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I understand that the solar system oscillates in its motion around the Milky Way. However at any given time, there must be a specific angle between the current direction of the solar apex and the galactic plane. This is the angle - in degrees - I am looking for.

To be specific. I'm not looking for the angle of the solar system with respect to the galactic plane, nor the angle between the celestial pole and the galactic plane, nor the angle between the plane of the ecliptic and the galactic plane. I am only looking for the current angle of motion between the solar apex and the galactic plane.

Also, is there more clear terminology for this? I've spent a few hours searching and it seems that I can't find an answer to what I would imagine is a relatively obvious question. Am I searching with the wrong terminology?

Thanks.

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    $\begingroup$ Related: astronomy.stackexchange.com/q/28639/16685 The Wikipedia article linked in the answer there says that the latitude in galactic coordinates of the solar apex is 22.54°. That's for the apex determined by visual observations. If you use the solar apex determined by radioastronomy, the galactic latitude is 17.72°. As Wikipedia says, "The evaluation of movement of Solar system within local neighborhood is involved; look at Talk page for some actual links". $\endgroup$
    – PM 2Ring
    Dec 30, 2019 at 7:46

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In an answer to this question, you can see that I describe the trajectory of the Sun around the Galaxy. Its current motion can be defined in a number of ways, according to your preferred frame of reference. From your question I judge that you want it in a "Galactocentric" frame, so we take the velocity of the Sun with respect to the "Local Standard of Rest" (LSR), which is the Sun's motion with respect to an average of local stars, and add something to it to account for the fact that local stars are orbiting the Galactic centre every 220 million years or so.

The velocity of the LSR is 10 km/s towards the Galactic centre, 5 km/s faster than an average star at the Sun's location tangentially to a line towards the Galactic centre, and 7 km/s upwards out of the Galactic plane (see for example here).

Thus, if we take this frame of reference, the component of the velocity of the Sun in the Galactic plane is $\sqrt{10^2 + 5^2} = 11.2$ km/s amd the component out of the Galactic plane is just 7 km/s. The angle this trajectory makes with the Galactic plane is $\arctan(7/11.2) = 32$ degrees. Note that there are uncertainties/disagreements about the exact value of the Sun's motion with respect to the LSR at the level of 1-2 km/s, so this final angle is also somewhat uncertain.

However, this rather steep angle does not take into account that the LSR is moving around the Galaxy at rather high speed. This speed is around 250 km/s in the Galactic plane and at right angles to the Galactic centre (e.g. Reid et al. 2014). There may be a more modern figure, but I cannot immediately locate it, but it is probably safe to assume an uncertainty of at least 5-10 km/s in this. With this we now have velocity components of 11.2 km/s out of the plane and 255 km/s in the plane. The angle you are seeking is then $\arctan(11.2/255) = 2.5$ degrees, with an uncertainty of only $\sim 0.3$ degrees. So if you were trying to make a drawing of the solar system motion with respect to the Galaxy, that is the angle you would use. Note that the Sun is currently above the Galactic plane by about 20 pc.

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