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I'm encountering some difficulty understanding the concept of line of sight inclination. Typically, we assume that the observer is positioned perpendicular to the system being observed. Consequently, the orbital inclination (which is crucial for calculating the radial velocity of the system) is represented by the angle 'i,' as illustrated in the figure below. enter image description here

My question is this: What happens if the observer's line of sight (LOS) is tilted at an angle? Moreover, what would occur if both the stellar system and the observer's line of sight are tilted?

I would deduce that the orbital inclination ($i$) can be calculated using the equation $$i = \arccos\left(\cos\left(i_{\text{LOS}} \right) \cos\left(i_{\text{star}} \right) \right) \,.$$ Here, $i_{\text{LOS}}$ represents the inclination on the line of sight plane, and $i_{\text{star}}$ denotes the inclination on the star's plane. Or you just sum the inclinations?

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Hard to follow your query. Maybe if you labelled $i_{\rm star}$ and $i_{\rm LOS}$. The inclination angle is defined in exactly the way you show - but the diagram is not unique.

Perhaps the source of confusion is that just knowing the inclination angle does not uniquely identify$^{\dagger}$ the orientation of a binary system in space. i.e. The vector that you have labelled $\vec{N}$, which could be the orbital angular momentum vector, can be rotated around the line of sight without changing observables like the measured radial velocity of the binary components. Furthermore, most observables are also agnostic to whether the inclination is $i$ or $180^\circ - i$.

$\dagger$ Except that an inclination of zero or 180 degrees does uniquely define the orientation.

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