radial velocity
Definition: Radial velocity denotes the physical quantity, process, or inference concept captured by this lecture-specific statement: In a two-body system, both the star (mass M*) and planet (mass Mp) orbit their common center of mass; by the center-of-mass condition M* r* = Mp rp, the star’s orbital radius r* = rp (Mp/M*) is much smaller than the planet’s for Mp << M*, but not zero -- it is this small stellar reflex...
Why it matters here: Every topic in this course -- demographics, architectures, atmospheres, habitability, biosignatures -- depends on first having detected and characterized a planet. This lecture derives the method (radial velocity) that opened the field in 1995 and remains the primary way planet masses are measured today.
Relationships: It is used with Doppler spectroscopy to interpret the lecture model. The quantitative connection is the working relation \(K = \left(\dfrac{2\pi G}{P}\right)^{1/3}\dfrac{M_p\sin i}{(M_\ast+M_p)^{2/3}}\dfrac{1}{\sqrt{1-e^2}}\).
Boundary or common confusion: Reporting a radial-velocity-derived planet mass as if it were the true mass without stating the sin(i) assumption. This unit-and-completeness error matters: the true mass equals the minimum mass only if the orbit happens to be observed edge-on (i=90 degrees); for a typical randomly oriented orbit, sin(i)<1 and the true mass is larger than the quoted minimum...
Doppler spectroscopy
Definition: Doppler spectroscopy denotes the physical quantity, process, or inference concept captured by this lecture-specific statement: In a two-body system, both the star (mass M*) and planet (mass Mp) orbit their common center of mass; by the center-of-mass condition M* r* = Mp rp, the star’s orbital radius r* = rp (Mp/M*) is much smaller than the planet’s for Mp << M*, but not zero -- it is this small stellar reflex...
Why it matters here: Every topic in this course -- demographics, architectures, atmospheres, habitability, biosignatures -- depends on first having detected and characterized a planet. This lecture derives the method (radial velocity) that opened the field in 1995 and remains the primary way planet masses are measured today.
Relationships: It is used with radial velocity, center of mass (barycenter) to interpret the lecture model. The quantitative connection is the working relation \(K = \left(\dfrac{2\pi G}{P}\right)^{1/3}\dfrac{M_p\sin i}{(M_\ast+M_p)^{2/3}}\dfrac{1}{\sqrt{1-e^2}}\).
Boundary or common confusion: Reporting a radial-velocity-derived planet mass as if it were the true mass without stating the sin(i) assumption. This unit-and-completeness error matters: the true mass equals the minimum mass only if the orbit happens to be observed edge-on (i=90 degrees); for a typical randomly oriented orbit, sin(i)<1 and the true mass is larger than the quoted minimum...
center of mass (barycenter)
Definition: Center of mass (barycenter) denotes the physical quantity, process, or inference concept captured by this lecture-specific statement: In a two-body system, both the star (mass M*) and planet (mass Mp) orbit their common center of mass; by the center-of-mass condition M* r* = Mp rp, the star’s orbital radius r* = rp (Mp/M*) is much smaller than the planet’s for Mp << M*, but not zero -- it is this small stellar reflex...
Why it matters here: Every topic in this course -- demographics, architectures, atmospheres, habitability, biosignatures -- depends on first having detected and characterized a planet. This lecture derives the method (radial velocity) that opened the field in 1995 and remains the primary way planet masses are measured today.
Relationships: It is used with Doppler spectroscopy, semi-amplitude K to interpret the lecture model. The quantitative connection is the working relation \(K = \left(\dfrac{2\pi G}{P}\right)^{1/3}\dfrac{M_p\sin i}{(M_\ast+M_p)^{2/3}}\dfrac{1}{\sqrt{1-e^2}}\).
Boundary or common confusion: Reporting a radial-velocity-derived planet mass as if it were the true mass without stating the sin(i) assumption. This unit-and-completeness error matters: the true mass equals the minimum mass only if the orbit happens to be observed edge-on (i=90 degrees); for a typical randomly oriented orbit, sin(i)<1 and the true mass is larger than the quoted minimum...
semi-amplitude K
Definition: Semi-amplitude k denotes the physical quantity, process, or inference concept captured by this lecture-specific statement: Combining Kepler’s third law (relating orbital period P to the total mass and semimajor axis a) with the projection of the star’s orbital velocity onto the line of sight (which depends on the orbital inclination i, defined so i=90 degrees is edge-on) gives the full radial-velocity...
Why it matters here: Every topic in this course -- demographics, architectures, atmospheres, habitability, biosignatures -- depends on first having detected and characterized a planet. This lecture derives the method (radial velocity) that opened the field in 1995 and remains the primary way planet masses are measured today.
