Lab 3 measuring creater sizes (2)
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Journeying Across the Cosmos: Investigating Distant Exoplanets Lab 4 Astronomy 101, L1F Professor: Aaron Boley TA: Justine Obidowski November 30
th
,2023 By: Bashar Adamat Student #: 22501787 Group members: Jimmy Gev, Ansh Garv
Introduction
When gazing into space, one frequently finds planets in star systems other than our own. These planets named "exoplanets"
—
can be studied in terms of both their unique features and those of their host star to explore and speculate on the possibility that they could support life on Earth. In this lab, we first use a combination of stellar spectroscopy to estimate the approximate temperature and mass of the star, and a light curve that shows the exoplanet's transits around a host star in terms of the star's observed flux over time. Following that, the acquired values will be utilized to compute details of the host star-exoplanet system, including the approximate size of the star and the primary lab objectives, which are the orbital period, semi-major axis, radius, mass, and temperature of the planet. We address the topic of whether human existence is possible on this exoplanet, focusing on these planetary values. We also speculate about potential features of other planets that might be present in the same star system.
Methods/Observations
By the end of the lab, we will determine: •
Orbital period of the Exoplanet •
Orbital Semi Major Axis of the Exoplanet •
Planetary Radius •
Planetary mass •
Planetary Temperature We first observe a light curve plotting the flux received from a distant host over time (see Figure 1). Along the graph are multiple dips in the flux value, taken to be due to the transit of the exoplanet across the visible surface of the host star to Earth. (Figure 1: A section of the known light curve, depicting flux received from the host star. Dips in the graph shows periods of time where the Exoplanet transits) For our purposes, and as a measure to reduce uncertainty, we take 5 arbitrary consecutive dips of the light curve and record information for each drop in flux, including the depth of the drop, the horizontal length of the drop (in hours, where each tick of the x-axis is 1 hour), and the time since the center of the previous drop. We observe the following: From these observations, we can easily determine the following: •
Average Transit Depth: 0.019
•
Average Transit Duration: 3 hours •
Average Time since previous transit: 84.5 hours Since multiple transits signify the movement of the exoplanet across a similar region of its orbit (the portion in front of its host star) the time since the previous transit directly taken from the light curve also yields to us that: •
Orbital Period of the Exoplanet = 84.5 hours We also again note that the average transit duration is 3 hours in length; this will be important for later. Next, we use stellar spectroscopy to determine a classification for the host star of this system. We compare a spectrum of the host star to that of a known classification to infer its approximate temperature. (Figure 2a: The recorded spectrum (Figure 2b: The spectrum of a star of the GOV for the observed star) classification) We can see that the two graphs comparing the host star to a general classification are identical. As such, the host star for our exoplanet's system is a GOV star. This gives us that: • Temperature of the host star (T
star
) = 6000 K Also, by spectroscopy, we also know that for a G-class star, the distance to it, which is the distance from which Earth receives its flux, is approximately 31.1 lightyears, or. 31.1 x (9.5 x 10
15
m) = 2.9545 x 10
17
m And that the flux received from a G-class star is: 2.14 x 10
−10
W/ m The following procedure is used to calculate the remaining information pertinent to our discussion: 1.
Determine the mass of the host star using the flux from and distance to a G-class star to first find its luminosity, as per:
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F= 𝐿
(4𝜋𝐷
2
)
→ 𝐿 = 𝐹 × 4𝜋𝐷
2
Where D = 2.9545 x
10
17
𝑚
, as shown through the stellar spectroscopy and subsequently applying the luminosity-mass relation: Where M
sun = 1.99 × 10
30
𝑘𝑔
and L
sun = 3.828 × 10
26
𝑊
2.
Determine the orbital semi major axis of the exoplanet using Kepler's Third Law (Harmonic law) using standardized units of seconds and meters: Where G is the gravitational constant 6.67 × 10
−11
(
𝑚
3
𝑘𝑔×𝑠
2
)
because the planetary mass is expected to be of lesser magnitude than the mass of the host star, it may be considered negligible, for a simplification: 3.
