavoidable changes in the spectrograph (from, for instance, slow changes in the crystalline structure of the metals involved or irregular thermal outputs from the detector electronics) are tracked via emission sources such as laser frequency combs, which are locked to atomic clocks and provide essentially perfect wavelength references.

Today, the state of the art is represented by the HARPS (Queloz et al 2001b) and ESPRESSO (Pepe et al 2010) spectrographs of ESO, which are stable below the 1 m/s level (the latter aspires to 10 cm/s precision.)

So it looks like Jupiter, Saturn and Uranus might possibly be detectable with long term measurements (of order 1 century) but the inner planets would be difficult.

Remember that the maximum radial velocity is also scaled by the cosine of the inclination of the orbit with respect to the viewing direction. If someone is looking normal to the plane of the ecliptic, they will see only a very tiny fraction of the radial velocity that would be seen from the plane of the ecliptic.

Transit photometry (Eclipse)

The Kepler telescope sensitivity threshold varies depending on the brightness of the star and the period of observation. But a threshold of order 10 to 100 ppm might be estimated.

In this case the results are similar. The outer planets (Jupiter through Neptune) would be readily observable but the inner ones would pose a significant challenge.

Constraints on angle are much more severe, there's a much smaller chance that the orientation of the solar system would result in a geometrical transit, so on average, transits are highly unlikely to be observable from a random position in space.