Relationships: It is used with center of mass (barycenter), minimum mass (Mp sin i) to interpret the lecture model. The quantitative connection is the working relation \(K = \left(\dfrac{2\pi G}{P}\right)^{1/3}\dfrac{M_p\sin i}{(M_\ast+M_p)^{2/3}}\dfrac{1}{\sqrt{1-e^2}}\).
Boundary or common confusion: Reporting a radial-velocity-derived planet mass as if it were the true mass without stating the sin(i) assumption. This unit-and-completeness error matters: the true mass equals the minimum mass only if the orbit happens to be observed edge-on (i=90 degrees); for a typical randomly oriented orbit, sin(i)<1 and the true mass is larger than the quoted minimum...
minimum mass (Mp sin i)
Definition: Minimum mass (mp sin i) denotes the physical quantity, process, or inference concept captured by this lecture-specific statement: Because only the projected velocity is observed, radial velocity alone cannot separate a small, edge-on-viewed planet from a larger, more face-on one producing the same observed K; the equation therefore only ever returns the combination Mp sin(i), the minimum mass, until an independent...
Why it matters here: Every topic in this course -- demographics, architectures, atmospheres, habitability, biosignatures -- depends on first having detected and characterized a planet. This lecture derives the method (radial velocity) that opened the field in 1995 and remains the primary way planet masses are measured today.
Relationships: It is used with semi-amplitude K, inclination degeneracy to interpret the lecture model. The quantitative connection is the working relation \(K = \left(\dfrac{2\pi G}{P}\right)^{1/3}\dfrac{M_p\sin i}{(M_\ast+M_p)^{2/3}}\dfrac{1}{\sqrt{1-e^2}}\).
Boundary or common confusion: Reporting a radial-velocity-derived planet mass as if it were the true mass without stating the sin(i) assumption. This unit-and-completeness error matters: the true mass equals the minimum mass only if the orbit happens to be observed edge-on (i=90 degrees); for a typical randomly oriented orbit, sin(i)<1 and the true mass is larger than the quoted minimum...
inclination degeneracy
Definition: Inclination degeneracy denotes the physical quantity, process, or inference concept captured by this lecture-specific statement: Because only the projected velocity is observed, radial velocity alone cannot separate a small, edge-on-viewed planet from a larger, more face-on one producing the same observed K; the equation therefore only ever returns the combination Mp sin(i), the minimum mass, until an independent...
Why it matters here: Every topic in this course -- demographics, architectures, atmospheres, habitability, biosignatures -- depends on first having detected and characterized a planet. This lecture derives the method (radial velocity) that opened the field in 1995 and remains the primary way planet masses are measured today.
Relationships: It is used with minimum mass (Mp sin i), mass function to interpret the lecture model. The quantitative connection is the working relation \(K = \left(\dfrac{2\pi G}{P}\right)^{1/3}\dfrac{M_p\sin i}{(M_\ast+M_p)^{2/3}}\dfrac{1}{\sqrt{1-e^2}}\).
Boundary or common confusion: Reporting a radial-velocity-derived planet mass as if it were the true mass without stating the sin(i) assumption. This unit-and-completeness error matters: the true mass equals the minimum mass only if the orbit happens to be observed edge-on (i=90 degrees); for a typical randomly oriented orbit, sin(i)<1 and the true mass is larger than the quoted minimum...
mass function
Definition: Mass function denotes the physical quantity, process, or inference concept captured by this lecture-specific statement: Solving the mass function numerically for Mp sin(i) given the measured K = 55.9 m/s: Mp sin(i) = 0.477 Jupiter masses, matching the published minimum mass (0.469 MJup) computed independently by the discovery team.
Why it matters here: Every topic in this course -- demographics, architectures, atmospheres, habitability, biosignatures -- depends on first having detected and characterized a planet. This lecture derives the method (radial velocity) that opened the field in 1995 and remains the primary way planet masses are measured today.
Relationships: It is used with inclination degeneracy to interpret the lecture model. The quantitative connection is the working relation \(K = \left(\dfrac{2\pi G}{P}\right)^{1/3}\dfrac{M_p\sin i}{(M_\ast+M_p)^{2/3}}\dfrac{1}{\sqrt{1-e^2}}\).
Boundary or common confusion: Reporting a radial-velocity-derived planet mass as if it were the true mass without stating the sin(i) assumption. This unit-and-completeness error matters: the true mass equals the minimum mass only if the orbit happens to be observed edge-on (i=90 degrees); for a typical randomly oriented orbit, sin(i)<1 and the true mass is larger than the quoted minimum...