Determine the radius of the host star using an approximate relation involving the transit duration (Hour):
simplified, > where R
sun = 696340 km or 6.9634 × 10
4
𝑚
4.
Calculate the planetary radius using an equation that relates the areas to the transit depth (previously recorded as 0.019) as the flux from the host star diminishes or lowers to an area of the Earth-facing surface being covered by an exoplanet transit:
5.
Determine the estimated planetary mass using the Lissauer et al. 2011 mass-radius relationship: Where R
earth = 6378 km or 6.378 × 10
6
𝑚
and M
earth
= 5.98 × 10
24
𝑘𝑔
6.
Finally, determine the planetary temperature by assuming radiative equilibrium balance, per equation: Where D is the Semi Major Axis, “A” is the bond albedo, and 𝜀
is the emissivity: T
star as per stellar spectroscopy of GOV stars (6000k). Analysis and Results We analyze the transit data to directly determine (repeated here for convenience) Transit Depth = 0.019 Transit Duration = 3 hours - Orbital Period of the Exoplanet = 84.5 hours We proceed with the remaining procedure as specified in section "Observation and Methods" (for constant values and details, please see the corresponding step in the above section) 1. Luminosity of the host star Mass of the host star
2. Orbital Semi Major Axis of the Exoplanet
The orbital period of the Exoplanet = 84.5 x 60 minutes / hour x 60 seconds / minute = 304200 seconds 3. Radius of the host star (transit duration t
d = 3 from transit data)
4. Planetary Radius 5. Planetary Mass 6. Planetary Temperature
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Comparing our calculated values of mass and radius for our exoplanet to other planets within our local Solar System, we find that they are like those of Jupiter. As such, we take the Bond Albedo = 0.34; Emissivity unspecified so = 1 Discussion
We identify several sources of uncertainty for this lab, especially with the data that are derived straight from the light curve. There may be a small amount of discrepancy in the values taken for the orbital period and transit depth in our subsequent calculations due to a recurring fitter found in the recorded flux across the entire length of the curve - evidenced by the image of the light curve that quickly "zig-zags" - even with the mitigation through taking an average value across multiple transits on the curve. Based on an arbitrary number of significant figures derived from our results, we may also find slight variations in our computed numbers; nevertheless, we anticipate this difference to be almost insignificant. Given that our flux and distance data are provided by the stellar spectroscopy model provided to us, we are susceptible to any inaccuracies in the values specified for the flux from the host star and the distance to that star. In relation to our topic of discussion, we discover that the exoplanet's surface temperature is a very high 1003.39°C. As a result, it seems improbable that human life could readily survive on this planet because it is simply too hot for humans to tolerate without deliberately changing the temperature. We suggest that this exoplanet has a profile like that of what has been referred to as a "hot Jupiter" based on the values computed for the planet's mass and size and accounting for the previously indicated surface temperature. This also aligns with our observations of a very short orbital period. If we were to compare this star system to our own local Solar System, it is plausible that there are other huge planets and even hotter terrestrial planets located farther out from the host star, with this planet acting as the "Jupiter" of the system. In this scenario, the terrestrial planets will have significantly shorter orbital periods, maybe about 1-2 Earth days, with larger durations for the outer planets. If our light curve appears to be impacted only by transits of our original exoplanet of interest, we may also conclude that
other planets may have their orbital planes inclined so that they do not transit against the host star.
Conclusion
In this lab, we used stellar spectroscopy and a light curve of the flux of the host star to evaluate the possibility of human life on an exoplanet of interest. This provided us with the exoplanet's orbital period (84.5 hours) and other details about the host star, such as its estimated temperature and, consequently, its mass and radius. We next carry out computations that produce the orbital semi-major axis (
6.498 × 10
9
𝑚
), planetary mass (
1.73 × 10
27
𝑘𝑔
), and planetary temperature (1277.5 K = 1003.39 C). As a "hot Jupiter" in its star system, the planet is thought to not be able to support life on Earth in its natural state.
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