The technical specifications for the planned SIM PlanetQuest...

The technical specifications for the planned SIM PlanetQuest mission call for the ability to resolve two point sources with an accuracy of better than $0.000004^{\prime \prime}$ for objects as faint as $20 \mathrm{th}$ magnitude in visible light. This will be accomplished through the use of optical interferometry. (a) Assuming that grass grows at the rate of $2 \mathrm{cm}$ per week, and assuming that SIM could observe a blade of grass from a distance of $10 \mathrm{km}$, how long would it take for SIM to detect a measurable change in the length of the blade of grass? (b) Using a baseline of the diameter of Earth's orbit, how far away will SIM be able to determine distances using trigonometric parallax, assuming the source is bright enough? (For reference, the distance from the Sun to the center of the Milky Way Galaxy is approximately 8 kpc. (c) From your answer to part (b), what would the apparent magnitude of the Sun be from that distance? (d) The star Betelgeuse (in Orion) has an absolute magnitude of -5.14 . How far could Betelgeuse be from SIM and still be detected? (Neglect any effects of dust and gas between the star and the spacecraft.

The technical specifications for the planned SIM PlanetQuest mission call for the ability to resolve two point sources with an accuracy of better than $0.000004^{\prime \prime}$ for objects as faint as $20 \mathrm{th}$ magnitude in visible light. This will be accomplished through the use of optical interferometry.
(a) Assuming that grass grows at the rate of $2 \mathrm{cm}$ per week, and assuming that SIM could observe a blade of grass from a distance of $10 \mathrm{km}$, how long would it take for SIM to detect a measurable change in the length of the blade of grass?
(b) Using a baseline of the diameter of Earth's orbit, how far away will SIM be able to determine distances using trigonometric parallax, assuming the source is bright enough? (For reference, the distance from the Sun to the center of the Milky Way Galaxy is approximately 8 kpc.
(c) From your answer to part (b), what would the apparent magnitude of the Sun be from that distance?
(d) The star Betelgeuse (in Orion) has an absolute magnitude of -5.14 . How far could Betelgeuse be from SIM and still be detected? (Neglect any effects of dust and gas between the star and the spacecraft.

Key Concepts

Optical Interferometry

Optical interferometry is a technique in astronomy that combines the light collected by two or more telescopes to effectively simulate a much larger aperture. This method significantly enhances the angular resolution, allowing the observation of extremely small details in distant objects, which is critical when resolving minute angular separations or small changes over time.

Angular Resolution

Angular resolution refers to the smallest angular separation between two objects that an instrument can distinguish. It is a key parameter in high-precision astrometry because it determines the instrument’s capability to detect small positional shifts or changes over vast distances, thereby enabling the observation of minute movements and separations in celestial sources.

Baseline

The baseline is the separation distance between individual telescopes in an interferometric array. A larger baseline increases the resolving power of the system by effectively creating a virtual aperture as large as the distance between telescopes. In addition, in techniques like trigonometric parallax, the baseline (such as the diameter of Earth's orbit) directly influences the measurable parallax angles, which are used to determine distances.

Trigonometric Parallax

Trigonometric parallax is a geometric method used to measure the distances to nearby stars by observing the apparent displacement of a star against distant background objects as the observer's vantage point changes. The technique relies on having a known baseline, such as the diameter of Earth's orbit, and precise angle measurements to calculate the distance to the star with high accuracy.

Magnitude System (Apparent and Absolute Magnitude)

The magnitude system is a logarithmic scale used to quantify the brightness of astronomical objects. Apparent magnitude measures how bright an object appears from Earth, while absolute magnitude defines the intrinsic brightness of an object as if it were located at a standard distance. This system is essential for assessing the detectability of objects at various distances, connecting observable brightness with distance through relationships like the distance modulus.

Distance Modulus

The distance modulus is a formula that relates the apparent magnitude, absolute magnitude, and distance of an astronomical object. It provides a quantitative means to determine one of these parameters if the other two are known, thus enabling astronomers to estimate distances and compare intrinsic luminosities across celestial bodies.

Transcript

00:01 Here we're going to investigate the idea of angular size.

00:06 So if you are looking at an object, you can't really measure its size with your eyeball.

00:14 What you're actually measuring is how much of the space occupied by the sphere in front of your face, how much of that sphere is covered by that object.

00:28 And that's the idea of angular size.

00:32 And it depends on distance.

00:35 So here i've shown a wedge with the object, with a real diameter, s, we're going to call it s, but it's really the diameter of the object.

00:49 And it is occupying a certain arc length.

00:53 This doesn't work so well if the object is brought close to your eye.

00:57 It's not rounded like an arc length would be.

01:02 But basically, we're going to use the circular geometry that s is equal to r times theta.

01:12 This works if the theta is in a degree unit, in an angular unit called a radian.

01:23 So notice that that s would be the circumference of a circle if theta were 2 pi, radians.

01:33 Radians are a natural unit, unlike some of the units that we develop as humans.

01:41 In any event, we're going to take a look at two cases.

01:45 Case one is we're going to be sort of mimicking the resolution.

01:53 So here we're talking about how accurately you can measure the parallax of a nearby star.

02:01 We don't need to get into this, but the resolution of the hipparchos mission, that resolution was 0 .001 seconds of arc.

02:15 And we are going to worry about a dime.

02:18 Okay, i can't draw a dime very well, but it's got lincoln's head on it, et cetera.

02:23 And it is roughly 1 .9 centimeters across.

02:31 So yeah, that's a dime, 10 cents.

02:38 And we'll make that s is our actual object, 0 .019 meters.

02:45 And we'd like to know how far away are you from the dime so that it has the angular size equal to the resolution of the hipparchose.

02:59 So that is s over theta.

03:02 And so we need to convert one second of arc into radiance.

03:08 So the way to do that is there are 360, sorry, 3 ,600 arc seconds in a degree, kind of like the same thing as the number of seconds in an hour.

03:27 And then there are pi radians in 180 degrees, would be one conversion factor you could use.

03:39 So notice that we get rid of the units we don't want, and are left with radiance.

03:55 Okay, so that's a very small amount of radiance, and now we can solve for our...

04:02 We'll take the diameter of the dime in meters and divide by 4 .85 times 10 to the minus 6 radiance.

04:12 And this is the thing about radiance, is you can ignore them because they're natural units.

04:20 It's kind of looking, like talking about five apples.

04:26 Five of anything is five of anything.

04:32 Working that out, that winds up to be about 3 .9.

04:37 We'll say 3 .92 kilometers.

04:42 So that is typically the size of a small town.

04:46 So imagine taking a dime and moving it to the other side of the town in which you live.

05:00 That would be the idea of how small the dime would appear to your eye.

05:09 Case two, we are going to imagine looking at grass grow with a fairly small resolution...

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