[hamdani yusuf (OP)]

Parker Solar Probe's orbital period is similar to Mercury's. But the eccentricity is significantly different. How does it affect its precession?

https://en.wikipedia.org/wiki/Parker_Solar_Probe
Semi-major axis   0.388 AU (58.0 million km; 36.1 million mi)
Perihelion altitude   0.046 AU (6.9 million km; 4.3 million mi; 9.86 R☉)[note 1]
Aphelion altitude   0.73 AU (109 million km; 68 million mi)[5]
Inclination   3.4°
Period   88 days

https://en.wikipedia.org/wiki/Mercury_(planet)
Aphelion   0.466697 AU;  69,816,900 km
Perihelion   0.307499 AU;  46,001,200 km
Semi-major axis   0.387098 AU;  57,909,050 km
Eccentricity   0.205630[3]
Orbital period (sidereal)   87.9691 d;  0.240846 yr

The perihelion precession of Mercury is 5,600 arcseconds (1.5556°) per century relative to Earth, or 574.10±0.65 arcseconds per century[132] relative to the inertial ICRF.

[Halc]

Parker Solar Probe's orbital period is similar to Mercury's.

Hardly. OK, it has a similar period, likely coincidental. Parker gets close enough to touch the sun and take samples. Mercury is nowhere near that.
Current orbital period is about 96 days, with aphelion of about 0.066 AU. I think the 88+ days is an eventual figure near end of mission after two more Venus encounters.

But the eccentricity is significantly different. How does it affect its precession?

Parker isn't in any kind of stable orbit and it isn't particularly meaningful to talk about its precession until it is. It regularly gets its trajectory altered by small powered boosts and by regular gravitational boosts from Venus. This gives it a non-elliptical trajectory. I suppose one could talk about its precession given the absence of the inner planets and the probe actually becoming inertial like Mercury is, but right now neither of those things is the case.

[evan_au]

Parker Solar Probe's orbital period is similar to Mercury's

NASA planned the orbit of the Parker Probe so that it will periodically meet up with Venus.
- That means the aphelion is just outside the orbit of Venus (allowing 2 potential encounters)
- They ensure that the orbital period of the probe is a simple fraction of the orbital period of Venus (but this fraction changes on every encounter).
- This is so they can use a gravitational slingshot from Venus to reduce the perihelion of the Parker probe.
- It takes a huge delta-V to bring the orbit close to the Sun, and it is impractical to provide this much delta-V with chemical propellants

How does it affect its precession?

It would be interesting if NASA used Einstein's corrections to Newton's equations in planning the orbit of this space probe.
- For trips to Mercury, Venus, the Moon and farther out, Newton's equations are fine.
- Einstein's corrections apply on the order of centuries, while the planned lifetime of this space probe is just 7 years

https://en.wikipedia.org/wiki/Parker_Solar_Probe#Mission

[evan_au]

Google found this: https://physics.stackexchange.com/questions/348854/parker-solar-probe-passing-extremely-close-to-the-sun-what-relativistic-effects

NASA did take relativistic effects into account. Apparently the biggest effect is the Shapiro delay on communications: This is the effect of a change in the speed of radio signals while passing through a deep gravitational field.

[hamdani yusuf (OP)]

Is there any measurable effect of solar wind?

[evan_au]

Is there any measurable effect of solar wind?

The Solar Wind has a visible affect in deflecting the gas and dust shed by a comet.
- The Solar Wind would be much denser, and moving faster as you get closer to the Sun, so I am sure that there would be some effect
- But a 500kg space probe is much denser than a dust grain!

NASA monitors the orbit of the probe very closely, and does orbit correction burns to keep it on the intended course despite random gusts of solar wind....
https://en.wikipedia.org/wiki/Solar_wind